A NUMERICAL SCHEME FOR THREE-DIMENSIONAL FRONT

PROPAGATION AND CONTROL OF JORDAN MODE

K. R. ARUN

Abstract. As an example of a front propagation, we study the propagation of a three-dimensionalnonlinear wavefront into a polytropic gas in a uniform state and at rest. The successive positions andgeometry of the wavefront are obtained by solving the conservation form of equations of a weaklynonlinear ray theory. The proposed set of equations forms a weakly hyperbolic system of sevenconservation laws with an additional vector constraint, each of whose components is a divergence-free condition. This constraint is an involution for the system of conservation laws and it is termedas geometric solenoidal constraint. The analysis of a Cauchy problem for the linearised systemshows that when this constraint is satisfied initially, the solution does not exhibit any Jordan mode.For the numerical simulation of the conservation laws we employ a high resolution central scheme.The second order accuracy of the scheme is achieved by using MUSCL type reconstructions andRunge-Kutta time discretisations. A constrained transport type technique is used to enforce thegeometric solenoidal constraint. The results of several numerical experiments are presented, whichconfirm the efficiency and robustness of the proposed numerical method and the control of Jordanmode.

1. Introduction

In various fields of science one often encounters the propagation of a curved front. Calculation ofthe successive positions of such a front is of relevance in many practical problems and a variety ofmethods is available in the literature, e.g. the level set method [30], the front tracking method [19],to name a few. The kinematical conservation laws (KCL) is a system of conservation equations,derived in specially defined ray coordinates, which governs the evolution of a front in two or threespace dimensions [18, 29] and hence forms another method to track fronts. The KCL is a puregeometric result and the dynamics of the moving front is not included in its derivation. As aresult, the KCL gives rise to an under-determined system of equations. The analysis and variousapplications of the two-dimensional (2-D) KCL can be found in the references [7, 8, 9, 28, 35].Though the three-dimensional (3-D) KCL system was formally derived in [18], its analysis wascompleted only in [3, 4].

In [3, 4], the authors have used the 3-D KCL theory to model the propagation of a nonlinearwavefront in a polytropic gas. A nonlinear wavefront, across which the field variables are continuous,can be visualised as a propagating surface in a gaseous medium, which satisfies the high-frequencyapproximation. An equation representing the transport of energy for such a wavefront [32], whencoupled to the under-determined 3-D KCL system, leads to a complete system of conservationlaws which can be used to trace the history of the wavefront. The system of conservation laws

Date: January 27, 2012.2010 Mathematics Subject Classification. Primary 35L60, 35L65, 35L67, 35L80; Secondary 58J47, 65M06.Key words and phrases. kinematical conservation laws, kink, ray theory, nonlinear wavefront, polytropic gas, meancurvature, weakly hyperbolic system.At the time of this work, the author was supported by the Council of Scientific and Industrial Research (CSIR),Government of India, under grant-09/079(2084)/2006-EMR-1. The Department of Mathematics is partially fundedby the University Grants Commission (UGC) under DSA-SAP, Phase IV.

1

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thus obtained has been designated as the conservation laws of a 3-D weakly nonlinear ray theory(WNLRT); see [3] for more details.

A curved nonlinear wavefront, during its evolution, exhibits a very complex phenomenon ofpossessing curves of discontinuity (singularities), called kinks [29], across which the normal to thefront and the amplitude distribution on it are discontinuous. Based on the results of Prasad and hiscollaborators on nonlinear wavefronts, see [33] for a survey, we strongly believe that the geometryof a nonlinear wavefront is qualitatively similar to that of a weak shock front1. The extensivenumerical experiments reported in [8, 28, 35], using 2-D KCL, demonstrate that both 2-D nonlinearwavefronts and weak shock fronts develop kinks during their evolution. It has to be noted thatthe formation of kinks on shocks were also observed in the shock focusing experiments due toSturtevant and Kulkarny [43] and the analytical and numerical investigations due to Whitham andhis collaborators [20, 38, 39, 40, 44, 45] using geometrical shock dynamics (GSD). The modellingof the motion of a nonlinear wavefront or a shock front using the KCL theory clearly shows thatthe formation of kinks on fronts is due to the formation of shocks in the corresponding system ofgoverning conservation laws in ray coordinates [29]. As a consequence, it can also be inferred thatthe kinks are geometric in nature and they can appear not only on wavefronts and shock fronts butalso on other propagating surfaces.

The goal of the present work is to develop a high resolution Godunov type finite volume schemefor the numerical approximation of the conservation laws of 3-D WNLRT and to present the re-sults of extensive numerical simulations. We also intent to study the geometrical features of 3-Dnonlinear wavefronts and analyse the complex structure of kinks on them. It has been proved in[3, 4] that the system of conservation laws of 3-D WNLRT gives rise to a weakly hyperbolic sys-tem; in the sense that it has a multiple eigenvalue with an incomplete eigenspace. In addition,the system of conservation laws has to be complemented by a stationary vector constraint, each ofwhose three components is analogous to the solenoidal constraint in the equations of ideal magne-tohydrodynamics. We shall refer to this new divergence-free type condition as ‘geometric solenoidalconstraint’. It is well known from the literature that the appearance of δ-waves and δ-shocks in thesolution of weakly hyperbolic systems generally makes their numerical approximation very complex;see [13, 15, 26]. Another major challenge in the numerical modelling of 3-D WNLRT is that a nu-merical approximation of the system of conservation laws may not respect the geometric solenoidalconstraint and this can produce spurious numerical solutions.

The solution to a Cauchy problem for a weakly hyperbolic system, with a deficiency of eigenspaceby one, typically contains a mode which grows linearly in time. This mode, so-called the Jordanmode, is in the direction of a generalised eigenvector. However, it can be seen in [10, 11, 12, 31]that certain weakly hyperbolic systems with a convex entropy and additional stationary differentialconstraints can be symmetrised and hence the solution to a Cauchy problem will not contain theJordan mode. Due to the complexity of the system of equations of 3-D WNLRT and the lack ofan entropy function, it is difficult to establish this result for 3-D WNLRT. Therefore, we study theCauchy problem for a linearised 3-D WNLRT system and our results show that if the geometricsolenoidal constraint is satisfied at t = 0, then the Jordan mode of this weakly hyperbolic linearised

1At this point, we shall like to make a clear distinction between a wavefront and a shock front; see also [33] for a moredetailed explanation. The distinction can be very clearly seen in a pulse produced by the motion of a one-dimensionalpiston in an undisturbed polytropic gas. When the piston accelerates forward with initially zero velocity, there is acontinuous pulse for a short time. The different points of the pulse, moving with their local sound velocities, constitutea succession of nonlinear wavefronts. Later, a shock appears in the pulse, which is followed by the nonlinear wavefronts.These wavefronts interact with the shock and modify its evolution. The amplitude of the pulse varies continuouslyacross a wavefront, whereas it is discontinuous across a shock front. Depending on the number of compression andexpansion phases (due to acceleration and consequent deceleration of the piston) in the pulse, one or more shockswould appear in the pulse.

A NUMERICAL SCHEME FOR 3-D FRONT PROPAGATION 3

system does not appear at any time. We strongly believe this to be true also for the full nonlinearequations of 3-D WNLRT.

In [2] we have reported some preliminary numerical investigations on 3-D weakly WNLRT usingstaggered central schemes. The results in [2] clearly show the efficacy of 3-D WNLRT to produceseveral physically realistic geometrical features of nonlinear wavefronts. However, it well known thatthe staggered central schemes suffer from a large amount numerical diffusion and they have a lesserstability range; see [24] for more details. Therefore, in this paper we employ the high-resolutioncentral scheme of Kurganov and Tadmor (KxT) [24] for the numerical simulation of the conserva-tion laws of 3-D WNLRT. A main advantage of the KxT scheme is that it admits a semi-discreteformulation, and hence it can easily be extended to any high order accuracy in space and timeusing standard recovery procedures and total variation diminishing (TVD) Runge-Kutta time step-ping procedures. In addition, like other central schemes, the KxT scheme also has the advantagethat it does not require complicated and time consuming Riemann solvers. This is particularlyimportant, as the 3-D WNLRT system is only weakly hyperbolic. Moreover, the system is multi-dimensional, where there is no exact Riemann solver. However, the high resolution central schemeneed not respect the geometric solenoidal constraint and therefore we use a constrained transporttype technique to enforce it. Our numerical experiments confirm the efficiency and robustness ofthe numerical method and we verify that constrained transport technique preserves the constraintup to machine round off error. The numerical results reveal many fascinating geometrical featuresof 3-D nonlinear wavefronts and the non-appearance of the Jordan mode.

