High-order ADER schemes for hyperbolic equations: a...
Transcript of High-order ADER schemes for hyperbolic equations: a...
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HighHigh--order ADER schemes for order ADER schemes for hyperbolic equations: a reviewhyperbolic equations: a review
Eleuterio TOROLaboratory of Applied Mathematics
University of Trento, ItalyWebsite: www.ing.unitn.it/toro
Email: [email protected]
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I am indebted to my collaborators:
Michael Dumbser and Svetlana Tokareva (Trento, Italy)Arturo Hidalgo (Madrid, Spain)
Carlos Pares, Manuel Castro and Alberto Pardo (Malaga, Spain)
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Part A:
FORCE-type centred, monotone schemes for hyperbolic equations in multiple space dimensions on general meshes
Part B:
ADER high-order methods
Part C:Sample applications: the 3D Euler equations, the 3D Baer-Nunziato equations for compressible multiphase flows on unstructured meshes
CONTENTS
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We are interested in developing numerical methods to solve
)()()()()( QDQSQHQGQFQ zyxt
)()()()()( QDQSQQCQQBQQAQ zyxt
Source terms S(Q) may be stiff
Advective terms may not admit a conservative form(nonconservative products)
Meshes are assumed unstructured
Very high order of accuracy in both space and time
May use upwind or centred approaches for numerical fluxes
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Design constraints:
We want the schemes to be
Conservative (in view of the Lax-Wendroff theorem)
Non-linear(in view of the Godunov’s theorem)
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In view of Godunov’s theorem our numerical approach relies on:
1.A monotone numerical flux (at most first order accurate)
2.A framework to construct non-linear numerical methods of high-order of accuracy in both space and time in which
The monotone flux is the bluilding block
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Part A:
FORCE-type centred, monotone schemes for hyperbolic equations in multiple space dimensions solved on general meshes
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Conservative schemes in 1D
2/1i2/1ini
1ni FF
xtQQ
0)Q(FQ xt
Task: define numerical flux
2/1iF
Basic property required: MONOTONICITY
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2/1in
1i
ni
xt
F0x if Q
0x if Q)0,x(Q
0)Q(FQ
There are two approaches:
)Q,Q(HF n1i
ni2/1i
I: Upwind approach (Godunov). Solve the Riemann problem
II: Centred approach. The numerical flux is
(Toro E F. Riemann solvers and numerical methods for fluid dynamics. Springer, 3° edition 2009)
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The FORCE flux (First ORder CEntred)
Brief review of 1D case
TORO E F, BILLETT S J. (2000). Centred TVD schemes for hyperbolic conservation laws. IMA JOURNAL OF NUMERICAL
ANALYSIS. vol. 20, pp. 47-79 ISSN: 0272-4979.
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Glimm’s method on a staggered mesh
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Recall that the integral form of the conservation laws
0)Q(FQ xt
in a control volume 21RL t,tx,x
R
L
R
L
x
x
t
tL
t
tRxx
x
xx dttxQFdttxQFdxtxQdxtxQ
2
1
2
1
)),(()),((),(),( 11
12
1
is
FORCE: replace random state by averaged state
Averaging operator
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)()(21
21
112/1
2/1ni
ni
ni
ni
ni QFQF
xtQQQ
)Q(F)Q(Fxt
21QQ
21Q n
1ini
ni
n1i
2/1n2/1i
)()(21
21 2/1
2/12/1
2/12/1
2/12/1
2/11
ni
ni
ni
ni
ni QFQF
xtQQQ
Step I
Step II
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force2/1i
force2/1i
ni
1ni FF
xtQQ
with numerical flux
Step III: One-step conservative form
)(21
2/12/12/1LF
iLW
iforce
i FFF
)Q(F)Q(Fxt
21QQ
21Q),Q(FF n
in
1in
1ini
lw2/1i
lw2/1i
LW2/1i
ni
n1i
n1i
ni
LF2/1i QQ
tx
21)Q(F)Q(F
21F
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)FF(21F LF
2/1iLW
2/1iforce
2/1i
The numerical flux is in fact
)Q(F)Q(Fxt
21QQ
21Q),Q(FF n
in
1in
1ini
lw2/1i
lw2/1i
LW2/1i
with
ni
n1i
n1i
ni
LF2/1i QQ
tx
21)Q(F)Q(F
21F
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Properties of the FORCE scheme
flforce
xforcext
ccx
qqqequationModifiedMonotone
cStable
21)1(
41
1||0
2
)2(
Proof of convergence of FORCE scheme in:
Chen C Q and Toro E F.Centred schemes for non-linear hyperbolic equations.
