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1 The ADER high-order approach for solving evolutionary PDEs Eleuterio TORO Laboratory of Applied Mathematics University of Trento, Italy www.ing.unitn.it/toro [email protected]
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The ADER high-order approach for solving evolutionary PDEs
Eleuterio TORO Laboratory of Applied Mathematics
University of Trento, Italy www.ing.unitn.it/toro
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Introduction
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I am indebted to many collaborators:
Richard Millington Mauricio Caceres
Tomas Schwarzkopff Claus-Dieter Munz
Vladimir Titarev Yoko Takakura
Michael Dumbser Martin Kaeser Cedric Enaux
Cristobal Castro Giovanni Russo
Carlos Pares Manuel Castro Arturo Hidalgo
Gianluca Vignoli Giovanna Grosso Matteo Antuono
Alberto Canestrelli Annunziato Siviglia
Gino Montecinos Lucas Mueller Junbo Cheng
.......
.......
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)Q(D)Q(S)Q(AQt +=+∂
We are interested in developing numerical methods for approximating
time-dependent partial differential equations of the form
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AIM:
solve to high (arbitrary) accuracy in space and time
non-linear systems of hyperbolic balance laws
with stiff/non-stiff source terms in multiple space dimensions on structured/unstructured meshes
in the frameworks of Finite Volume and
Discontinuous Galerkin Finite Element Methods
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Two basic design constraints
Ø Methods must be conservative (because of the Lax-Wendroff theorem, 1960)
Ø Methods must be non-linear (because of the Godunov theorem, 1959)
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The ADER approach
First results for linear equations in:
E. F. Toro, R. C. Millington and L. A. M. Nejad.
Towards Very High–Order Godunov Schemes. In Godunov Methods: Theory and Applications. Edited Review.
E. F. Toro (Editor), pages 905–937. Kluwer Academic/Plenum Publishers, 2001
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Key features of ADER:
High-order non-linear spatial reconstruction
+ high-order Riemann problem
(also called the Generalized Riemann problem)
This generalized Riemann problem has initial conditions with a high-order (spatial) representation, such as polynomials,
and source terms are included
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High accuracy.
But why ?
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Collaborators: M. Dumbser, T. Schwartzkopff, and C.-D. Munz. Arbitrary high order finite volume schemes for linear wave
propagation. Book Series Notes on Numerical Fluid mechanics and Multidisciplinary Design. Springer Berlin / Heidelberg ISSN 1612-2909, Volume 91/2006
Test for linear acoustics ADER
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ADER in 1D Finite volume formulation
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⎪⎪⎪⎪
⎭
⎪⎪⎪⎪
⎬
⎫
=
=
=
∫ ∫
∫
∫
Δ
ΔΔ
Δ
+Δ+
Δ
+
−
+
−
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,(
ττ
Exact relation between integral averages
[ ] i2/1i2/1ini
1ni tSFF
xtQQ Δ+−
ΔΔ
−= −++ Exact relation
)Q(S)Q(FQ xt =∂+∂
],0[],[ 2/12/1 txx ii Δ×+−Integration in space and time
on control volume
Integral averages
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Data reconstruction:
M. Dumbser, M. K¨aser, V. A. Titarev, and E. F. Toro. Quadrature-Free Non-Oscillatory Finite Volume Schemes on Unstructured Meshes for Nonlinear Hy- perbolic Systems. J. Comput. Phys., 226(8):204–243, 2007.
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Data variation across interface
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In 3D
The numerical flux requires the calculation of an integral in space along the volume/element interface and in time.
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A key ingredient:
the high-order (or generalized)
Riemann problem
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The high-order (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
Time
-10 0 100
3
6
9
12
15
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A solver for the generalized Rieman problem
∑=
++ ∂+=K
k
kk
tLR kQQQ
1
)(
!)0,0()0,0()( τ
τ
The leading term and
higher-order terms
Extension of work of Ben-Artzi and Falcovitz, 1984, see also Raviart and LeFloch 1989
See also the related work of Harten et al, 1987.