2. Governing Equations

Consider a one parameter family of surfaces in R3 such that it represents the successive positions

of a moving surface Ωt as time varies. Associated with the family, we have a ray velocity vector χat any point (x1, x2, x3) on the surface Ωt. We consider only the isotropic evolution of Ωt so thatwe take χ to be in the direction of the unit normal n to Ωt, i.e. χ=mn, where m is the normalvelocity of propagation of Ωt. We introduce a ray coordinate system (ξ1, ξ2, t) such that t = constis Ωt with (ξ1, ξ2) as the surface coordinates. The two parameter family of curves (ξ1 = const, ξ2 =const), along which t varies, represents the rays which are orthogonal to the successive positionsof Ωt. Let u and v be respectively unit tangent vectors to the curves (ξ2 = const, t = const) and(ξ1 = const, t = const) on Ωt, and let n be the unit normal to Ωt given by

(2.1) n =u× v

‖u× v‖.

Let an element of distance along a curve (ξ2 = const, t = const) be g1dξ1. Analogously, denote byg2dξ2, the element of distance along a curve (ξ1 = const, t = const) and by mdt, the element ofdistance along a ray (ξ1 = const, ξ2 = const). Based on geometrical considerations we can derivethe 3-D KCL [3, 18]

(g1u)t − (mn)ξ1 = 0,(2.2)

(g2v)t − (mn)ξ2 = 0,(2.3)

subject to the condition

(2.4) (g2v)ξ1 − (g1u)ξ2 = 0.

We refer the reader to [3] for the analysis of the conservation laws (2.2)-(2.4). It follows immedi-ately from (2.2)-(2.3) that the quantity (g2v)ξ1 − (g1u)ξ2 does not depends on t. The existence ofcoordinates (ξ1, ξ2) on Ω0 guarantees that the condition (2.4) is satisfied at t = 0. The evolutionequations (2.2)-(2.3) then shows that it is satisfied for all t. In other words, the constraint (2.4) is

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an involution for the system of conservation laws (2.2)-(2.3); i.e. once fulfilled at the initial time, itis fulfilled for all times. Note that the vector constraint (2.4) can be recast as three divergence-freeconditions. Let us introduce three vectors Bk ∈ R

2, k = 1, 2, 3, via

(2.5) Bk := (g2vk,−g1uk).

The constraint (2.4) can then be rewritten in the equivalent form

(2.6) div(Bk) = 0, k = 1, 2, 3.

Thus, the three vectors Bk are divergence-free at any time t if they are so at t = 0. Since the threescalar equations in (2.6) are analogous to the solenoidal condition in the equations of 2-D idealmagnetohydrodynamics, we shall refer to (2.4) (or (2.6)), as ‘geometric solenoidal constraint’.

The 3-D KCL is a system of six scalar evolution equations (2.2)-(2.3). Since ‖u‖ = ‖v‖ = 1,they both can have only two independent components. Thus, there are seven dependent variablesin (2.2)-(2.3): two independent components of each of u and v, the front velocity m of Ωt andg1 and g2. Hence, KCL is an under-determined system and it can be closed only with the helpof additional relations or equations, which would follow from the nature of the surface Ωt and thedynamics of the medium in which it propagates. Following [3], we use the closure obtained by theenergy propagation along the rays of a WNLRT and conservation of energy in a ray-tube. We referthe reader to [33] for a comprehensive treatment of WNLRT. The energy transport equation ofWNLRT for a polytropic gas, initially at rest and in uniform state, can be written in a conservationform [3]

(2.7)((m− 1)2e2(m−1)g1g2 sinψ

)t= 0,

where ψ is the angle between the vectors u and v. The system of equations (2.2)-(2.3) and (2.7)is the complete set of conservation laws of 3-D WNLRT, describing the evolution of a nonlinearwavefront.

Assuming smooth solutions, a quasi-linear form of the system of equations (2.2)-(2.3) and (2.7)can be obtained as

(2.8) AVt +B(1)Vξ1 +B(2)Vξ2 = 0,

where V = (u1, u2, v1, v2,m, g1, g2)T and the Jacobian matrices A,B(1) and B(2) are defined in

[3]. The eigenvalues of the quasilinear system (2.8) can be obtained by solving the characteristicequation

(2.9) det(e1B

(1) + e2B(2) − λA

)= 0,

where (e1, e2) ∈ R2 with e21 + e22 = 1. It has been proved in [4] that the roots of (2.9) are precisely

λ1, λ2(= −λ1), λ3 = · · · = λ7 = 0, where

(2.10) λ1 =

m− 1

2 sin2 ψ

(e21g21

−2e1e2g1g2

cosψ +e22g22

) 1

2

.

Remark 2.1. At this point, it has to be mentioned that there are only four independent eigenvectorscorresponding to the multiple eigenvalue zero; see [3, 4] for more details. The factor in braces in(2.10) is a quadratic form in e1/g1 and e2/g2 and it is clearly positive definite. Hence, the eigenvalueλ1 is real for m > 1 and pure imaginary for m < 1. Therefore, the system (2.8) is only weaklyhyperbolic when m > 1. The goal of this paper is to consider only the case when m > 1.

A NUMERICAL SCHEME FOR 3-D FRONT PROPAGATION 5

3. Numerical Approximation

In this section we proceed to the numerical approximation of the conservation laws of 3-DWNLRT, using high resolution Godunov type finite volume schemes. Our goal is to study theevolution of weakly nonlinear wavefronts in R

3 and the complex structure of kinks formed on them.First, we note that the system of conservation laws of 3-D WNLRT, (2.2)-(2.3) and (2.7), can berecast in the usual divergence form

(3.1) Wt + F1(W )ξ1 + F2(W )ξ2 = 0,

where the vector of conserved variables W and the flux-vectors F1(W ) and F2(W ) are given by

(3.2)

W =(g1u, g2v, (m− 1)2e2(m−1)g1g2 sinψ

)T,

F1(W ) = (mn,0, 0)T ,

F2(W ) = (0,mn, 0)T .

3.1. Semi-discrete Central Scheme. In order to numerically solve (3.1), we first discretise thegiven computational domain. In our computations we use a rectangular grid with mesh sizes h1 andh2 respectively in ξ1- and ξ2-directions. We will denote by Ci,j , the cell centred around the point(ξ1i, ξ2j), i.e.

(3.3) Ci,j :=

[ξ1i −

h12, ξ1i +

h12

]×

[ξ2j −

h22, ξ2j +

h22

].

Let us denote by W i,j , the cell average of W at time t taken over Ci,j , i.e.

(3.4) W i,j(t) :=1

h1h2

∫

Ci,j

W (ξ1, ξ2, t)dξ1dξ2.

The time step ∆t is chosen by the CFL condition

(3.5) ∆tmaxi,j

(ρ1i,jh1

,ρ2i,jh2

)= ν,

where ρ1 and ρ2 are respectively the maximum of the absolute values of the generalised eigenvaluesof the matrices B(1) and B(2) with respect to A, cf. (2.8) and ν is the CFL number. Using the giventhe cell averages W

ni,j at time tn a piecewise linear interpolant is reconstructed, resulting in

(3.6) W (ξ1, ξ2, tn) =

∑

i,j

(W

ni,j +W ′

i,j(ξ1 − ξ1i) +W 8

i,j(ξ2 − ξ2j))1i,j(ξ1, ξ2),

where 1i,j is the characteristic function of the cell Ci,j andW′

i,j andW8

i,j are respectively the discreteslopes in ξ1- and ξ2-directions. A possible computation of these slopes, which results in an overallnon-oscillatory scheme is given by a smooth central weighted essentially non-oscillatory (CWENO)limiter [21]

(3.7)

W ′

i,j = CWENO

(W

ni+1,j −W

ni,j

h1,W

ni,j −W

ni−1,j

h1

),

W 8

i,j = CWENO

(W

ni,j+1 −W

ni,j

h2,W

ni,j −W

ni,j−1

h2

),

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where the CWENO function is defined by

(3.8) CWENO(a, b) =ω(a) · a+ ω(b) · b

ω(a) + ω(b), ω(a) =

(ǫ+ a2

)−2, ǫ = 10−6.

Using the reconstruction (3.6) we compute the extrapolated values on the boundary of a cell

(3.9) WL(R)i,j =W

ni,j ∓

h12W ′

i,j , WB(T )i,j =W

ni,j ∓

h22W 8

i,j .

A starting point for the construction of the numerical scheme is a semi-discrete discretisation of(3.1). An integration of the balance law (3.1) over the cell Ci,j yields

(3.10)dW i,j

dt= −

F1i+ 1

2,j −F1i− 1

2,j

h1−

F2i,j+ 1

2

−F2i,j− 1

2

h2,

where the quantities F1i+1/2,j and F2i,j+1/2 are respectively the numerical fluxes at the cell interfacesi + 1/2, j and i, j + 1/2. In a Riemann solver based upwind scheme, these fluxes are obtained bysolving local Riemann problems. The central type schemes we employ here completely avoid thesolving of Riemann problems.