Journal of Hyperbolic Differential. Equations 1 (1), pp 531-566, 2004.
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The FORCE flux for the scalar case:more general averaging.
10,F)1(FF LF2/1i
LW2/1i2/1i
(GFORCE) c1
1(FORCE) 2/1
Wendroff)-(Lax 1)Friedrichs-(Lax 0
Special cases:
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Related works on centred schemes include
H Nessyahu and E Tadmor. Non-oscillatory central differencing for hyperbolic conservation Laws. J. Computational Physics, Vol 87, pp 408-463, 1990.
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Recent extensions of FORCE
• Multiple space dimensions
• Unstructured meshes
• Path-conservative version
• High-order non-oscillatory (FV, DG, PC)
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Toro E F, Hidalgo A and Dumbser M.FORCE schemes on unstructured meshes I:
Conservative hyperbolic systems.(Journal of Computational Physics, Vol. 228, pp 3368-3389, 2009)
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Illustration in 2D
Consider, for example, triangular meshes in 2 space dimensions
Compact stencil
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Definition of secondary mesh for intercell fluxes
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Averaging operator applied on edge-base control volume gives
Initial condition: integral averages at time n niQ
j
j
j
j
nSV
V
Portion of j edge-base volume inside cell i
Portion of j edge-base volume outside cell i
Area of face j (between cells i and j)
Unit outward normal vector to of face j
Stage I
),,( HGFF
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Averaging operator applied on primary mesh gives
Initial condition: integral averages at time n+1/22/1n2/1jQ
Stage II
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Stage III: one-step conservative scheme
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Numerical flux in conservative scheme
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The Cartesian case
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Three space dimensions
0)Q(H)Q(G)Q(FQ zyxt
2/1k,j,i2/1k,j,ik,2/1j,ik,2/1j,ik,j,2/1ik,j,2/1in
k,j,i1nk,j,i HH
ztGG
ytFF
xtQQ
Task: to define numerical fluxes
Conservative one-step scheme:
2/1k,j,ik,2/1j,ik,j,2/1i H;G;F
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)Q(F)Q(Fxt
21QQ
21Q
),Q(FF
nk,j,i
nk,j,1i
nk,j,1i
nk,j,i
lwk,j,2/1i
lwj,2/1i
lwj,2/1i
Lax-Wendroff type flux
nk,j,i
nk,j,1i
nk,j,1i
nk,j,i
lfk,j,2/1i QQ
tx
21)Q(F)Q(F
21F
3
)FF(21F lf
k,j,2/1ilw
k,j,2/1iforce
k,j,2/1i
The FORCE flux in 3D is
Lax-Friedrichs type flux
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)()(21
21
)(
112/1
2/12/1
ni
ni
ni
ni
lwi
lwi
lwi
QFQFxtQQQ
QFF
ni
n1i
n1i
ni
lf2/1i QQ
tx
21)Q(F)Q(F
21F
parameter:
)FF(21F lf
2/1ilw
2/1iforce
2/1i
One-dimensional interpretation
0)Q(FQ xt
2/1i2/1ini
1ni FF
xtQQ
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One-dimensional interpretation
0)t,x(q)t,x(q xt
2/1i2/1ini
1ni ff
xtqq
The scheme is monotone and linearly stable under the condition
12c
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Numerical resultsfor the 1D Euler equations
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35.0c
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Analysis of the FORCE schemes in 1,2 and 3 space
dimensions
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Consider model hyperbolic equation
;q)q(h;q)q(g;q)q(f0)q(h)q(g)q(fq
321
zyxt
k,j,2/1ik,j,2/1ik,j,2/1ik,j,2/1ik,j,2/1ik,j,2/1in
k,j,i1nk,i,i hh
ztgg
ytff
xtqq
Monotonicity, linear stability and numerical viscosity
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Stability and monotonicity results
FORCE-type fluxes
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Part A. Summary:
I have presented a new conservative, centred monotone scheme for hyperbolic systems in three space dimensions, solved on general unstructured meshes
The flux can be used in the following schemes:
• Finite volume• Discontinuous Galerkin FE methods (ADER, RK)• Path-conservative methods (Pares et al. 2006)
The flux is the building block of high-order non- linear methods
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Part B:
High-Order Methods
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The ADER approach
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Toro E F, Millington R and Nejad L (1999). Towards Very High-Order Godunov Schemes. In Godunov Methods: Theory and
Applications, Edited Review, E F Toro (Editor), Kluwer/Plenum Academic Publishers, 2001, pp 907-940.
