Non-oscillatory methods for relaxation approximation of Hamilton–Jacobi equations

Applied Mathematics and Computation 183 (2006) 170–183

www.elsevier.com/locate/amc

Non-oscillatory methods for relaxation approximationof Hamilton–Jacobi equations

Mapundi Banda a,*, Mohammed Seaıd b

a School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag X01, Scottsville 3209, South Africab Fachbereich Mathematik, Technische Universitat Kaiserslautern, D–67663 Kaiserslautern, Germany

Abstract

In this paper, a class of high order non-oscillatory methods based on relaxation approximation for solving Hamilton–Jacobi equations is presented. The relaxation approximation transforms the nonlinear weakly hyperbolic equations to asemilinear strongly hyperbolic system with linear characteristic speeds and stiff source terms. The main ideas are to applythe weighted essentially non-oscillatory (WENO) reconstruction for the spatial discretization and an implicit–explicitmethod for the temporal integration. To illustrate the performance of the method, numerical results are carried out onseveral test problems for the two-dimensional Hamilton–Jacobi equations with both convex and nonconvex Hamiltonians.� 2006 Elsevier Inc. All rights reserved.

Keywords: Hamilton–Jacobi equations; Relaxation approximation; Non-oscillatory schemes; WENO reconstruction; Implicit–explicitmethods

1. Introduction

In this paper we propose a class of non-oscillatory numerical methods for the Cauchy problem forHamilton–Jacobi equations

0096-3

doi:10

* CoE-m

otuþ HðruÞ ¼ 0; x 2 Rd ; t > 0;

uðt ¼ 0; xÞ ¼ u0ðxÞ; x 2 Rd ;ð1:1Þ

where u(t,x) is the unknown function, u0(x) is given initial data, H is a Lipschitz continuous Hamiltonian, and$ denotes the gradient with respect to x = (x1, . . . ,xd)T defined by ru ¼ ðox1

u; . . . ; oxd uÞT. The Hamilton–Jacobi equations arise in many applications such as optimal control, image processing, geometric optics,differential games, calculus of variations, and front propagation problems. It is well known that after a finitetime the problem (1.1) develops discontinuous derivatives even if the initial data u0 is sufficiently smooth.Hence a solution of (1.1) has to be understood in an appropriate weak sense. Such a weak formulation is

003/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

.1016/j.amc.2006.05.066

rresponding author.ail addresses: [email protected] (M. Banda), [email protected] (M. Seaıd).

mailto:[email protected]

mailto:[email protected]

M. Banda, M. Seaıd / Applied Mathematics and Computation 183 (2006) 170–183 171

referred to by viscosity solutions, see [9] for more details. It is also known that the equations (1.1) can be trans-formed to a d-dimensional system of conservation laws

otpþrHðpÞ ¼ 0; x 2 Rd ; t > 0;

pðt ¼ 0; xÞ ¼ p0ðxÞ; x 2 Rd ;ð1:2Þ

where p(t,x) = $u(t,x) and p0(x) = $u0(x). It is clear that, if p is known, then u is recovered by integrating theordinary differential equation

otuþ HðpÞ ¼ 0: ð1:3Þ

A way to regularize the hyperbolic system (1.2) is the so-called relaxation approximation, see for example[14,13]. It consists of replacing the equations (1.2) with the following system

otpþrv ¼ 0;

otvþ Ar � p ¼ � 1

eðv� HðpÞÞ;

otuþ v ¼ 0;

ð1:4Þ

where e is the relaxation rate, v is the relaxation variable and A is a positive diagonal matrix. Note that, wehave replaced a nonlinear equation by a semi-linear system with the main advantage that it can be solvednumerically without introducing Riemann solvers. The relaxation system as defined above were first proposedin [14]. There a first order upwind scheme and a second order MUSCL scheme was used for the space discret-ization and a second order implicit–explicit (IMEX) Runge–Kutta scheme for the time integration. The mainadvantage in considering relaxation methods is that Riemann solvers are completely avoided in their recon-struction. Numerical relaxation schemes have been extensively studied in [4,21,5,3] for hyperbolic systemsof conservation laws and convection-diffusion problems.