In [24], Kurganov and Tadmor first derive a fully discrete central scheme by constructing anintermediate mesh of variable cell length, making use of the local wave speeds ai+1/2,j and ai,j+1/2

at the cell interfaces, where

(3.11)

ai+ 1

2,j := max

ρ

(∂F1

∂W

(WR

i,j

)), ρ

(∂F1

∂W

(WL

i+1,j

)),

ai,j+ 1

2

:= max

ρ

(∂F2

∂W

(W T

i,j

)), ρ

(∂F2

∂W

(WB

i,j+1

)).

Here, ρ(A) := maxk|λk(A)| with λk(A) being the eigenvalues of the matrix A. Letting the time-step tends to zero, the fully discrete scheme yields the semi-discrete formulation (3.10) which isvery simple and robust; see [24] for the derivation and more details. The numerical flux functionsFi+1/2,j and Fi,j+1/2 in (3.10) can be finally obtained as

(3.12)F1i+ 1

2,j

(WR

i,j ,WLi+1,j

)=

1

2

(F1

(WL

i+1,j

)+ F1

(WR

i,j

))−ai+ 1

2,j

2

(WL

i+1,j −WRi,j

),

F2i,j+ 1

2

(W T

i,j ,WBi,j+1

)=

1

2

(F2

(WB

i,j+1

)+ F2

(W T

i,j

))−ai,j+ 1

2

2

(WB

i,j+1 −W Ti,j

).

Note that the numerical fluxes F1i+1/2,j and F2i,j+1/2 depend only on the local speeds of propagationai+1/2,j and ai,j+1/2 and due to this simple and general form, the scheme (3.10) can easily beextended to achieve any high order spatial accuracy for smooth solutions.

To improve the temporal accuracy and to gain second order accuracy in time we use a TVD Runge-Kutta scheme [41] to numerically integrate the system of ordinary differential equations (ODEs) in(3.10). Denoting the right hand side of (3.10) by Li,j(W ), the second order Runge-Kutta schemeupdates W through the following two stages

(3.13)W

(1)i,j =W

ni,j +∆tLi,j

(W

n),

Wn+1i,j =

1

2W

ni,j +

1

2W

(1)i,j +

1

2∆tLi,j

(W (1)

).

A NUMERICAL SCHEME FOR 3-D FRONT PROPAGATION 7

3.2. Constrained Transport. The geometric solenoidal constraint (2.6) of 3-D KCL needs specialattention because if it is not fulfilled exactly, it can produce non-physical solutions. Especially nearthe discontinuities, the error due to numerical discretisation can produce very large divergences ofBk. To avoid this, three additional transport equations have been introduced to the central scheme,by means of which the geometric solenoidal constraint can be enforced. This additional equationsare relations for the three potential functions Ak, k = 1, 2, 3 which can be derived as follows.

We convert the six evolution equations (2.2)-(2.3) of the 3-D KCL system into three equationsfor the potentials Ak. The existence of the potentials Ak follow from the fact that the divergenceof Bk is equal to zero, i.e.

(3.14) div(Bk) ≡ (g2vk)ξ1 + (−g1uk)ξ2 = 0.

Therefore, it follows that

(3.15)g1uk = Akξ1 ,

g2vk = Akξ2 .

In the light of (3.15), the 3-D KCL system (2.2)-(2.3) reads

(3.16)Akξ1t − (mnk)ξ1 = 0,

Akξ2t − (mnk)ξ2 = 0

and thus, we obtain

(3.17) Akt −mnk = 0.

From the definition (2.1) of the unit normal n, it follows that

n =g1u× g2v

‖g1u× g2v‖

=Aξ1 × Aξ2

‖Aξ1 × Aξ2‖.(3.18)

Note that here we have denoted A = (A1,A2,A3). Therefore, from (3.17)-(3.18) we obtain thefollowing evolution equations for A1,A2,A3

A1t −m

(A2ξ1A3ξ2 − A3ξ1A2ξ2

)√A2ξ1A2ξ2− (Aξ1 · Aξ2)

2= 0,(3.19)

A2t −m

(A3ξ1A1ξ2 − A1ξ1A3ξ2

)√A2ξ1A2ξ2− (Aξ1 · Aξ2)

2= 0,(3.20)

A3t −m

(A1ξ1A2ξ2 − A2ξ1A1ξ2

)√A2ξ1A2ξ2− (Aξ1 · Aξ2)

2= 0.(3.21)

It is interesting to note that (3.19)-(3.21) forms a coupled system of three fully nonlinear partialdifferential equations, each of them is of the Hamilton-Jacobi type. The basic idea behind theconstrained transport is to make use of A1,A2,A3 to get the values of g1u and g2v. These correctedvalues should fulfil a discrete version of the geometric solenoidal constraint.

In each time step, the system of conservation laws (3.1) is approximated by the central scheme(3.10). Using the solution thus obtained we compute the updated values of A1,A2,A3. The correctedvalues of g1u and g2v are then obtained from (3.15). It is the discretisation of the derivatives in(3.15) which enforces the geometric solenoidal constraint. The decisive point is how to approximatethis derivatives. In this paper, we follow the procedure described by Rossmanith [36]. In [36] it is

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proposed to use a staggered grid and to arrange the variables on the grids as depicted in Figure 1.We use the following strategy:

original gridstaggered grid

data stored on original grid:g1u, g2v

(m− 1)2e2(m−1)g1g2 sinχ

data stored on staggered grid:A1,A2,A3

g2v

g1u

Figure 1. Arrangement of the variables on the grids.

• The potentials A1,A2,A3 are computed at the midpoints of the staggered grid which coincidewith the corners of the original grid.

• The vector g1u lies at the east and west edges of the staggered grid, which are the northand south edges of the original grid.

• The vector g2v lies at the north and south edges of the staggered grid, which are the eastand west edges of the original grid.

With this arrangement, the derivatives of A1,A2,A3 in the equation (3.15) can be approximated bycentral differences and the corrected values of the vectors g1u and g2v are obtained as

(3.22)

[g1uk]n+1i,j+ 1

2

=1

h1

(Ak

n+1i+ 1

2,j+ 1

2

− Akn+1i− 1

2,j+ 1

2

),

[g2vk]n+1i+ 1

2,j=

1

h2

(Ak

n+1i+ 1

2,j+ 1

2

− Akn+1i+ 1

2,j− 1

2

).

We now consider a cell of the original grid. The corrected values of g1u and g2v lie on the edges ofthat cell. In the discrete formula for the geometric solenoidal constraint (2.6), the derivatives areapproximated by central differences of g1u and g2v values on the edges to get

(3.23) [ div(Bk)]n+1i,j =

[g2vk]n+1i+ 1

2,j− [g2vk]

n+1i− 1

2,j

h1−

[g1uk]n+1i,j+ 1

2

− [g1uk]n+1i,j− 1

2

h2.

Replacing the values of g1uk and g2vk on the cell edges by the expressions in (3.22), it can be seenthat the right hand side of (3.23) vanishes. Thus, we have devised a method to get the correctedvalues of g1u and g2v so that a discrete version of the geometric solenoidal constraint is satisfied.

It remains to be discussed how to get the updated values of A1,A2,A3. There are several dif-ferent approaches for the constrained transport in the MHD literature; e.g. see [6, 14, 36, 37] andthe references therein. We have already seen from (3.19)-(3.21) that the evolution equations forA1,A2,A3 form a coupled system of nonlinear equations. Hence, directly discretising them could be

A NUMERICAL SCHEME FOR 3-D FRONT PROPAGATION 9

tedious and time consuming. The decisive step is to use instead the simple and equivalent form in(3.17) to get the staggered update for A1,A2,A3. Integrating and averaging (3.17) over a staggeredgrid yields the semi-discrete approximation

(3.24)d

dt[Ak]i+ 1

2,j+ 1

2

= [mnk]i+ 1

2,j+ 1

2

.

The above equation (3.24) can be used to get the updated values of the staggered averages of thepotentials A1,A2,A3. However, in order to use (3.24) we need to have the values of mnk at thecorners of the grid cells. At this point, we note that mnk is precisely the nonzero element in theflux functions F1 and F2 of the system of conservation laws (3.1). Since the numerical flux functionsF1 and F2 at the interfaces are already available from the central scheme, following the approachof Ryu et al. [37], we take

(3.25) [mnk]i+ 1

2,j+ 1

2

=1

4

([mnk]i+ 1

2,j + [mnk]i+ 1

2,j+1 + [mnk]i,j+ 1

2

+ [mnk]i+1,j+ 1

2

).

