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On Arbitrary-Lagrangian-Eulerian One-Step WENO Schemes for Stiff Hyperbolic Balance Laws

Published online by Cambridge University Press:  03 June 2015

Michael Dumbser ,
Ariunaa Uuriintsetseg  and
Olindo Zanotti
Michael Dumbser*
Affiliation:
Laboratory of Applied Mathematics, University of Trento, I-38123 Trento, Italy
Ariunaa Uuriintsetseg
Affiliation:
Laboratory of Applied Mathematics, University of Trento, I-38123 Trento, Italy
Olindo Zanotti
Affiliation:
Laboratory of Applied Mathematics, University of Trento, I-38123 Trento, Italy
*
Corresponding author.Email:[email protected]
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Abstract

In this article we present a new family of high order accurate Arbitrary Lagrangian-Eulerian one-step WENO finite volume schemes for the solution of stiff hyperbolic balance laws. High order accuracy in space is obtained with a standard WENO reconstruction algorithm and high order in time is obtained using the local space-time discontinuous Galerkin method recently proposed in [20]. In the Lagrangian framework considered here, the local space-time DG predictor is based on a weak formulation of the governing PDE on a moving space-time element. For the space-time basis and test functions we use Lagrange interpolation polynomials defined by tensor-product Gauss-Legendre quadrature points. The moving space-time elements are mapped to a reference element using an isoparametric approach, i.e. the space-time mapping is defined by the same basis functions as the weak solution of the PDE. We show some computational examples in one space-dimension for non-stiff and for stiff balance laws, in particular for the Euler equations of compressible gas dynamics, for the resistive relativistic MHD equations, and for the relativistic radiation hydrodynamics equations. Numerical convergence results are presented for the stiff case up to sixth order of accuracy in space and time and for the non-stiff case up to eighth order of accuracy in space and time.

Keywords

65Mxx35LxxArbitrary Lagrangian-Eulerianfinite volume schememoving meshhigh order WENO reconstructionlocal space-time DG predictormoving isoparametric space-time elementsstiff relaxation source termsEuler equationsresistive relativistic MHD equationsrelativistic radiation hydrodynamics
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 2 , August 2013 , pp. 301 - 327
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Balsara, D.Total variation diminishing scheme for relativistic magnetohydrodynamics. The Astrophysical Journal Supplement Series, 132:83–101, 2001.Google Scholar
[2]Balsara, D. and Shu, C.W.Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. Journal of Computational Physics, 160:405–452, 2000.CrossRefGoogle Scholar
[3]Ben-Artzi, M. and Falcovitz, J.A second-order godunov-type scheme for compressible fluid dynamics. Journal of Computational Physics, 55:1–32, 1984.CrossRefGoogle Scholar
[4]Benson, D.J.Computational methods in lagrangian and eulerian hydrocodes. Computer Methods in Applied Mechanics and Engineering, 99:235–394, 1992.Google Scholar
[5]Bourgeade, A., Le, P.Floch, and Raviart, P.A.An asymptotic expansion for the solution of the generalized Riemann problem. Part II: application to the gas dynamics equations. Annales de l’institut Henri Poincaré (C) Analyse non linéaire, 6:437–480, 1989.Google Scholar
[6]Caramana, E.J., Burton, D.E., Shashkov, M.J., and Whalen, P.P.The construction of compatible hydrodynamics algorithms utilizing conservation of total energy. Journal of Computational Physics, 146:227–262, 1998.Google Scholar
[7]Carreé, G., Del Pino, S., Despreés, B., and Labourasse, E.A cell-centered lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension. Journal of Computational Physics, 228:5160–5183, 2009.CrossRefGoogle Scholar