The objective of this paper is to extend the multidimensional relaxation approach for Hamilton–Jacobiequations to higher-order accuracy. In extending this approach to higher-order we would like to maintainthe advantages of the approach as presented in [14,13] i.e., produce entropy solution in the zero relaxationlimit and design a higher-order numerical scheme that is Riemann solver free. In [14] only first-order and sec-ond-order MUSCL schemes were introduced. Further to the above objective, we would also like to maintainthe simplicity of relaxation schemes for Hamilton–Jacobi equations as presented in [13]. This property is pre-served for any space dimension, see [13] for a more detailed discussion. The other advantage of relaxationapproach to solving the Hamilton–Jacobi equations is that the structure of the derived relaxation system isstrongly hyperbolic. It has been pointed out that two important properties which make this approach workis that in the first-order zero relaxation limit one yields a Lax–Friedrichs type scheme. Such schemes have beendemonstrated to work well in weakly hyperbolic systems [7,11]. With the relaxation approach, the weaklyhyperbolic equations are written as strongly hyperbolic systems which are well-posed. Hence in the numericalapproach a strongly well-posed system is discretized in such a way that in the zero relaxation limit a Lax–Friedrichs method is obtained for first-order. In our presentation we demonstrate the correct use of numericalviscosity as designed in [14] sufficient for higher-order approaches. The spurious oscillations do not appear inhigher-order approaches if the same approach for introducing numerical viscosity is applied. This is also thecase in any space dimensions.

Our goal in this paper is also to generalize the relaxation approach to higher-order reconstructions for anyspace dimensions. The reconstruction applied is based on the weighted essentially non-oscillatory (WENO)approach in [2,12] where schemes of higher-order spatial accuracy were presented. For the purposes of thispresentation fifth-order reconstruction will be considered. Extensions to higher-order can be done analo-gously. In [18] it was observed that higher-order schemes have advantages over second-order schemesespecially for computing solutions containing both discontinuities and complex solution features. Suchhigh-order reconstructions are integrated with the relaxation formulation. The idea here is to preserve allthe advantages of the relaxation approach [15]. For the time discretization a third-order IMEX scheme isapplied [1,17]. The choice of such schemes is done in a novel way in order to have a scheme that is strong sta-bility preserving in the zero relaxation limit [8]. Further, on the choice of parameters like the characteristic

172 M. Banda, M. Seaıd / Applied Mathematics and Computation 183 (2006) 170–183

speeds, the relaxation approach gives practitioners a variety of choices to make. The constraint that mustbe observed is the subcharacteristic condition [15,6] which also influences the definition of the Courant–Friedrichs–Lewy (CFL) condition. The simplicity of such an approach even in the numerical sense makes itvery attractive for solving a wide variety of practical problems. A higher-order approach based on the WENOreconstruction is new. To demonstrate its practical applicability, numerical results have been presented in Sec-tion 4.

The structure of this paper is as follows. In Section 2 we briefly review the relaxation approximation for theHamilton–Jacobi equations. Non-oscillatory relaxation methods are formulated in Section 3. This sectionincludes a fifth-order WENO reconstruction for the spatial discretization and a third-order IMEX methodfor the time integration. Numerical illustrations are presented for two-dimensional problems in Section 4.Concluding remarks summarize this paper in Section 5.

2. Relaxation approximation of Hamilton–Jacobi equations

To simplify the discussion and the presentation, we focus our attention on the two-dimensional Hamilton–Jacobi equations, although the methods presented in this paper extend to general problems without majorconceptual modifications. In two space dimensions, x = (x,y)T, p = (p,q)T and $ = (ox,oy)T, the equations(1.4) reduce to

otp þ oxv ¼ 0; ð2:1aÞotqþ oyv ¼ 0; ð2:1bÞ

otvþ aoxp þ boyq ¼ � 1

eðv� Hðp; qÞÞ; ð2:1cÞ

otuþ v ¼ 0: ð2:1dÞ

Formally, when e! 0, the solution of the relaxation system (2.1) converges to the solution of the conser-vation law (1.2). In fact, from Eq. (2.1c) we can write

v ¼ Hðp; qÞ � eðotvþ aoxp þ boyqÞ: ð2:2Þ
Also, differentiating Eq. (2.1d) with respect to x and y gives
otðoxuÞ þ oxv ¼ 0; ð2:3aÞotðoyuÞ þ oyv ¼ 0: ð2:3bÞ