Using (3.25) in (3.24) and discretising the resulting system of ODEs using the same Runge-Kuttamethod (3.13) gives the staggered updates for A1,A2,A3.

Thus, we have devised a way to compute the values of g1u and g2v so that the discrete divergenceof the three vectors Bk equals zero. An algorithm, in which the solution of the 3-D WNLRTequations calculated using the central scheme (3.10) is corrected accordingly in each time step, canbe written in the following form.

(1) The system of conservation laws (3.1) is solved using the central scheme, giving the valuesof the conserved variableW . The vectors Bk are not yet divergence free, it will be correctedin the next steps.

(2) The potentials A1,A2,A3 are updated by solving (3.24).(3) The spatial derivatives of A1,A2,A3 are calculated and the corrected values of g1u and g2v

are obtained using (3.22).(4) The values of g1u and g2v on the cell edges are averaged, to get the values at the cell centres

of the original grid

(3.26)

[g1u]n+1i,j =

1

2

([g1u]

n+1i,j− 1

2

+ [g1u]n+1i,j+ 1

2

),

[g2v]n+1i,j =

1

2

([g2v]

n+1i− 1

2,j+ [g2v]

n+1i+ 1

2,j

).

3.3. Formulation of Initial and Boundary Conditions. To complete the algorithm, the ini-tialisation of the data on the respective grids and the implementation of appropriate boundaryconditions are to be done. The initial data for the system (3.1) can be formulated as follows.

Let the initial position of a weakly nonlinear wavefront Ωt be given in a parametric form

(3.27) Ω0 : x = x0(ξ1, ξ2).

Given the representation (3.27), the initial values of the metrics g1 and g2 and unit tangent vectorsu and v can be taken as

g10 = ‖x0ξ1‖, g20 = ‖x0ξ2‖,(3.28)

u0 =x0ξ1

‖x0ξ1‖, v0 =

x0ξ2

‖x0ξ2‖.(3.29)

Note that

(3.30)(g20v0)ξ1 − (g10u0)ξ2 = x0ξ2ξ1 − x0ξ1ξ2

= 0.

10 ARUN

Thus, the geometric solenoidal constraint (2.6) is satisfied by the initial values of g1u and g2v. Theunit normal n0 of Ω0 can be obtained using

(3.31) n0 =u0 × v0

‖u0 × v0‖.

Let us assume that the distribution of the front velocity m on Ω0 be given as

(3.32) m = m0(ξ1, ξ2).

Given the initial values of g1, g2,u,v and m, the initial value of the conserved variable W can beeasily calculated. The potentials A1,A2,A3 need to be initialised on the staggered grids. Here carehas to be taken, because the relation (3.15) has to be fulfilled. It is easy to check that the choice

(3.33) Ak0(ξ1, ξ2) = x0k(ξ1, ξ2), k = 1, 2, 3

will serve the purpose. Hence, formulation of the initial data is completed.We now proceed to the discussion of the boundary conditions. First, it is to be noted that the

boundaries in all the problems considered in this paper are artificial boundaries, i.e. they are in-troduced only to truncate the computational domain and they do not correspond to any physicalboundaries. Here, we have implemented two types boundary conditions, namely the non-reflectingboundary conditions and periodic boundary conditions, depending on the problem under consid-eration. The non-reflecting boundary conditions do not introduce any reflections of the outgoingwaves. They are also known as absorbing boundary conditions in the literature, see [25], since theyare supposed to absorb completely all the waves that hit them.

We have implemented the boundary conditions by extending the computational domain by ghost-cells, whose values are set at the beginning of each time-step. Since we use a second order accuratescheme, we use two layers of ghost-cells on all the boundaries of the computational domain. Letwi,j denotes any grid function. We implement the non-reflecting boundary conditions for w by azero order extrapolation to the ghost cells which are adjacent to the boundary of the computationaldomain. Thus, we have at a left hand vertical edge

(3.34) w−1,j = w0,j , w−2,j = w0,j , j ∈ Z,

where a negative index denotes the values in the ghost cells to the left of the computational domain.The other boundaries are treated in an analogous manner. The zero order extrapolation boundaryconditions of this type gives a reasonable absorbing boundary conditions; see also the text book byLeVeque [25] for a detailed discussion. In any of our numerical experiments reported in section 5we did not observe any spurious waves generated/reflected from the boundary after implementingthis boundary condition. The periodic boundary conditions are implemented by setting, e.g. at lefthand and right hand vertical edges

(3.35) w−1,j = wK1−1,j , w−2,j = wK1−2,j , wK1,j = w0,j , wK1+1,j = w1,j , j ∈ Z,

where K1 denotes the number of mesh cells in the ξ1-direction. It has to be noted that the data inthe boundary cells of the staggered cells needs special attention. The staggered grid at each side ishalf a cell larger than the original grid. Therefore, the boundary conditions have to be implementedin order to be able to calculate the data in the boundary cells of the staggered grid.

There are two ways of proceeding. In the first method, a ghost cell layer boundary conditionfor the potentials Ak could be implemented. In this case, the values of g1u and g2v are calculatedaccording to (3.15) in all points where they are needed. This method has the advantage that thediscrete geometric solenoidal constraint is satisfied at the boundary cells too. However, as mentionedabove, when a ghost cell layer boundary condition is implemented, the values in the boundary cellsare copies of the values in the neighbouring cells. As a result, the spatial derivatives of Ak in (3.15)

A NUMERICAL SCHEME FOR 3-D FRONT PROPAGATION 11

vanish at the boundary cells. This in turn leads to zero values for g1u and g2v at the boundary,independent of their physically exact values. This can produce oscillations in the solution.

We can alternatively implement a boundary condition for the variables g1u and g2v. This canbe done by an absorbing boundary condition or a periodic boundary condition depending on theproblem. In this case the potentials Ak need not be calculated in the boundary cells and theboundary values of g1u and g2v will much more conform with those that are physically correct.However, notice that (3.15) is now no longer used and the discrete geometric solenoidal constraintis not enforced at the boundary cells. Nevertheless, the constraint is satisfied in all the interiorcells of the computational domain. In all our numerical experiments we follow this approach andwe have not observed any spurious oscillations in the solution if this alternative is chosen.

3.4. Method of Construction of Successive Fronts. We now proceed to explain the methodof calculation of the successive positions of the nonlinear wavefront Ωt. At any time t, given thevalues of g1, g2,u,v and m, we first compute the value of the conserved variable W . Using thesevalues we numerically solve the system (3.1) to get the updated value of W at time t +∆t. Since‖u‖ = ‖v‖ = 1, from the first six components of W , the values of g1, g2,u and v can be computedvery easily. The unit normal n is then given by (2.1). To get the updated value of the normalvelocity m we proceed as follows. Notice that, say,

(3.36) (m− 1)2e2(m−1) =W7

g1g2 sinψ≡ κ.

We now solve the nonlinear equation

(3.37) ϑ(m) ≡ (m− 1)2e2(m−1) − κ = 0

for m using Newton-Raphson method. The monotonicity of the function ϑ in (1,∞) ensures theuniqueness of the solution of (3.37).

At any time t, we approximate the front Ωt by a discrete set of points xi,j(t), where

(3.38) xi,j(t) := x(ξ1i, ξ2j , t).

To get the successive positions of Ωt, we numerically solve the system of ODEs

(3.39)dxi,j(t)

dt= mi,j(t)ni,j(t),

wheremi,j(t) and ni,j(t) are the corresponding values ofm and n obtained fromW i,j(t). A pictorialrepresentation of our time-marching scheme is depicted in Figure 2. The numerical integration ofthe system of ODEs in (3.39) is done along with that of the system conservation laws, using thesame TVD Runge-Kutta time stepping scheme (3.13). This enables us to get the updated values of(x1, x2, x3) and hence the successive positions of the front Ωt.

4. Analysis of Linearised Cauchy Problem

It is well known from the literature [16, 22, 23, 27, 46] that weakly hyperbolic systems give rise toa Jordan mode. The solution to a Cauchy problem for a weakly hyperbolic system with a deficiencyof eigenspace by one exhibits a Jordan mode which grows linearly in time. The Jordan mode is inthe direction of the corresponding generalised eigenvector. Even though the 3-D WNLRT systemis weakly hyperbolic, its numerical solution does not exhibit any such component. The reason forthe disappearance of this mode is the geometric solenoidal constraint (2.6), which is inherent to thesystem of equations. Due to the nonlinearity and complexity of the equations, we are unable toestablish this result for the full nonlinear system. Nevertheless, in order to support our assertion,we prove the result for a linearised 3-D WNLRT system. We strongly believe the same result tohold also for the full nonlinear system.