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),0(lim)0,0( 0 tQQ t +−+ =
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Available information at time t=0
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)t/x(D )0(Solution:
)0(D)0,0(Q )0(=+Leading term:
Computing the leading term in
Solve the classical RP
Take Godunov state at x/t=0
∑=
++
τ∂+=τ
K
1k
k)k(
tLR !k)0,0(Q)0,0(Q)(Q
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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
q)(qq)(q
qq0qq
)m(x
m)m(t
)2(x
2)2(t
xt
xt⎪⎩
⎪⎨
⎧
∂λ−=∂
∂λ−=∂
∂λ−=∂
⇒=∂λ+∂
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.
∑=
++
τ∂+=τ
K
1k
k)k(
tLR !k)0,0(Q)0,0(Q)(Q
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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…cont..
0q)(q)(0qq =∂∂+∂∂⇒=∂+∂ xxxtxt λλ
Note:
∑=
++
τ∂+=τ
K
1k
k)k(
tLR !k)0,0(Q)0,0(Q)(Q
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)/(:
0 )0(0 )0(
)0,(
0)())0,0(()()(
)(
)()(
)()(
txDsolxifQxifQ
xQ
QQAQk
Rkx
Lkxk
x
kxx
kxt
⎪⎭
⎪⎬
⎫
⎩⎨⎧
>∂
<∂=∂
=∂∂+∂∂ +
)0(D)0,0(Q )k()k(x =∂ +
All spatial derivatives at x=0 are now defined
Evaluate solution at x/t=0
For each k solve classical Riemann problem:
Computing the higher-order terms…cont..
∑=
++
τ∂+=τ
K
1k
k)k(
tLR !k)0,0(Q)0,0(Q)(Q
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))0,0(Q),....,0,0(Q(G)0,0(Q )k(x
)0(x
)k()k(t +++ ∂∂=∂
All time derivatives at x=0 are then defined
!k)0,0(Q)0,0(Q)(Q
kK
1k
)k(tLR
τ∂+=τ ∑
=++
Solution of DRP is:
GRP-K = 1( non-linear RP) + K (linear RPs)
Options: state expansion and flux expansion
Computing the higher-order terms…cont..
∑=
++
τ∂+=τ
K
1k
k)k(
tLR !k)0,0(Q)0,0(Q)(Q
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Complete ADER scheme
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ττ∫Δ
+Δ+ =t
iti dQFF0
2/11
2/1 ))((
ADER finite volume method for
[ ] i2/1i2/1ini
1ni tSFF
xtQQ Δ+−
ΔΔ
−= −++
)Q(S)Q(FQ xt =∂+∂
Numerical flux:
Numerical source:
∫ ∫Δ
ΔΔ
+
−
=t x
xixti
i
i
dxdttxQSS0
112/1
2/1
)),((
One-step scheme
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More solvers for the generalized Riemann problem:
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.
Summary of ADER schemes
)()()()( QSQHQGQFQ zyxt =∂+∂+∂+∂
one-step fully discrete schemes for
Accuracy in space and time is arbitrary
Unified framework
Finite volume, DG finite elements
General meshes
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The Cauchy-Kowalewski procedure:
A Fortran Example Code for the Cauchy-Kowalewski Procedure for the 3D Euler Equations
M. Dumbser, M. Käser, V.A. Titarev, E.F. Toro. Quadrature-free non-oscillatory finite
volume schemes on unstructured meshes for nonlinear hyperbolic systems. Journal of Computational Physics. Vol. 226, Issue 1, Pages 204-243,
10 September 2007.
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How to avoid the Cauchy-Kowalewski procedure:
Numerical evolution of data (related to Harten’s method, 1987).
See
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.