Subtracting (2.3a) from (2.1a) and (2.3b) from (2.1b) gives

otðoxu� pÞ ¼ 0 and otðoyu� qÞ ¼ 0; ð2:4Þ
respectively. Using the initial conditions p0 = oxu0 and q0 = oyu0, Eq. (2.4) yield
oxuðt; x; yÞ ¼ pðt; x; yÞ; oyuðt; x; yÞ ¼ qðt; x; yÞ; t > 0: ð2:5Þ
Back-substituting (2.5) into (2.2) gives
v ¼ Hðoxu; oyuÞ � eðotvþ ao2xxuþ bo

2yyuÞ: ð2:6Þ

A Chapman–Enskog expansion in (2.6) for small e gives

otuþ Hðoxu; oyuÞ ¼ e ða� ðopHÞ2Þo2xxuþ ðb� ðoqHÞ2Þo2

yyu� 2ðopHoqHÞo2xyu

h i: ð2:7Þ

Hence, a sufficient condition for the coefficient of the dissipation in (2.7) to be positive is

ðopHÞ2

aþ ðoqHÞ2

b6 1: ð2:8Þ

This condition, known as a sub-characteristic condition [15,6,14,13], ensures that the characteristic speedsof the relaxation system (2.1) are at least as large as the characteristic speeds of the original equation (1.2).


Remark 2.1. To avoid initial layer in the relaxation system (2.1), the initial condition for the variable v isselected to be consistent with the local equilibrium v = H(p,q), i.e.

vðt ¼ 0; x; yÞ ¼ Hðp0; q0Þ; ð2:9Þ
where p0(x,y) and q0(x,y) are initial data defined by
p0ðx; yÞ ¼ oxu0ðx; yÞ and q0ðx; yÞ ¼ oyu0ðx; yÞ: ð2:10Þ

3. Non-oscillatory relaxation method

The relaxation system (2.1) is numerically solved using a splitting operator where the advection stage andrelaxation stage are discretized separately using the method of lines. In this section, we first discuss a fifth-order WENO reconstruction for the spatial discretization. Then a third-order IMEX time integration schemeis formulated for the semi-discrete relaxation system.

3.1. WENO reconstruction for spatial discretization

For simplicity we consider a uniform control volume [xi�1/2,xi+1/2] · [yj�1/2,yj+1/2] with mesh sizesDx = xi+1/2 � xi�1/2 and Dy = yj+1/2 � yj�1/2 in x- and y-direction, respectively. Integrating (2.1) over the con-trol volume and keeping the time continuous we obtain the following semi-discrete relaxation system

dpi;j

dtþ viþ1=2;j � vi�1=2;j

Dx¼ 0; ð3:1aÞ

dqi;j

dtþ vi;jþ1=2 � vi;j�1=2

Dy¼ 0; ð3:1bÞ

dvi;j

dtþ ai;j

piþ1=2;j � pi�1=2;j

Dxþ bi;j

qi;jþ1=2 � qi;j�1=2

Dy¼ � 1

eðvi;j � Hðp; qÞi;jÞ; ð3:1cÞ

dui;j

dtþ vi�1=2;j þ vi;j�1=2 þ viþ1=2;j þ vi;jþ1=2

4¼ 0; ð3:1dÞ

where wi,j is the space average of a generic function w in the cell [xi�1/2,xi+1/2] · [yj�1/2,yj+1/2], wi+1/2j andwij+1/2 are the numerical fluxes at (xi+1/2,yj) and (xi,yj+1/2), i.e.