First written account of schemes for linear problems in 1D, 2D and 3D on structured meshes.
Further developments:
Toro, Titarev, Schwartzkopff , Munz, Dumbser, Kaeser, Takakura, Castro, Vignoli, Hidalgo, Enaux,Grosso, Russo,
Felcman,…
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High accuracy.But why ?
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CPU Time106 107 108 109 1010 101110-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
O2O3O4O5O6O15O16O23O24
iso-error
iso
-CP
UT
ime
0.8 orders
3.8
orde
rs
3.3
ord
ers
Collaborators: Munz, Schwartzkoppf (Germany), Dumbser (Trento)
Test for acoustics ADER
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t x
xixti
t
iti
x
xx
ni
i
i
i
i
dxdttxQSS
dQFF
dxxQQ
0
11
02/1
12/1
1
2/1
2/1
2/1
2/1
)),((
))((
)0,(
Exact relation between integral averages
i2/1i2/1ini
1ni tSFF
xtQQ
Exact relation
)Q(S)Q(FQ xt
],0[],[ 2/12/1 txx ii Integration in space and time
on control volume
Integral averages
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t
LRti dQFF0
12/1 ))((
Godunov’s finite volume scheme in 1D
)(QLR
2/1i2/1ini
1ni FF
xtQQ
Conservative formula
Godunov’s numerical flux
: Solution of classical Riemann problem
x
x
t
q(x,0)
qL
qR
QL QR
x=0
x=0
Classical Riemann Problem
0 0
)0,(
0)(
1 xifQxifQ
xQ
QFQ
ni
ni
xt
)/()0( txQ
))0(())0(( )0(
0
)0(12/1 QFdQFF
t
ti
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t x
xixti
t
iti
x
xx
ni
i
i
i
i
dxdttxQSS
dQFF
dxxQQ
0
11
02/1
12/1
1
2/1
2/1
2/1
2/1
)),((
))((
)0,(
Illustration of ADER finite volume method
i2/1i2/1ini
1ni tSFF
xtQQ
Update formula
)Q(S)Q(FQ xt
],0[],[ 2/12/1 txx ii
Integral average at time n
Control volume in computational domain
Numerical flux
Numerical source
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ADER on 2D unstructured meshes
The numerical flux requires the calculation of an integral in space alongThe volume/element interface and in time.
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Local Riemann problems from high-order representation of data
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Key ingredient:
the high-order (or generalized)
Riemann problem
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The high-order (or derivative, or generalized) Riemann problem:
KGRP 0)(0)(
)0,(
)()(
xifxQxifxQ
xQ
QSQFQ
R
L
xt
Initial conditions: two smooth functions
)x(Q),x(Q RL
For example, two polynomials of degree K
The generalization is twofold: (1) the intial conditions are two polynomials of arbitrary degree(2) The equations include source terms
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x
Tim
e
-10 0 100
3
6
9
12
15
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Four solvers:
C E Castro and E F Toro. Solvers for the high-order Riemann problem for hyperbolic balance laws. Journal Computational Physics Vol. 227, pp 2482-2513,, 2008
M Dumbser, C Enaux and E F Toro. Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws . Journal of Computational Physics, Vol 227, pp 3971-4001, 2008.