12 ARUN

Ωtn

Ωtn+∆t

xi,j(tn)

xi,j(tn +∆t)

Wni,j

Wn+1i,j

Figure 2. Geometrical representation of the time marching scheme. The dottedlines denote the rays.

For notational convenience in this section2 we shall denote U = g1u,V = g2v. The 3-D WNLRTsystem (2.2)-(2.3) and (2.7) with this new notations reads

U t − (mn)ξ1 = 0,(4.1)

V t − (mn)ξ2 = 0,(4.2)((m− 1)2e2(m−1)‖U × V ‖

)t= 0.(4.3)

Let us consider a planar wavefront given by

(4.4) Ωt : x1 = ξ1, x2 = ξ2, x3 = m0t,

where we assume m0 > 1. The unknown variables U ,V and m assume constant values on Ωt andthey can easily be obtained as

(4.5) U = (1, 0, 0), V = (0, 1, 0), m = m0.

2These notations are used only in this section.

A NUMERICAL SCHEME FOR 3-D FRONT PROPAGATION 13

We linearise the 3-D WNLRT system (4.1)-(4.3) about the constant state (4.5). After simplificationsthe linearised equations can be obtained as

(4.6)

U1

U2

U3

V1V2V3m

t

+

m0U3

m0V3−m000

− (m0−1)2 U3

ξ1

+

000

m0U3

m0V3−m

− (m0−1)2 V3

ξ2

= 0.

Let W = (U1, U2, U3, V1, V2, V3,m)T and we write the linearised system (4.6) in the usual matrixform

(4.7) Wt +A1Wξ1 +A2Wξ2 = 0,

where

(4.8) A1 =

0 0 m0 0 0 0 00 0 0 0 0 m0 00 0 0 0 0 0 −10 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

0 0 − (m0−1)2 0 0 0 0

, A2 =

0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 m0 0 0 0 00 0 0 0 0 m0 00 0 0 0 0 0 −1

0 0 0 0 0 − (m0−1)2 0

.

Consider (ν1, ν2) ∈ R2 with ν21 + ν22 = 1. The matrix pencil A := ν1A1 + ν2A2 has the eigenvalues

(4.9) λ1 =

√m0 − 1

2, λ2 = −λ1, λ3 = · · · = λ7 = 0.

We note that like the original system (4.1)-(4.3), the linearised system (4.6) is also degenerate asthe eigenvalue zero has multiplicity five but the associated eigenspace is only four-dimensional. Wenow proceed to solve a Cauchy problem for (4.6). Let an initial data for (4.6) be given by

(4.10) W (ξ1, ξ2, 0) =W0(ξ1, ξ2).

The main result is the following theorem.

Theorem 4.1. The solution to the Cauchy problem (4.7) and (4.10) does not contain any linearlygrowing Jordan mode when the constraint

(4.11) V3ξ1 − U3ξ2 = 0

is satisfied at t = 0.

Proof. The solution of the Cauchy problem (4.6) and (4.10) can be obtained using the Fouriertransform method. The Fourier transform of W with respect to the space variables ξ1, ξ2 is definedvia

(4.12) W (k1, k2, t) :=

∫∞

−∞

∫∞

−∞

W (ξ1, ξ2, t)e−i(k1ξ1+k2ξ2)dξ1dξ2.

Taking the Fourier transform of (4.7) yields

(4.13) Wt + i (k1A1 + k2A2) W = 0.

14 ARUN

The initial data for (4.13) is obtained by taking Fourier transform of (4.10)

(4.14) W (k1, k2, 0) = W0(k1, k2).

The solution of the Cauchy problem (4.13)-(4.14) is given by

(4.15) W (k1, k2, t) = e−it(k1A1+k2A2)W0(k1, k2).

Let k =√k21 + k22 so that (k1, k2) = k(ν1, ν2) with ν21 + ν22 = 1. Reducing the matrix pencil A to

Jordan canonical form yields

(4.16) J =

0 1 0 0 0 0 00 0 0 0 0 0 00 0 c 0 0 0 00 0 0 −c 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

,

where c :=√(m0 − 1)/2. Let P be the non-singular matrix of the right eigenvectors of A including

the generalised eigenvector. We note that P satisfies

(4.17) AP = PJ.

Using the Jordan matrix (4.16) the columns of P can be obtained by solving the eigenvalue problems

(4.18) Ap1 = 0, Ap2 = p1, Ap3 = cp3, Ap4 = −cp4, Ap5 = Ap6 = Ap7 = 0,

where p2 is the generalised eigenvector. After algebraic computations we can obtain the transfor-mation matrix

(4.19) P =

−ν1ν2m0 0 −2ν2

1m0

m0−1 −2ν2

1m0

m0−1 1 0 0

ν21m0 0 −2ν1ν22m0

m0−1 −2ν1ν22m0

m0−1 0 1 0

0 −ν2

√2

m0−1ν1 −√

2m0−1ν1 0 0 0

−ν22m0 0 −2ν1ν22m0

m0−1 −2ν1ν22m0

m0−1 0 0 1

ν1ν2m0 0 −2ν2

2m0

m0−1 −2ν2

2m0

m0−1 0 0 0

0 ν1

√2

m0−1ν2 −√

2m0−1ν2 0 0 0

0 0 1 1 0 0 0

.

The matrix exponential in the equation (4.15) can be computed as

(4.20) Pe−iktJP−1 =

1 0m0tk1k22

k30 0 −

m0tk21k2k3

2m0

m0−1k21(1−cos(ktc))

k2

0 1 −m0tk21k2

k30 0

m0tk31k3

2m0

m0−1k1k2(1−cos(ktc))

k2

0 0k21cos(ktc)+k2

2

k20 0 −k1k2(1−cos(ktc))

k20

0 0m0tk32k3

1 0 −m0tk1k22

k32m0

m0−1k21(1−cos(ktc))

k2

0 0 −m0tk1k22

k30 1

m0tk21k2k3

2m0

m0−1k22(1−cos(ktc))

k2

0 0 −k1k2(1−cos(ktc))k2

0 0k21+k2

2cos(ktc)k2

0

0 0 0 0 0 0 cos(ktc)

A NUMERICAL SCHEME FOR 3-D FRONT PROPAGATION 15

Using (4.20) in (4.15) we obtain the final solution

U1(k1, k2, t) = U10(k1, k2)−m0tk1k2k3

(k1V30 − k2U30

)(k1, k2)

+2m0

m0 − 1

k21(1− cos(ktc))

k2m0(k1, k2),(4.21)

U2(k1, k2, t) = U20(k1, k2) +m0tk21k3

(k1V30 − k2U30

)(k1, k2)

+2m0

m0 − 1

k1k2(1− cos(ktc))

k2m0(k1, k2),(4.22)

U3(k1, k2, t) =1

k2(k21 cos(ktc) + k22

)U30(k1, k2)−

k1k2k2

(1− cos(ktc))V30(k1, k2),(4.23)

V1(k1, k2, t) = V10(k1, k2)−m0tk22k3

(k1V30 − k2U30

)(k1, k2)

+2m0

m0 − 1

k1k2(1− cos(ktc))

k2m0(k1, k2),(4.24)

V2(k1, k2, t) = U0(k1, k2) +m0tk1k2k3

(k1V30 − k2U30

)(k1, k2)

+2m0

m0 − 1

k22(1− cos(ktc))

k2m0(k1, k2),(4.25)

V3(k1, k2, t) = −k1k2k2

(1− cos(ktc))U30(k1, k2) +1

k2(k21 + k22 cos(ktc)

)V30(k1, k2),(4.26)

m(k1, k2, t) = cos(ktc)m0(k1, k2).(4.27)

From the equations (4.21)-(4.27) we infer that the expressions for U1, U2, V1 and V2 contain a term

with t multiplied by a factor k1V30 − k2U30. It is very crucial that k1V30 − k2U30 is the preciselyFourier transform of V30ξ1−U30ξ2 . From the system (4.6) one can observe that the quantity Vξ1−Uξ2

remains constant in time. Hence, when the constraint

(4.28) V3ξ1 − U3ξ2 = 0 at t = 0

is satisfied the term in (4.21)-(4.22) and (4.24)-(4.25), which grows linearly with t vanishes. Theunknowns U ,V and m are obtained by taking the inverse Fourier transform of (4.21)-(4.27) inwhich t appears only as a parameter. Definitely, the inversion will not introduce any additional termgrowing with t. Thus, we conclude that the Jordan mode in the solutions (4.21)-(4.27) disappearsunder the constraint (4.28).

Remark 4.2. It is worth noting that (4.28) is only one component of the vector constraint

(4.29) V ξ1 −U ξ2 = 0 at t = 0.