wi;j ¼1

Dx1

Dy

Z xiþ1=2

xi�1=2

Z yjþ1=2

yj�1=2

wðt; x; yÞdxdy;

wiþ1=2;j ¼ wðt; xiþ1=2; yjÞ; wi;jþ1=2 ¼ wðt; xi; yjþ1=2Þ:

The spatial discretization of the relaxation system is complete when a numerical reconstruction of the fluxes in(3.1) is chosen. Since the system (2.1) has linear characteristics given by

F� ¼ v� ap; G� ¼ v� bq; ð3:2Þ
upwind schemes can be easily implemented. For example, the first-order upwind scheme [14,13] applied to Fþ
and F� gives

Fþiþ1=2;j ¼Fþ

i;j; F�iþ1=2;j ¼F�

iþ1;j: ð3:3Þ

The second-order reconstruction [14,13] is achieved by applying the van Leer’s MUSCL scheme to Fþ andF�

Fþiþ1=2;j ¼Fþ

i;j þ1

2Dxrþi;j; F�

iþ1=2;j ¼F�iþ1;j þ

1

2Dxr�iþ1;j; ð3:4Þ

where rþi;j and r�i;j are respectively, the slopes of Fþ and F� on the control volume [xi�1/2,xi+1/2] · [yj�1/2,yj+1/2]. They are defined as

r�i;j ¼1

DxðF�

iþ1;j �F�i;jÞUðh

�i;jÞ; h�i;j ¼

F�i;j �F�

i�1;j

F�iþ1;j �F�

i;j

;


where U(h) defines the van Leer’s slope limiter function

UðhÞ ¼ jhj þ h1þ jhj :

The reconstruction of the fluxes G�i;jþ1=2 can be done in a similar manner. We should mention that a third-order relaxation scheme has also been proposed by the authors in [5] and intensively tested in [4,3,19–21] for awide hyperbolic systems of conservation laws. The third-order method in these references reconstructs thefluxes in (3.1) using linear combination of small stencils such that the overall combination preserves thethird-order accuracy. In the current work, we formulate a fifth-order reconstruction for the numerical fluxesin (3.1). The key idea is to replace the linear weights in the third-order reconstruction by nonlinear weightscapable of reducing the spurious oscillations that usually develop in regions with discontinuous derivatives.Furthermore, the fifth-order reconstruction is applicable directly to the linear characteristic variables (3.2)without relying on Riemann solvers. In what follows we only formulate pi+1/2,j and qi+1/2,j, and the expressionsfor other fluxes are obtained in an entirely analogous manner.

Since the variables Fþ and F� travel along constant characteristics with speed +a and �a, respectively, aWENO reconstruction can be easily applied to them. Thus, the numerical fluxes F�

iþ1=2;j at the cell boundary(xi+1/2,yj) are defined as left and right extrapolated values Fþ;L

iþ1=2;j and F�;Riþ1=2;j by

Fþiþ1=2;j ¼Fþ;L

iþ1=2;j; F�iþ1=2;j ¼F�;R

iþ1=2;j: ð3:5Þ

These extrapolated values are obtained from cell averages by means of high-order WENO polynomialreconstruction. In the present work we use the fifth-order WENO reconstruction proposed in [12]. Higher-order reconstructions are also possible, such as those developed in [2]. For a generic function w(x,y) thefifth-order accurate left boundary extrapolated value wL

iþ1=2;j is defined as

wLiþ1=2;j ¼ x0V0 þ x1V1 þ x2V2; ð3:6Þ

where Vr is the extrapolated value obtained from cell averages in the rth stencil (i � r, i � r + 1,i � r + 2) inx-direction

V0 ¼1

6ð�wiþ2;j þ 5wiþ1;j þ 2wi;jÞ;

V1 ¼1

6ð�wi�1;j þ 5wi;j þ 2wiþ1;jÞ;

V2 ¼1

6ð2wi�2;j � 7wi�1;j þ 11wi;jÞ;

and xr, r = 0,1,2, are nonlinear WENO weights given by

xk ¼akP2l¼0al

; a0 ¼3

10ðsþ IS0Þ2; a1 ¼

3

5ðsþ IS1Þ2; a2 ¼

1

10ðsþ IS2Þ2: ð3:7Þ

Here the parameter s is introduced to guarantee that the denominator does not vanish and is empirically takento be 10�6. The smoothness indicators ISr, r = 0,1,2, are given by [12]