[8]Casulli, V.Semi-implicit finite difference methods for the two-dimensional shallow water equations. Journal of Computational Physics, 86:56–74, 1990.Google Scholar
[9]Casulli, V. and Cheng, R.T.Semi-implicit finite difference methods for three-dimensional shallow water flow. International Journal of Numerical Methods in Fluids, 15:629–648,1992.Google Scholar
[10]Cheng, J. and Shu, C.W.A high order ENO conservative Lagrangian type scheme for the compressible Euler equations. Journal of Computational Physics, 227:1567–1596, 2007.CrossRefGoogle Scholar
[11]Cheng, J. and Shu, C.W.A cell-centered Lagrangian scheme with the preservation of symmetry and conservation properties for compressible fluid flows in two-dimensional cylindrical geometry. Journal of Computational Physics, 229:7191–7206, 2010.CrossRefGoogle Scholar
[12]Cheng, J. and Shu, C.W.Improvement on spherical symmetry in two-dimensional cylindrical coordinates for a class of control volume Lagrangian schemes. Communications in Computational Physics, 11:1144–1168,2012.Google Scholar
[13]Courant, R., Isaacson, E., and Rees, M.On the solution of nonlinear hyperbolic differential equations by finite differences. Comm. Pure Appl. Math., 5:243–255, 1952.Google Scholar
[14]Dedner, A., Kemm, F., Kroöner, D., Munz, C.D., Schnitzer, T., and Wesenberg, M.Hyperbolic divergence cleaning for the MHD equations. Journal of Computational Physics, 175:645– 673, 2002.Google Scholar
[15]Zanna, L. Del and Bucciantini, N.An efficient shock-capturing central-type scheme for multidimensional relativistic flows. I. Hydrodynamics. Astron. Astroph., 390:1177–1186,August 2002.Google Scholar
[16]Del Zanna, L., Zanotti, O., Bucciantini, N., and Londrillo, P.ECHO: a Eulerian conservative high-order scheme for general relativistic magnetohydrodynamics and magnetodynamics. Astronomy & Astrophysics, 473:11–30, October 2007.CrossRefGoogle Scholar
[17]Despreés, B. and Mazeran, C.Lagrangian gas dynamics in two-dimensions and lagrangian systems. Archive for Rational Mechanics and Analysis, 178:327–372, 2005.Google Scholar
[18]Dumbser, M.Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations. Computers & Fluids, 39:60–76, 2010.Google Scholar
[19]Dumbser, M., Balsara, D.S., Toro, E.F., and Munz, C.D.A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes. Journal of Computational Physics, 227:8209–?253, 2008.Google Scholar
[20]Dumbser, M., Enaux, C., and Toro, E.F.Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. Journal of Computational Physics, 227:3971–?001, 2008.Google Scholar
[21]Dumbser, M. and Kaäser, M.Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. Journal of Computational Physics, 221:693–723, 2007.Google Scholar
[22]Dumbser, M., Kaäser, M., Titarev, V.A, and Toro, E.F.Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. Journal of Computational Physics, 226:204–243, 2007.Google Scholar
[23]Dumbser, M. and Toro, E.F.On universal Osher-type schemes for general nonlinear hyperbolic conservation laws. Communications in Computational Physics, 10:635–671, 2011.CrossRefGoogle Scholar
[24]Dumbser, M. and Zanotti, O.Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations. Journal of Computational Physics, 228:6991–7006, 2009.Google Scholar
[25]Farris, B. D., Li, T. K., Liu, Y. T., and Shapiro, S. L.Relativistic radiation magnetohydrodynamics in dynamical spacetimes: Numerical methods and tests. Phys. Rev. D, 78(2):024023, July 2008.Google Scholar
[26]Fedkiw, R., Aslam, T., Merriman, B., and Osher, S.A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). Journal of Computational Physics, 152:457–492, 1999.Google Scholar
[27]Fedkiw, R.P., Aslam, T., and Xu, S.The Ghost Fluid method for deflagration and detonation discontinuities. Journal of Computational Physics, 154:393–427, 1999.CrossRefGoogle Scholar
[28]Ferrari, A.SPH simulation of free surface flow over a sharp-crested weir. Advances in Water Resources, 33:270–276, 2010.Google Scholar