E F Toro and V A Titarev. Soloution of the generalized Riemann problem for advection-reaction equations. Proc. Royal Society of London, A, Vol. 458, pp 271- 281, 2002.
E F Toro and V A Titarev. Derivative Riemann solvers for systems of conservation laws and ADER methods. Journal Computational Physics Vol. 212, pp 150-165,2006
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K
1k
k)k(
tLR !k)0,0(Q)0,0(Q)(Q
The leading termand
higher-order terms
Solver 1Toro E. F. and Titarev V. A. Proc. Roy. Soc. London. Vol. 458, pp 271-281, 2002
Toro E. F. and Titarev V. A. J. Comp. Phys. Vol. 212, No. 1, pp. 150-165, 2006.
(Based on work of Ben-Artzi and Falcovitz, 1984, see also Raviart and LeFloch 1989)
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Initial conditions
q
q
q
q
q
q
q
q
q
(3)
(2)
(K)
(1)
(2)
(3)
(K)
q (1)L
L
L
L
R
R
R
R
R
L(x)
(x)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
x=0x
q(x,0)
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0x if )0(Q0x if )0(Q
)0,x(Q
0)Q(FQ
R
L
xt
)t/x(D )0(Solution:
)0(D)0,0(Q )0(Leading term:
Computing the leading term:Solve the classical RP
Take Godunov state at x/t=0
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)Q,....,Q(G)t,x(Q )k(x
)0(x
)k()k(t
Computing the higher-order terms:
First use the Cauchy-Kowalewski (*) procedure yields
q)(qq)(q
qq0qq
)m(x
m)m(t
)2(x
2)2(t
xt
xt
Example:
Must define spatial derivatives at x=0 for t>0
(*) Cauchy-Kowalewski theorem. One of the most fundamental results in the theory of PDEs. Applies to problems in which all functions involved are analytic.
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Then construct evolution equations for the variables:
)t,x(Q)k(x
)Q,...,Q,Q(H)Q()Q(A)Q( )k(x
)1(x
)0(x
)k()k(xx
)k(xt
For the general case it can be shown that:
Neglecting source terms and linearizing we have
0)Q())0,0(Q(A)Q( )k(xx
)k(xt
Computing the higher-order terms
0q)(q)(0qq xxxtxt Note:
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0x if )0(Q0x if )0(Q
)0,x(Q
0)Q())0,0(Q(A)Q(
R)k(
x
L)k(
x)k(x
)k(xx
)k(xt
)t/x(D )k(Similarity solution
)0(D)0,0(Q )k()k(x
All spatial derivatives at x=0 are now defined
Evaluate solution at x/t=0
Computation of higher-order termsFor each k solve classical Riemann problem:
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))0,0(Q),....,0,0(Q(G)0,0(Q )k(x
)0(x
)k()k(t
Computing the higher-order termsAll time derivatives at x=0 are then defined
!k)0,0(Q)0,0(Q)(Q
kK
1k
)k(tLR
Solution of DRP is
GRP-K = 1( non-linear RP) + K (linear RPs)
Options: state expansion and flux expansion
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t x
xixti
t
iti
x
xx
ni
i
i
i
i
dxdttxQSS
dQFF
dxxQQ
0
11
02/1
12/1
1
2/1
2/1
2/1
2/1
)),((
))((
)0,(
Illustration of ADER finite volume method
i2/1i2/1ini
1ni tSFF
xtQQ
Update formula
)Q(S)Q(FQ xt
],0[],[ 2/12/1 txx ii
Integral average at time n
Control volume in computational domain
Numerical flux
Numerical source
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The latest solverM Dumbser, C Enaux and E F Toro. Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. Journal of Computational Physics, Vol 227, pp 3971-4001, 2008.