IS0 ¼13

12ðwi;j � 2wiþ1;j þ wiþ2;jÞ

2 þ 1

4ð3wi;j � 4wiþ1;j þ wiþ2;jÞ

2;

IS1 ¼13

12ðwi�1;j � 2wi;j þ wiþ1;jÞ

2 þ 1

4ðwi�1;j � wiþ1;jÞ

2;

IS2 ¼13

12ðwi�2;j � 2wi�1;j þ wi;jÞ

2 þ 1

4ðwi�2;j � 4wi�1;j þ 3wi;jÞ

2:

The right value wRiþ1=2;j is obtained by symmetry as

wRiþ1=2;j ¼ x0V0 þ x1V1 þ x2V2; ð3:8Þ


where

V0 ¼1

6ð2wiþ3;j � 7wiþ2;j þ 11wiþ1;jÞ;

V1 ¼1

6ð�wiþ2;j þ 5wiþ1;j þ 2wi;jÞ;

V2 ¼1

6ð�wi�1;j þ 5wi;j þ 2wiþ1;jÞ:

The corresponding weights and smoothness indicators are given by (3.7) with

IS0 ¼13

12ðwiþ1;j � 2wiþ2;j þ wiþ3;jÞ

2 þ 1

4ð3wiþ1;j � 4wiþ2;j þ wiþ3;jÞ

2;

IS1 ¼13

12ðwi;j � 2wiþ1;j þ wiþ2;jÞ

2 þ 1

4ðwi;j � wiþ2;jÞ

2;

IS2 ¼13

12ðwi�1;j � 2wi;j þ wiþ1;jÞ

2 þ 1

4ðwi�1;j � 4wi;j þ 3wiþ1;jÞ

2:

Once Fþiþ1=2;j and F�

iþ1=2;j are reconstructed in (3.5), the numerical fluxes pi+1/2,j and vi+1/2,j in the relaxationsystem are obtained from (3.2) by

piþ1=2;j ¼1

2ai;jðFþ

iþ1=2;j �F�iþ1=2;jÞ; viþ1=2;j ¼

1

2ðFþ

iþ1=2;j þF�iþ1=2;jÞ: ð3:9Þ

Analogously, the numerical fluxes qi,j+1/2 and vi,j+1/2 are obtained from Gþi;jþ1=2 and G�i;jþ1=2 as

qi;jþ1=2 ¼1

2bi;jðGþi;jþ1=2 þ G�i;jþ1=2Þ; vi;jþ1=2 ¼

1

2ðGþi;jþ1=2 þ G�i;jþ1=2Þ: ð3:10Þ

We should point out that if the discretization of the variable v in (3.1d) is discretized as vi,j instead of averagingthe values at all cell boundaries, one may not recover the correct viscosity solution for the Hamilton–Jacobiequation. We refer to [14] for more discussion on numerical passage from conservation laws to Hamilton–Jacobi equations via relaxation approximation.

Remark 3.1. The approximation of the flux function H(p,q)i,j in (3.1) can be performed using the high-orderSimpson quadrature rule as in [5]. However, for small values of e it suffices to project H(p,q)ij into the localequilibrium H(pi,j,qi,j).

3.2. Implicit–explicit methods for time integration

The semi-discrete relaxation system (3.1) can be rewritten as a system of autonomous ordinary differentialequations

dY

dt¼ FðYÞ � 1

eGðYÞ; ð3:11Þ

where the time-dependent vector functions

Y ¼

pi;j

qi;j

vi;j

ui;j

0BBB@

1CCCA; FðYÞ ¼

� viþ1=2;j�vi�1=2;j

Dx

� vi;jþ1=2�vi;j�1=2

Dy

�ai;jpiþ1=2;j�pi�1=2;j

Dx � bi;jqi;jþ1=2�qi;j�1=2

Dy

0

0BBBB@

1CCCCA; GðYÞ ¼

0

0

vi;j �Hðp; qÞi;je

vi�1=2;jþvi;j�1=2þviþ1=2;jþvi;jþ1=2

4

0BBBB@

1CCCCA:

It is clear that, when e! 0, the equations (3.11) become highly stiff and any explicit treatment of the righthand side in (3.11) requires extremely small time stepsizes. This fact might restrict any long term simulationin (2.1). On the other hand, integrating the equations (3.11) by an implicit scheme, either linear or nonlinearalgebraic equations have to be solved at every time step of the computational process. To find solutions of


such systems is computationally very demanding. In this paper we consider an approach based on IMEXRunge–Kutta splitting. The non-stiff stage of the splitting for F is treated by an explicit Runge–Kutta scheme,while the stiff stage for G is approximated by a diagonally implicit Runge–Kutta scheme, compare [1,17] formore details.

Let Dt be the time step and Yn denote the approximate solution at time t = nDt. The implementation of anIMEX method to solve (3.11) can be carried out in the following steps:

1. For l = 1, . . . , s,(a) Evaluate K�l as

K�l ¼ Yn þ DtXl�1

m¼1

~almFðKmÞ �Dte

Xl�1

m¼1

almGðKmÞ: ð3:12aÞ

(b) Solve for Kl

Kl ¼ K�l �Dte

allGðKlÞ: ð3:12bÞ

2. Update Yn+1 as

Ynþ1 ¼ Yn þ DtXs

l¼1

~blFðKlÞ �Dte

Xs

l¼1

blGðKlÞ: ð3:12cÞ

The s · s matrices ~A ¼ ð~almÞ, ~alm ¼ 0 for m P l and A = (alm) are chosen such that the resulting scheme is expli-cit for F, and implicit for G. The s-vectors ~b and b are the canonical coefficients which characterize the IMEXs-stage Runge–Kutta scheme [17]. They can be given by the standard double tableau in Butcher notation,

Here, ~c and c are s-vectors used in non autonomous cases. For instance, the second-order time integrationprocedure in [15] can be formulated as (3.12) with the explicit and implicit Runge–Kutta tables given by

Note that, using the above relaxation scheme neither linear algebraic equation nor nonlinear source terms canarise. In addition, since the source terms are treated implicitly and the advection terms are treated explicitly,the high order relaxation scheme is stable independently of e, so the choice of Dt is based only on the usualCFL condition

CFL ¼ maxDth; aij

DtDx; bij

DtDy

� �6 1; ð3:13Þ

where h = max(Dx,Dy) denotes the maximum cell size.In this paper we use the third order IMEX scheme studied in [17]. Its associated double tableau can be rep-

resented as

where c ¼ 3þffiffi3p

6. Other IMEX schemes of third and higher order are also discussed in [1,17]. It is clear that, at

the limit (e! 0) the time integration procedure tends to a TVD time integration scheme of the limit equationsbased on the explicit scheme given by the left table.


Remark 3.2. It is easy to verify that, from the expansion (2.7), the characteristic speeds a and b are related tothe numerical diffusion in the relaxation method. In general, a and b can be chosen large enough such that thesub-characteristic condition (2.8) is satisfied. However, the numerical diffusion increases with their values andfor accuracy reasons a and b should be chosen as small as possible. In the current work, the characteristicspeeds in (3.1) are calculated locally at each control volume as

TableError-

Gridpo

20 · 2040 · 4080 · 80160 · 1

ai;j ¼ maxfjopHðpLiþ1=2;j; q

Li;jþ1=2Þj; jopHðpR

iþ1=2;j; qRi;jþ1=2Þjg;

bi;j ¼ maxfjoqHðpLiþ1=2;j; q

Li;jþ1=2Þj; joqHðpR

iþ1=2;j; qRi;jþ1=2Þjg:

ð3:14Þ

For more discussions on the selection of characteristic speeds in general relaxation methods we refer to[4,5,3,19–21].