[29]Ferrari, A., Dumbser, M., Toro, E.F., and Armanini, A.A New Stable Version of the SPH Method in Lagrangian Coordinates. Communications in Computational Physics, 4:378–404, 2008.Google Scholar
[30]Ferrari, A., Dumbser, M., Toro, E.F., and Armanini, A.A new 3D parallel SPH scheme for free surface flows. Computers & Fluids, 38:1203–1217, 2009.CrossRefGoogle Scholar
[31]Ferrari, A., Fraccarollo, L., Dumbser, M., Toro, E.F., and Armanini, A.Three-dimensional flow evolution after a dambreak. Journal of Fluid Mechanics, 663:456–477, 2010.Google Scholar
[32]Ferrari, A., Munz, C.D., and Weigand, B.A high order sharp interface method with local timestepping for compressible multiphase flows. Communications in Computational Physics, 9:205–230, 2011.Google Scholar
[33]Floch, P. Le and Raviart, P.A.An asymptotic expansion for the solution of the generalized Riemann problem. Part I: General theory. Annales de l’institut Henri Poincaré (C) Analyse non linéaire, 5:179–207, 1988.Google Scholar
[34]Giacomazzo, B. and Rezzolla, L.The exact solution of the Riemann problem in relativistic magnetohydrodynamics. Journal of Fluid Mechanics, 562:223–259, 2006.CrossRefGoogle Scholar
[35]Harten, A., Engquist, B., Osher, S., and Chakravarthy, S.Uniformly high order essentially non-oscillatory schemes, III. Journal of Computational Physics, 71:231–303, 1987.Google Scholar
[36]Hidalgo, A. and Dumbser, M.ADER schemes for nonlinear systems of stiff advectiondiffu-sionreaction equations. Journal of Scientific Computing, 48:173–189, 2011.Google Scholar
[37]Hirt, C., Amsden, A., and Cook, J.An arbitrary Lagrangian-Eulerian computing method for all flow speeds. Journal of Computational Physics, 14:227–253, 1974.CrossRefGoogle Scholar
[38]Hui, W.H.The unified coordinate system in computational fluid dynamics. Communications in Computational Physics, 2:577–610, 2007.Google Scholar
[39]Jia, Zupeng and Zhang, Shudao. A new high-order discontinuous galerkin spectral finite element method for lagrangian gas dynamics in two-dimensions. Journal of Computational Physics, 230:2496–2522, 2011.Google Scholar
[40]Jiang, G.S. and Shu, C.W.Efficient implementation of weighted ENO schemes. Journal of Computational Physics, pages 202–228, 1996.Google Scholar
[41]Komissarov, S. S.Multidimensional numerical scheme for resistive relativistic magnetohy-drodynamics. Mon. Not. Roy. Astr. Soc., 382:995–1004, December 2007.Google Scholar
[42]Lentine, M., Grétarsson, Joón Toómas, and Fedkiw, R.An unconditionally stable fully conservative semi-lagrangian method. Journal of Computational Physics, 230:2857–2879, 2011.Google Scholar
[43]Liu, W., Cheng, J., and Shu, C.W.High order conservative Lagrangian schemes with LaxWen-droff type time discretization for the compressible Euler equations. Journal of Computational Physics, 228:8872–8891, 2009.CrossRefGoogle Scholar
[44]Maire, P.H.A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes. Journal of Computational Physics, 228:2391–2425, 2009.Google Scholar
[45]Maire, P.H.A high-order one-step sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids. Computers and Fluids, 46(1):341–347, 2011.Google Scholar
[46]Maire, P.H.A unified sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids. International Journal for Numerical Methods in Fluids, 65:1281–1294, 2011.Google Scholar
[47]Maire, P.H., Abgrall, R., Breil, J., and Ovadia, J.A cell-centered Lagrangian scheme for two-dimensional compressible flow problems. SIAM Journal on Scientific Computing, 29:1781–1824, 2007.Google Scholar
[48]Maire, P.H. and Breil, J.A second-order cell-centered Lagrangian scheme for two-dimensional compressible flow problems. International Journal for Numerical Methods in Fluids, 56:1417–1423, 2007.Google Scholar