With this solver we can deal with stiff source terms
Work in progress to simplify solvers for the generalized Riemann problem with stiff source terms
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Work in progress to simplify solvers for the generalized Riemann problem with stiff source terms
Evaluation/comparison of four currently available
generalized Riemann problem solvers:
G I Montecinos, C E Castro, M Dumbser and E F Toro. Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms.
Journal of Computational Physics, 2012. Available on line. DOI: http://dx.doi.org/10.1016/j.jcp.2012.06.011
Main applications so far
1, 2, 3D Euler equations on unstructured meshes 3D compressible Navier-Stokes equations Reaction-diffusion (parabolic equations) Dispersive systems Sediment transport in water flows (single phase) Two-phase sediment transport (Pitman and Le model) Two-layer shallow water equations Aeroacoustics in 2 and 3D Seismic wave propagation in 3D Tsunami wave propagation Magnetohydrodynamics 3D Maxwell equations 3D compressible two-phase flow, etc.
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Sample numerical results
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Linear advection
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WENO-5
ADER-3
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2D and 3D Euler equations
2D Euler equations: reflection from triangular object
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3D Euler equations: reflection from cone
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2D and 3D Baer-Nunziato equations
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3D Baer-Nunziato equations for compressible two-phase flow
11 nonlinear hyperbolic PDES Stiff source terms: relaxation terms
Michael Dumbser, Arturo Hidalgo, Manuel Castro, Carlos Parés and Eleuterio F. Toro. FORCE schemes on unstructured meshes II: Non-conservative hyperbolic systems . Computer methods in Applied Science and Engineering, Vol. 199, Issues 9-12, pp 625-647, January 2010.
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3D Baer-Nunziato equations: spherical explosion
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Double Mach reflection for the 2D Baer-Nunziato equations
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Convergence rates
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2D non-linear Euler equations on unstructured meshes
M. Dumbser, M. K¨aser, V. A. Titarev, and E. F. Toro. Quadrature-Free Non-Oscillatory Finite Volume Schemes on Unstructured Meshes for Nonlinear Hy- perbolic Systems. J. Comput. Phys., 226(8):204–243, 2007.
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6.
Convergence rates
Non-linear Euler equations with very stiff source terms
M. Dumbser, C. Enaux, and E. F. Toro. Finite Volume Schemes of Very High Order of Accuracy for Stiff Hyperbolic Balance Laws. J. Comput. Phys., 227(8):3971–4001, 2008.
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Diffusion-reaction equations
E. F. Toro and A. Hidalgo. ADER Finite Volume Schemes for Diffusion–Reaction Equations. Applied Numerical Mathematics, 59:73–100, 2009.
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Convergence rates for the Baer-Nunziato equations in 2D unstructured meshes
Michael Dumbsera, , , Arturo Hidalgob, , Manuel Castroc, , Carlos Parésc, and Eleuterio F. Toroa, FORCE schemes on unstructured meshes II: Non-conservative hyperbolic systems . Computer methods in Applied Science and Engineering, Vol. 199, Issues 9-12, pp 625-647, January 2010.
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Summary and Concluding remarks
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Ø Schemes of arbitrary accuracy in space and time for solving time-dependent PDEs (eg hyperbolic balance laws with stiff source terms) on unstructured meshes have been presented
Ø Non-linear reconstruction + generalized Riemann problem
Ø One-step, fully discrete, conservative and non-linear
Ø Unified frame, all orders in single scheme
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Schemes are well established in two important scientific communities:
Acoustics
Seismology
Important advances in:
tsunami wave propagation
and astrophysics
Current work: further simplification of algorithms
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Introduction to ADER approach in chapter 19 and 20:
Eleuterio Toro. Riemann solvers and numerical methods for fluid dynamics.
A practical introduction. Third edition. Springer-Verlag, Berlin Heidelberg, 2009.
Book (724 pages). ISBN 978-3-540-25202- 3, 2009.
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Thank you