4. Numerical results and examples

Several test examples on the two-dimensional Hamilton–Jacobi equations have been selected to assess theperformance of the relaxation method proposed in this paper. Most of the examples are standard and havebeen used before in the literature, see [13,10,16,7] among others. In all the examples presented in this section:given a Hamiltonian H and an initial datum u0, the initial conditions for p, q and v are obtained from (2.10)and (2.9), respectively. The boundary conditions are periodic for all variables. In all the computations thecharacteristic speeds are chosen locally as (3.14). Moreover, we set e = 10�10, the CFL number is fixed to0.5 and we use variable time stepsizes Dt adjusted at each step according to (3.13) as

1norms for the linear advection equation (4.1)

ints L1-error Rate L1-error Rate

0.85504E�01 – 0.63718E�01 –0.21435E�01 3.207 0.13037E�01 3.5950.13121E�02 4.030 0.58060E�03 4.489

60 0.41836E�04 4.971 0.17296E�04 5.069

0

1

2

0

1

2

0

0.5

−1

−0.5

−2

−1

−2

−1

Fig. 1. Solution of the convex Hamiltonian 4.2 at time t = 1.5/p2 obtained using the fifth-order relaxation scheme.

Fig. 2.relaxat


Dt ¼ CFL mini;j

h;Dxai;j

;Dybi;j

� �:

The following test examples are selected:

4.1. Linear advection equation

The Hamiltonian and the initial data are given by

Hðoxu; oyuÞ ¼ oxuþ oyu; u0ðx; yÞ ¼ sinpðxþ yÞ

2

� �: ð4:1Þ

The computational domain is [�2,2] · [�2,2]. In Table 1 we show the L1-error and L1-error norms at thetime t = 1. These errors are measured by the difference between the pointvalues of the exact solution and thereconstructed pointvalues of the computed solution. As expected the scheme preserves the fifth-order ofaccuracy.

4.2. Convex hamiltonian


Hðoxu; oyuÞ ¼1

2ðoxuþ oyuþ 1Þ2; u0ðx; yÞ ¼ � cos

pðxþ yÞ2

� �: ð4:2Þ

The computational domain is [�2,2] · [�2,2] discretized into 50 · 50 control volumes. In this example theperformance of the fifth-order scheme is demonstrated. The solution of the problem develops a singularity. Wecompute the solution up to time t = 0.8/p2 before the singularity and time t = 1.5/p2 after the singularity. Theresults of the computation for the singular case are presented in Fig. 1. In Fig. 2 a slice of the solution as wellas a close-up view of the solution is presented. This clearly demonstrates the performance of a higher-orderrelaxation-based approach.

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

x

u

First−orderSecond−orderFifth−order

0 0.1 0.2 0.3

Zooming

−0.2

−0.4

−0.6

−0.8

−1

−2 −1.5 −1 − 0.5

−0.98

−1

−1.02

−1.04

−1.06

−1.08

Comparison of the results for the convex Hamiltonian 4.2 at time t = 1.5/p2 obtained using the first-, second- and fifth-orderion schemes.


4.3. Nonconvex hamiltonian


Fig. 4.relaxat

Hðoxu; oyuÞ ¼ � cosðoxuþ oyuþ 1Þ; u0ðx; yÞ ¼ � cospðxþ yÞ

2

� �: ð4:3Þ

0

1

2 2

−1

0

1

2

−0.5

0

0.5

1

−1

−2

Fig. 3. Solution of the nonconvex Hamiltonian 4.3 at time t = 2.5/p2 obtained using the fifth-order relaxation scheme.

0 0.5–0.5 1–1 1.5–1.5 2–2

0

0.2

–0.2

0.4

–0.4

0.6

–0.6

0.8

–0.78

–0.8

–0.82

–0.84

–0.86

–0.88

–0.8

1

1.2

x

u

First-orderSecond-orderFifth-order

0 0.1 0.2

Zooming

Comparison of the results for the nonconvex Hamiltonian 4.3 at time t = 2.5/p2 obtained using the first-, second- and fifth-orderion schemes.