Extends Harten’s method (1987)**
•Evolves data left and right prior to “time-interaction”•Evolution of data is done numerically by an implicit space- time DG method•The solution of the LOCAL generalized Riemann problem is therefore implicit•The scheme remains globally explicit•Stiff source terms can be treated adequately•Reconciles stiffness with high accuracy in both space and time
**C E Castro and E F Toro. Solvers for the high-order Riemann problem for hyperbolic balance laws. Journal Computational Physics Vol. 227, pp 2482-2513,2008
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Main features of ADER schemes
)(
)()()()(
QSQCQBQAQ
QSQHQGQFQ
zyxt
zyxt
One-step fully discrete schemesReconstruction done once per time step
Accuracy in space and time is arbitrary
Unified framework
Finite volume, DG finite element and Path-conservativeformulations
General meshes
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Part C:
APPLICATIONS
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Main applications so far
1, 2, 3 D Euler equations on unstructured meshes3D Navier-Stokes equationsReaction-diffusion (parabolic equations)Sediment transport in water flows (single phase)Two-phase sediment transport (Pitman and Le model)Two-layer shallow water equationsAeroacoustics in 2 and 3DSeismic wave propagation in 3DTsunami wave propagationMagnetohydrodynamics3D Maxwell equations3D compressible two-phase flow, etc.
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Two and three-dimensional Euler equations
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2D Euler equations on unstructured triangular meshes
Two space dimensions
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2D Euler equations: reflection from triangle
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3D Euler equations: reflection from cone
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Baer-Nunziato equations
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)()()( QSQTQFQ xxt
7
6
5
4
3
2
1
2
2
)( 0
0
)(
)ˆˆˆˆˆ(ˆˆˆˆˆˆ
ˆˆˆ0
)()(
ˆˆˆˆˆˆ
ˆˆ
sssssss
QS
VPP
VVPP
QT
pEupu
u
pEupu
u
QF
Eu
Eu
Q
ii
i
i
ii
i
The Baer-Nunziato equations in 1D
:Q Solid phase :Q̂ Gas phase
puVi ˆP i
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Application of ADER to the 3D Baer-Nunziato equations
12 PDESStiff source terms: relaxation terms
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EXTENSION TO NONCONSERVATIVE SYSTEMS:Path-conservative schemes
DUMBSER M, HIDALGO A, CASTRO M, PARES C, TORO E F. (2009).FORCE Schemes on Unstructured Meshes II: Nonconservative Hyperbolic Systems.
(Under review)
Also published (NI09005-NPA) in pre-print series of the Newton Institute for Mathematical Sciences
University of Cambridge, UK.
It can be downloaded fromhttp://www.newton.ac.uk/preprints2009.html
CASTRO M, PARDO A, PARES C, TORO E F (2009).ON SOME FAST WELL-BALANCED FIRST ORDER SOLVERS FOR
NONCONSERVATIVE SYSTEMS. MATHEMATICS OF COMPUTATION. ISSN: 0025-5718. Accepted.
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Three space dimensions
Unstructured meshes
Path-conservative method
Centred non-conservative FORCE is bluilding block
ADER: high-order of accuracy in space and time
(implemented upto 6-th order in space and time)
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Reference solutions to the BN equations
Exact Riemann solver of Schwendemann et al. (2006) (1D)
Exact smooth solution the 2D BN equations to be used in convergence rate studies (Dumbser et al. 2009)
Spherically symmetric 3D BN equations reduced to 1D system with geometric source terms. This is used to test 2 and 3 dimensional solutions with shocks (Dumbser et al. 2009)
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Convergence rates study in 2D unstructured meshes
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BN equations: spherical explosion test
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Double Mach reflection for the 2D Baer-Nunziato equations
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Double Mach reflection for the 2D Baer-Nunziato equations
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Summary:
Part A: FORCE scheme in 2 and 3D on general meshes.
Part B: ADER high-order schemes
Part C: Application to the 3D Euler and Baer- Nunziato two-phase flow equations
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Thank you
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