Had we taken the base state in (4.5) as U = (0, 1, 0), V = (0, 0, 1) (or U = (0, 0, 1), V = (1, 0, 0)),then the condition (4.28) would have been replaced by V1ξ1 − U1ξ2 = 0 at t = 0 (or V2ξ1 − U2ξ2 =0 at t = 0).

5. Numerical Case Studies

In order to illustrate the applicability of 3-D KCL for the modelling of evolution of nonlinearwavefronts, in this section we present many illustrating examples. These examples reveal severalgenuinely three-dimensional geometrical features of nonlinear wavefronts and complex structure ofkinks formed on them. We have set the CFL number ν = 0.9 in all our numerical experiments.

16 ARUN

5.1. Propagation of a Non-axisymmetric Nonlinear Wavefront. We choose the geometry ofthe initial wavefront Ω0 in a such a way that it is not axisymmetric. A motivation for the choice ofinitial geometry of the front is some of the shock focusing problems studied in [39]. The front Ω0

has a single smooth dip with the initial shape given as

(5.1) Ω0 : x3 =−κ

1 +x2

1

α2 +x2

2

β2

,

where the parameter values are set to be κ = 1/2, α = 3/2, β = 3. At time t = 0, the ray coordinates(ξ1, ξ2) are chosen to be ξ1 = x1 and ξ2 = x2 and this leads to the parametric representation of thethe initial wavefront in the form

(5.2) Ω0 : x1 = ξ1, x2 = ξ2, x3 =−κ

1 +ξ21

α2 +ξ22

β2

.

With the aid of (5.2), the initial values g1, g2,u and v are calculated, cf. section 3.3. The normalvelocity is prescribed as a constant m0 = 1.2 everywhere on the initial wavefront Ω0.

In order to verify that our numerical scheme has second order convergence, we compute theexperimental order of convergence (EOC) for a smooth solution. Although an exact solution is notavailable, we can still study the EOC in the following way. Let us consider three meshes of sizesK1,K2 = K1/2,K3 = K2/2, then the EOC can be computed as

(5.3) EOC = log2‖WK2

−WK3‖

‖WK1−WK2

‖.

Here WK is the numerical solution on a mesh with K × K cells. The computational domain[−20, 20] × [−20, 20] was successively divided into 20 × 20, 40 × 40, . . . , 320 × 320 cells. The finaltime was taken to be t = 0.1 so that the solution remains smooth. In tables 1 and 2 we showrespectively the EOCs computed in the L1 norm using the components of the vectors g1u and g2v.The tables shows that the order of convergence is two.

K error in g1u1 EOC error in g1u2 EOC error in g1u3 EOC

20 0.00007227 0.00006519 0.0007926140 0.00003291 1.1349 0.00001217 2.4213 0.00017182 2.205780 0.00001203 1.4519 0.00000343 1.8270 0.00004181 2.0340160 0.00000377 1.6740 0.00000103 1.7356 0.00001064 1.9743320 0.00000106 1.8305 0.00000028 1.8791 0.00000270 1.9785

Table 1. L1-error and EOC computed using the components of g1v.

K error in g2v1 EOC error in g2v2 EOC error in g2v3 EOC

20 0.00002668 0.00001068 0.0001858040 0.00000719 1.8917 0.00000354 1.5931 0.00005253 1.822580 0.00000173 2.0552 0.00000101 1.8094 0.00001350 1.9602160 0.00000044 1.9752 0.00000027 1.9033 0.00000342 1.9809320 0.00000012 1.8745 0.00000007 1.9475 0.00000086 1.9916

Table 2. L1-error and EOC computed using the components of g2v.

A NUMERICAL SCHEME FOR 3-D FRONT PROPAGATION 17

Next, we do the simulations for longer times, so that kinks appear on the wavefront. We considera mesh consisting of 401 × 401 cells and the final times are set to t = 2.0, 6.0, 10.0. We have usednon-reflecting boundary conditions on all the four boundaries.

Figure 3. The successive positions of the nonlinear wavefront Ωt with an initialsmooth dip which is not axisymmetric.

In Figure 3 we plot the initial wavefront Ω0 and the successive positions of the wavefront Ωt attimes t = 2.0, 6.0, 10.0. It can be seen that the wavefront has moved up in the x3-direction andthe dip has spread over a larger area in x1- and x2-directions. The lower part of the front movesup, leading to a change in shape of the initial front Ω0. It is very interesting to note that two dipsappear at the centre of the wavefront, which are clearly visible at t = 6.0 and t = 10.0. These twodips are separated by an elevation almost like a wall parallel to the x2-axis. There is a pair of kinklines, which are also parallel to the x2-axis and are more clearly seen in Figure 4.

In order to show the convergence of rays we give in Figure 4, the slices of the wavefronts inx2 = 0 section and x1 = 0 section from time t = 0.0 to t = 10.0. Due to the particular choice ofthe parameters α and β in the initial data (5.1), the section of the front Ω0 in x2 = 0 plane hasa smaller radius of curvature than that of the section in x1 = 0 plane. This results in a strongerconvergence of rays in the x2 = 0 plane compared to those in the x1 = 0 plane as evident fromFigure 4. In the diagram on the top in Figure 4, we clearly note a pair of kinks at times t = 3.0onwards in the x2 = 0 section. However, there are no kinks in the bottom diagram in Figure 4 inx1 = 0 section.

18 ARUN

−5

0

5

0 2 4 6 8 10 12

x 1−ax

is

x3−axis

Nonlinear wavefronts

−5

0

5

0 2 4 6 8 10 12

x 2−ax

is

x3−axis

Nonlinear wavefronts

Figure 4. The sections of the nonlinear wavefront at times t = 0.0, . . . , 10.0 with atime step 0.5. On the top: along x2 = 0 plane. Bottom: in x1 = 0 plane.

We give now the plots of the normal velocity m in (ξ1, ξ2) plane along ξ1- and ξ2-directions inFigure 5. It is observed that m has two shocks in the ξ1-direction which correspond to the twokinks in the x1-direction.

−10 −5 0 5 101.18

1.2

1.22

1.24

1.26

1.28

1.3

ξ1−axis

m

Normal velocity

t = 0.0

t = 2.0

t = 4.0

t = 6.0

t = 8.0

t = 10.0

−10 −5 0 5 101.18

1.2

1.22

1.24

1.26

1.28

1.3

ξ2−axis

m

Normal velocity

t = 0.0

t = 2.0

t = 4.0

t = 6.0

t = 8.0

t = 10.0

(a) (b)

Figure 5. The time evolution of the normal velocity m. (a): along ξ1-direction inthe section ξ2 = 0. (b): along ξ2-direction in the section ξ1 = 0.

A NUMERICAL SCHEME FOR 3-D FRONT PROPAGATION 19

We plot the divergence of B1 at time t = 10.0 computed using the formula (3.23) in Figure 6.It is evident from the plot that the constrained transport technique preserves geometric solenoidalconstraint up to an error of 10−15. The divergences of B2 and B3 also show the same trend.

Figure 6. The divergence of B1 at t = 10.0. The error is of the order of 10−15.

The analysis of the linearised Cauchy problem for 3-D WNLRT in section 4 shows that when thegeometric solenoidal constraint (2.6) is satisfied, the Jordan mode does not appear in the solution.More precisely, the results of section 4 show that the linearly growing terms appearing in thecomponents of the vectors g1u and g2v vanish when the geometric solenoidal constraint is satisfiedat time t = 0. Note that the choice of the initial values in section 3.3 is such that the constraint issatisfied at t = 0. The constrained transport technique preserves the geometric solenoidal constraintat any time up to machine round-off error; see Figure 6. We now proceed to give an evidence ofthe disappearance of the linearly growing Jordan with the aid of our numerical experiments. Forthis, we compute the L1 and L2 norms of the components of the vectors g1u and g2v as functionsof time t, for a long time. If there is any Jordan mode present, the numerical values of these normswould exhibit a linear growth with time. For any grid function w, the norms are computed usingthe formulae

(5.4)

‖w(t)‖L1 = ∆ξ1∆ξ2

K1∑

i=1

K2∑

j=1

|wi,j(t)|,

‖w(t)‖L2 =

√√√√∆ξ1∆ξ2

K1∑

i=1

K2∑

j=1

wi,j(t)2.

Here, K1 and K2 respectively denote the number of mesh points in ξ1- and ξ2-directions. In Figure 7we give the plots of L1 and L2 norms of the components of the vectors g1u and g2v up to timet = 200. It is clear from the figure that there is no linear growth with time for any of these quantities.This is in conformity with the results obtained in section 4 that the Jordan mode does not appearwhen the geometric solenoidal constraint is satisfied.