The computational domain is [�2,2] · [�2,2] discretized into 50 · 50 control volumes. Again the solutionwas computed for the smooth case t = 0.8/p2 as well as the discontinuous case at t = 2.5/p2. A three-dimen-sional profile of the solution has been presented in Fig. 3. A one-dimensional profile through the solution ispresented as well as a close-up view of the solution is presented in Fig. 4. The results again show very goodperformance of the fifth-order relaxation scheme.

–2

–1

0

1

2 –2

–1

0

1

2

–1.5

–1

–0.5

0

0.5

1

Fig. 5. Solution of the fully two-dimensional Hamiltonian 4.4 at time t = 1.5 obtained using the fifth-order relaxation scheme.

0 0.5–0.5

–0.15

–0.2

–0.25

1–1

–1.6 –1.4

1.5–1.5 2–2

0

0.5

–0.5

1

1.5

x

u

First–order

Second–order

Fifth–order

Zooming

Fig. 6. Comparison of the results for the fully two-dimensional Hamiltonian 4.4 at time t = 1.5 obtained using the first-, second- and fifth-order relaxation schemes.


4.4. Fully two-dimensional hamiltonian


Fig. 8.scheme

Hðoxu; oyuÞ ¼ ðoxuÞðoyuÞ; u0ðx; yÞ ¼ sinðxÞ þ cosðyÞ: ð4:4Þ
The computational domain is [�p,p] · [�p,p] discretized into 50 · 50 control volumes. The idea is to dem-
onstrate that the relaxation scheme with a dimension-by-dimension approach for such a fully-dimensional

–2–1

01

2

–2

–1

0

1

2–1.6

–1.55

–1.5

–1.45

Fig. 7. Solution of the Eikonal equation 4.5 at time t = 0.6 obtained using the fifth-order relaxation scheme.

2 1.5 1 0.5 0 0.5 1 1.5 21.6

1.55

1.5

1.45

1.4

x

u

First–order

Second–order

Fifth–order

0.2 0 0.2

1.42

1.4

1.38

Zooming

_

_

_

_

_

_

_

_

_ _ _ _

Comparison of the results for the Eikonal equation 4.5 at time t = 0.6 obtained using the first-, second- and fifth-order relaxations.


case gives reasonable results. Computations were done up to t = 1.5 and the obtained results are presented inFig. 5. Further a close-up view is presented in Fig. 6. One observes good performance of such a simple schemein both these figures.

4.5. Eikonal equation in geometric optics

The Hamiltonian and the initial data are given by [13]

Hðoxu; oyuÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ðoxuÞ2 þ ðoyuÞ2

q; u0ðx; yÞ ¼ �

1

4ðcosð2pxÞ � 1Þðcosð2pyÞ � 1Þ þ 1: ð4:5Þ

Here again results are demonstrated on a nonconvex Hamiltonian. In Fig. 7 a three-dimensional plot of theresults at time t = 0.6 are presented. Here the sharp corners that develop in the problem are well resolved. Wefurther present a slice of the solutions in Fig. 8. Again our fifth-order formulation gives good results with avery coarse grid of 50 · 50 and the computational domain is [0,1] · [0,1].

5. Conclusion

In this paper we have proposed a class of non-oscillatory methods for relaxation approximation of Ham-ilton–Jacobi equations. The spatial discretization is performed by a fifth-order WENO reconstruction whilethe time integration is carried out using a third-order IMEX scheme for time integration. These techniquescombine the attractive attributes of the two procedures to yield a numerical method for weakly hyperbolicsystems without relying on Riemann solvers or characteristic decompositions. Numerical results are presentedfor a wide collection of well-established examples on Hamilton–Jacobi equations in two space dimensions. Theperformance of our relaxation scheme is very attractive since the computed solution remains stable, monotoneand highly accurate even on coarse grids without solving Riemann problems or requiring special front track-ing procedures.

Acknowledgements

Research supported in part by the University of KwaZulu-Natal competitive research grant 2005 ‘‘FlowModels with Source Terms, Scientific Computing, Optimisation in Applications’’. M. Seaıd is deeply grate-ful to the University of KwaZulu-Natal at Pietermaritzburg for their hospitality during a research visitthere.

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Non-oscillatory methods for relaxation approximation of Hamilton–Jacobi equations

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