[49]Mavriplis, D. J. and Nastase, C. R.On the geometric conservation law for high order discontinuous galerkin discretizations on dynamically deforming meshes. Journal of Computational Physics, 230:4285–4300, 2011.Google Scholar
[50]Monaghan, J.J.Simulating free surface flows with SPH. Journal of Computational Physics, 110:399–406, 1994.Google Scholar
[51]Mulder, W., Osher, S., and Sethian, J.A.Computing interface motion in compressible gas dynamics. Journal of Computational Physics, 100:209–228,1992.Google Scholar
[52]Munz, C.D.On Godunov-type schemes for Lagrangian gas dynamics. SIAM Journal on Numerical Analysis, 31:17–42, 1994.Google Scholar
[53]Osher, S. and Sethian, J.A.Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics, 79:12–49, 1988.Google Scholar
[54]Palenzuela, C., Lehner, L., Reula, O., and Rezzolla, L.Beyond ideal MHD: towards a more realistic modeling of relativistic astrophysical plasmas. Mon. Not. R. Astron. Soc., 394:1727–1740, 2009.Google Scholar
[55]Peery, J.S. and Carroll, D.E.Multi-material ale methods in unstructured grids,. Computer Methods in Applied Mechanics and Engineering, 187:591–619, 2000.Google Scholar
[56]Qiu, J.M. and Shu, C.W.Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow. Journal of Computational Physics, 230:863–889, 2011.Google Scholar
[57]Roe, P.L.Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics, 43:357–372, 1981.Google Scholar
[58]Rusanov, V. V.Calculation of Interaction of Non-Steady Shock Waves with Obstacles. J. Comput. Math. Phys. USSR, 1:267–279, 1961.Google Scholar
[59]Smith, R.W.AUSM(ALE): a geometrically conservative arbitrary Lagrangian-Eulerian flux splitting scheme. Journal of Computational Physics, 150:268–286, 1999.Google Scholar
[60]Stroud, A.H.Approximate Calculation of Multiple Integrals. Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1971.Google Scholar
[61]Thorne, K. S.Relativistic radiative transfer - Moment formalisms. Mon. Not. R. Astron. Soc., 194:439–473, February 1981.Google Scholar
[62]Titarev, V.A. and Toro, E.F.ADER: Arbitrary high order Godunov approach. Journal of Scientific Computing, 17:609–618, 2002.Google Scholar
[63]Titarev, V.A. and Toro, E.F.ADER schemes for three-dimensional nonlinear hyperbolic systems. Journal of Computational Physics, 204:715–736, 2005.Google Scholar
[64]Toro, E. F. and Titarev, V. A.Derivative Riemann solvers for systems of conservation laws and ADER methods. Journal of Computational Physics, 212(1):150–165, 2006.Google Scholar
[65]Toro, E.F.Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, third edition, 2009.Google Scholar
[66]Toro, E.F. and Titarev, V. A.Solution of the generalized Riemann problem for advection-reaction equations. Proc. Roy. Soc. London, pages 271–281, 2002.Google Scholar
[67]Toro, E.F. and Titarev, V.A.ADER schemes for scalar hyperbolic conservation laws with source terms in three space dimensions. Journal of Computational Physics, 202:196–215, 2005.Google Scholar
[68]van der Vegt, J. J. W. and der Ven, H. van. Space?time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows I. general formulation. Journal of Computational Physics, 182:546?–585, 2002.CrossRefGoogle Scholar
[69]der Ven, H. van and van der Vegt, J. J. W.Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows II. efficient flux quadrature. Comput. Methods Appl. Mech. Engrg., 191:4747?–4780, 2002.Google Scholar
[70]Neumann, J. von and Richtmyer, R.D.A method for the calculation of hydrodynamics shocks. Journal of Applied Physics, 21:232–237, 1950.Google Scholar
[71]Del Zanna, L., Bucciantini, N., and Londrillo, P.An efficient shock-capturing central-type scheme for multidimensional relativistic flows II. magnetohydrodynamics. Astronomy and Astrophysics, 400:397–413, 2003.CrossRefGoogle Scholar
[72]Zanotti, O., Roedig, C., Rezzolla, L., and Zanna, L. Del. General relativistic radiation hydrodynamics of accretion flows - I. Bondi-Hoyle accretion. Mon. Not. Roy. Astr. Soc., 417:2899–2915, November 2011.Google Scholar