5.2. Corrugation Stability of a Nonlinear Wavefront. By corrugation stability, we mean thestability of a plane front to perturbations. The corrugation stability of plane shock fronts was first

20 ARUN

0 50 100 150 2000.9999

0.9999

1

1

1

1

1

1

1

time

norm

Norm of g1u

1

0 50 100 150 2000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

time

norm

Norm of g1u

2

0 50 100 150 2000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

time

norm

Norm of g1u

3

0 50 100 150 2000

0.005

0.01

0.015

time

norm

Norm of g2v

1

0 50 100 150 2001

1

1

1

1

1

1

1

time

norm

Norm of g2v

2

0 50 100 150 2000

0.01

0.02

0.03

0.04

0.05

0.06

time

norm

Norm of g2v

3

Figure 7. The L1 and L2 norms of the components of g1u and g2v. Solid linerepresents the L1 norm and dotted line is the L2 norm.

A NUMERICAL SCHEME FOR 3-D FRONT PROPAGATION 21

discussed by Gardner and Kruskal [17] in the context of magnetohydrodynamics. Whitham [45] usedhis GSD theory to study this problem. Anile and Russo [1] obtained an exact stability criterion forplane relativistic shock waves. The WNLRT is a very powerful method to study the corrugationstability of a nonlinear wavefront. It should be possible to use it to obtain some theoretical results,but the system being highly nonlinear, this appears to be too difficult. Therefore, we take help ofnumerical computations to establish the results. The extensive numerical computations by Prasadand Sangeeta [35] with 2-D WNLRT show that a planar nonlinear wavefront in 2-D is corrugationstable; see also [28] for a related discussion of corrugation stability of a 2-D shock front.

Here we intent to study the corrugation stability of a 3-D nonlinear wavefront using 3-D WNLRT.We give many illustrations showing corrugation stability. We choose the initial front to be of aperiodic shape

(5.5) Ω0 : x3 = κ(2− cos

(πx1a

)− cos

(πx2b

)),

with the parameters κ = 0.1, a = b = 2. The initial choice of the ray coordinates and the unknownvariables are done as in the previous problem. The initial velocity has a constant value m0 = 1.2throughout the front. The computational domain [−4, 4] × [−4, 4] is divided into 401 × 401 meshpoints. The simulations are for t = 20, 40, 60 and we have applied periodic boundary conditionseverywhere.

In Figure 8 we give the surface plots of the initial wavefront Ω0 and the wavefronts Ωt at timest = 20, 40, 60. The front Ωt moves up in the x3-direction and has developed several kink lines. Attime t = 20 we can observe eight kink lines on the wavefront, of which four of them are parallel tox1-axis and remaining four parallel to the x2-axis. Further, we can also observe the interaction ofthe kink lines and the front developing complex patterns. During the time evolution, the elevationsand depressions on the front diminishes, which indicates that the wavefront is tending to becomeplanar, showing corrugation stability.

In can be noted from Figure 8 that the wavefronts at times t = 40 and t = 60 show a very complexpattern of kink lines. In order to have a better visualisation of this phenomenon, in Figure 9 weplot a zoomed portion of the wavefront in one period at time t = 40.

As mentioned above, the elevations and depressions on the front decreases during its evolutionand as a result the height of wavefront decreases. The reduction in height with respect to time canbe seen in the following way. Let us first compute the highest and lowest altitudes on the front via

(5.6) x3max(t) := maxi,j

x3(ξ1i, ξ2j , t), x3min(t) := mini,j

x3(ξ1i, ξ2j , t).

The maximum height h(t) can then be defined by

(5.7) h(t) := x3max(t)− x3min(t).

In Figure 10(a) we give the plot of the height versus time, which shows that h reduces with time.Hence, it can easily be seen that the wavefront tends to become planar leading to corrugation stabil-ity. Moreover, this test also shows the efficiency of the central scheme to continue the computationsfor a very long time. In Figure 10(b) we have plotted the divergence of B1 at time t = 60. Thefigure shows that the divergence of B1 is zero up to machine round-off error even after a very longtime.

In those portions on the wavefront where the rays focus, the normal velocity m increases where asit decreases in the portions where the rays diverge. Nevertheless, in our numerical experiments, wehave observed a tendency that m tends to a constant value as time increases. This also an evidenceof the corrugation stability, since the front is ultimately becoming planar. We have observed thatthe normal velocity m is decaying to its initial mean value 1.2 along each ray. In order to see this,

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Figure 8. Nonlinear wavefront Ωt starting initially in a periodic shape with m0 =1.2. The front develops a complex pattern of kinks and ultimately tends to becomeplanar.

Figure 9. A zoomed portion of the periodic nonlinear wavefront Ωt in one periodat time t = 40. The wavefront shows four horizontal kink curves intersecting at onepoint and four other kink curves in background.

A NUMERICAL SCHEME FOR 3-D FRONT PROPAGATION 23

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

t

h

Maximum height

(a) (b)

Figure 10. (a): time variation maximum height of the wavefront which is initiallya smooth periodic pulse. (b): divergence of B1 computed at time t = 60.

we compute the maximum and minimum of m taken over the entire front at any time t, i.e.

(5.8) mmax(t) := maxξ1,ξ2

m(ξ1, ξ2, t), mmin(t) := minξ1,ξ2

m(ξ1, ξ2, t).

In Figure 11(a) we plot the distribution of mmax(t),mmin(t) with respect to time t. It can be notedthat mmax(t) attains a maximum value greater than mmax(0) = 1.2 at time t = 7 almost, which isapproximately the time when the kinks first appear. Similarly, mmin(t) attains its first minimumvalue at t = 5 just before the kinks form. The Figure 11(a) shows that mmax(t) and mmin(t) bothtend to 1.2 asymptotically. From Figure 11(b) we observe that mmax(t) −mmin(t) → 0 as t → ∞,confirming corrugation stability.

0 10 20 30 40 50 601

1.1

1.2

1.3

1.4

1.5

t

m

Maximum and minimum m

minmax

0 10 20 30 40 50 600

0.1

0.2

0.3

t

m

Maximum − minimum

(a) (b)

Figure 11. (a): variation of mmax(t) and mmin(t) with time from t = 0 to t = 60.(b): the difference mmax(t)−mmin(t) tends to zero as t→ ∞.

We give yet another illustration of corrugation stability using the test problem considered insection 5.1. In Figure 7 we have given the plots of the L1 and L2 norms of the components of the

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vectors g1u and g2v. The plots show that these norms tend to a constant value as time t → ∞.This is due to the fact that as t→ ∞, the front becomes planar and as a result g1u and g2v assumeconstant values.

5.3. Converging Wavefront Initially in the Shape of a Circular Cylinder. Next we presentthe results of simulation of a cylindrically converging wavefront. The motivation for this test problemis the one studied by Schwendeman in [39]. The initial geometry of the wavefront is a portion of acircular cylinder of radius two units and π units height, i.e.

(5.9) Ω0 : x21 + x22 = 4, −

π

2≤ x3 ≤

π

2.

Initially the ray coordinates (ξ1, ξ2) are chosen as ξ1 = x3 and ξ2 = θ, where θ is the azimuthalangle. Therefore, the initial wavefront Ω0 can be expressed in the parametric form

(5.10) Ω0 : x1 = 2 cos ξ2, x2 = 2 sin ξ2, x3 = ξ1, −π

2≤ ξ1 ≤

π

2, 0 ≤ ξ2 ≤ 2π.

Using the representation (5.10), the formulation of the initial values of the unknown variables canbe done as in the previous problems. As a result of this particular choice of the ray coordinates(ξ1, ξ2), the unit normal to Ω0 is n0 = (− cos ξ2, sin ξ2, 0), which points inward and hence the frontconverges. However, if we choose a uniform distribution of m on Ω0, the front Ωt at any successivetime t will remain as a circular cylinder with no interesting geometrical features. Therefore, theinitial distribution of the normal velocity m is taken as

(5.11) m0(ξ1, ξ2) = 1.2 + α cos(νξ2)

with the parameter values α = 0.05 and ν = 8. The computational domain [−π/2, π/2]× [0, 2π] isdivided into 301 × 601 cells. The simulations are done up to a time t = 1.0 and we have appliedperiodic boundary conditions for ξ2 and non-reflecting boundary conditions for ξ1.

In Figure 12 we give the plots of the initial wavefront Ω0 and the successive wavefronts Ωt at timest = 0.2, 0.4, 0.6, 0.8, 1.0. The wavefronts are coloured using the variation of the normal velocity mwith the colour-bar on the right indicating the values of m. Note that the initial normal velocitym0 in (5.11) has a periodic variation with respect to ξ2 with a maximum value 1.25 and a minimumvalue 1.15. Those portions of the front where m0 has maximum value moves inwards faster andit results in a distortion of the circular shape of Ω0. It can be observed that several vertical kinklines are starting to form on the wavefront at time t = 0.8, and as a result the front finally tends toassume the shape of a polygonal cylinder.

In order to show the rate of wavefront focusing with respect to time t, we compute the maximumand minimum of the radius r and velocity m at a section of the wavefront cut off by the planex3 = 0. In Figure 13(a)-(b) we give the plots of these quantities versus time. From the figurewe can notice a sudden increase in the maximum velocity after t = 0.6. On the other hand, theminimum of the normal velocity shows a gradual increase with time in Figure 13(a). It is to benoted that we should not continue the computations indefinitely as m− 1 becomes too large for theWNLRT to be valid. From the plot in Figure 13(b) we can infer a linear decay of the maximumand minimum radial distance over time. At time t = 1.0, the maximum and minimum radius hasreduced to more than half of their initial value r = 2.

In Figure 14(a) we give the plot of the normal velocity m with respect to ξ2 from t = 0 to t = 1.0with a time step of 0.1. Due to symmetry, in Figure 14(a) we have plotted m only in the range[0, π/2]. We notice that the difference mmax(t) − mmin(t) oscillates significantly. However, thisoscillation is quite small at time t = 0.8 due to the fact that mmax decreases and mmin increases.We can see that four shocks are trying to form in m at t = 1.0 in the interval [0, π/2]. Theseshocks in turn correspond to the kink lines forming on the wavefront. The plot in Figure 14(b)

A NUMERICAL SCHEME FOR 3-D FRONT PROPAGATION 25

Figure 12. Cylindrically converging wavefronts at times t = 0.0, 0.2, 0.6, 0.8, 1.0.The initial wavefront is in the shape of a circular cylinder. The colour bar on theright hand side indicates the intensity of the normal velocity m.

gives the divergence of B1 at time t = 1.0. The constrained transport method maintains thediscrete divergence to zero with machine round-off error even in problem with a very strong radialconvergence of the rays.

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0 0.2 0.4 0.6 0.8 11.15

1.2

1.25

1.3

1.35

t

m

Minimum and maximum m

minmax

0 0.2 0.4 0.6 0.8 10.6

0.8

1

1.2

1.4

1.6

1.8

2

t

r

Minimum and maximum r

minmax

(a) (b)

Figure 13. (a): time variation of maximum and minimum m at the section x3 = 0.(b): distribution of maximum and minimum radial distance with time.

0 0.5 1 1.51.1

1.2

1.3

1.4

1.5

ξ2

m

t = 0.0

t = 1.0

Normal velocity

(a) (b)

Figure 14. (a): variation of m with respect to ξ2 in the interval [0, π/2] from timet = 0.0 to t = 1.0. (b): div(B1) at time t = 1.0.

5.4. Spherically Converging Wavefront. In this test problem we consider the propagation of aspherically converging wavefront. The motivation for this problem is also an analogous case studiedby Schwendeman in [39]. The initial geometry of the wavefront is a sphere of radius two units.Initially the ray coordinates are chosen to be ξ1 = π−φ and ξ2 = θ, where θ is the azimuthal angleand φ is the polar angle. Therefore, the parametric representation of the initial wavefront Ω0 isgiven by

(5.12) Ω0 : x1 = 2 sin ξ1 cos ξ2, x2 = 2 sin ξ1 sin ξ2, x3 = −2 cos ξ1.

In order to avoid the singularities at φ = 0 and φ = π, we remove these points as also done in[39]. Therefore, our computational domain is [π/15, 14π/15]× [0, 2π]. We choose the initial velocitydistribution as

(5.13) m0(ξ1, ξ2) = 1.2 + α cos(ν1ξ1) cos(ν2ξ2)

A NUMERICAL SCHEME FOR 3-D FRONT PROPAGATION 27

with α = 0.05, ν1 = 4, ν2 = 8. The computational domain is divided into 301×601 mesh points andthe computations are done up to time t = 0.85. We have applied absorbing boundary conditions atξ1 = π/15 and ξ1 = 14π/15 and periodic boundary conditions at ξ2 = 0 and ξ2 = 2π.

The plots of the wavefronts at times t = 0.0, 0.5, 0.75, 0.85 are given in Figure 15. The wavefrontsare coloured using the variation of the normal velocity m with the colour-bar on the right indicatingthe values of m. It can be observed from the figure that as the front starts focusing it developsseveral kink curves and its spherical shape gets distorted. The final shape of the wavefront is apolygon with facets.

Figure 15. Spherically focusing wavefronts at times t = 0.0, 0.5, 0.75, 0.85. Thecolour bar on the right represents the distribution of m.

To show the intensity of wavefront focusing we compute the maximum and minimum radialdistance r of the points on Ωt from the centre (0, 0, 0) as a function of time. We also calculate the

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maximum and minimum of the normal velocity m, taken over the surface of the front, with respectto time. In Figure 16(a)-(b) we give the plots of these quantities versus time. From the plot inFigure 16(a) we can observe that the radial distance r reduces almost linearly in time. The maximumand minimum radii reduce to more than half of their initial value 2 at t = 0. The plot of normalvelocity in Figure 16(b) shows a sharp increase in the maximum m at around t = 0.6, indicatingthe formation of kinks. The minimum m has a gradual increase. The variation of minimum andmaximum m are qualitatively similar to those of the cylindrically converging wavefront problem insection 5.3.

0 0.15 0.3 0.45 0.6 0.750.8

1

1.2

1.4

1.6

1.8

2

t

r

Minimum and maximum r

minmax

0 0.15 0.3 0.45 0.6 0.751.1

1.2

1.3

1.4

1.5

1.6

t

m

Minimum and maximum m

minmax

(a) (b)

Figure 16. (a): time distribution of the maximum and minimum r. (b): maximumand minimum m.

6. Concluding Remarks

The KCL theory can be used to solve many problems where the evolution of an interface or othersurfaces appear. One needs to find out appropriate closure relations or equations depending on thenature of the surface. The present work is application of the theory to model the evolution of 3-Dweakly nonlinear wavefronts and the theory for a weak shock front has already been worked out [5].It is also possible to use the method to compute the evolution of a nonlinear wavefront or a shockfront of any arbitrary strength, following [34, 42].

We have demonstrated that the 3-D WNLRT is a powerful tool to numerically compute theevolution of 3-D nonlinear wavefronts. We have been able to develop a very robust and accuratenumerical approximation of the governing system of conservation laws, using a second order accu-rate Godunov type central finite volume scheme. Even though the system of conservation laws isdegenerate and with constraints, we find that the computations can be continued indefinitely fora very long time. In all the test problems considered, the constrained transport technique pre-serves the geometric solenoidal constraint up to very high accuracy. We do not observe any linearlygrowing Jordan modes or instabilities, in spite of the system being weakly hyperbolic.

In order to trace the history of a nonlinear wavefront, it is enough to know its initial geometryand the distribution of normal velocity. The kinks, whenever formed, are automatically tracked asthe images of shocks in the solution of the conservation laws. Using the present theory, we have beenable to describe the evolution of several nonlinear wavefronts, starting from a wide range of initialgeometries. The numerical simulations reveal many complex, yet realistic, geometrical features of

A NUMERICAL SCHEME FOR 3-D FRONT PROPAGATION 29

nonlinear wavefronts, particularly the structure of kinks formed on them. The results of long timecomputations, for a test problem with an initially periodic pulse modelling the smooth perturbationof a planar front, confirm the corrugation stability of nonlinear wavefronts. In addition, we haveobserved in this test problem that the normal velocity of the front approaches its initial mean valuealong each ray as t→ ∞.

It is difficult to compare the results of the present paper with any result obtained in past, asthere is no other theory to calculate successive positions of a 3-D nonlinear wavefront. Whitham’sGSD is for a shock front and hence we cannot compare our results with those of GSD. We refer thereader to [8] for a comparative study of the results of GSD, 2-D SRT and the Euler equations.

Acknowledgements

The author sincerely thanks Professors Phoolan Prasad and Maria Lukacova for their incisivesuggestions and generous help. He also wishes to thank the anonymous reviewers, as well as Pro-fessor Hans De Sterck whose useful suggestions and comments led to the improved version of themanuscript.

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Department of Mathematics, Indian Institute of Science, Bangalore - 560012, India.

E-mail address: arunkr@math.iisc.ernet.inCurrent address: Institut fur Geometrie und Praktische Mathematik, RWTH-Aachen, Templergraben-55, D-52056

Aachen, Germany.E-mail address: arun@igpm.rwth-aachen.de