Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-28T01:22:53.008Z Has data issue: false hasContentIssue false
  • English
  • Français

High Order Accurate Direct Arbitrary-Lagrangian-Eulerian ADER-MOOD Finite Volume Schemes for Non-Conservative Hyperbolic Systems with Stiff Source Terms

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  05 December 2016

Walter Boscheri  and
Raphaël Loubère
Walter Boscheri*
Affiliation:
Laboratory of Applied Mathematics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, I-38123 Trento, Italy
Raphaël Loubère*
Affiliation:
CNRS and Institut de Mathèmatiques de Toulouse (IMT) Université Paul-Sabatier, Toulouse, France
*
*Corresponding author. Email addresses: [email protected] (W. Boscheri), [email protected] (R. Loubère)
*Corresponding author. Email addresses: [email protected] (W. Boscheri), [email protected] (R. Loubère)
Get access

Abstract

In this paper we present a 2D/3D high order accurate finite volume scheme in the context of direct Arbitrary-Lagrangian-Eulerian algorithms for general hyperbolic systems of partial differential equations with non-conservative products and stiff source terms. This scheme is constructed with a single stencil polynomial reconstruction operator, a one-step space-time ADER integration which is suitably designed for dealing even with stiff sources, a nodal solver with relaxation to determine the mesh motion, a path-conservative integration technique for the treatment of non-conservative products and an a posteriori stabilization procedure derived from the so-called Multidimensional Optimal Order Detection (MOOD) paradigm. In this work we consider the seven equation Baer-Nunziato model of compressible multi-phase flows as a representative model involving non-conservative products as well as relaxation source terms which are allowed to become stiff. The new scheme is validated against a set of test cases on 2D/3D unstructured moving meshes on parallel machines and the high order of accuracy achieved by the method is demonstrated by performing a numerical convergence study. Classical Riemann problems and explosion problems with exact solutions are simulated in 2D and 3D. The overall numerical code is also profiled to provide an estimate of the computational cost required by each component of the whole algorithm.

Keywords

Direct Arbitrary-Lagrangian-Euleriana posteriori MOOD stabilizationBaer-Nunziato modelstiff source termsnon-conservative productsunstructured meshADERhigh order of accuracy in space and timehigh performance computing (HPC)hyperbolic conservation laws

MSC classification

Secondary: 65M08: Finite volume methods 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 1 , January 2017 , pp. 271 - 312
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Andrianov, N. and Warnecke, G.. The riemann problem for the baer-nunziato two-phase flow model. Journal of Computational Physics, 212:434–464, 2004.CrossRefGoogle Scholar
[2] Baer, M.R. and Nunziato, J.W.. A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. J. Multiphase Flow, 12:861–889, 1986.Google Scholar
[3] Balsara, D.. Second-order accurate schemes for magnetohydrodynamics with divergence-free reconstruction. The Astrophysical Journal Supplement Series, 151:149–184, 2004.Google Scholar
[4] 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.Google Scholar
[5] Balsara, D.S.. Self-adjusting, positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics. Journal of Computational Physics, 231:75047517, 2011.Google Scholar
[6] Barth, T.J. and Frederickson, P.O.. Higher order solution of the euler equations on unstructured grids using quadratic reconstruction. 28th Aerospace Sciences Meeting, pages AIAA paper no. 90–0013, January 1990.Google Scholar
[7] Barth, T.J. and Jespersen, D.C.. The design and application of upwind schemes on unstructured meshes. AIAA Paper 89-0366, pages 1–12, 1989.Google Scholar
[8] Boscheri, W., Balsara, D.S., and Dumbser, M.. Lagrangian ADER-WENO Finite Volume Schemes on Unstructured Triangular Meshes Based On Genuinely Multidimensional HLL Riemann Solvers. Journal of Computational Physics, 267:112–138, 2014.Google Scholar
[9] Boscheri, W. and Dumbser, M.. Arbitrary–Lagrangian–Eulerian One–Step WENO Finite Volume Schemes on Unstructured Triangular Meshes. Communications in Computational Physics, 14:1174–1206, 2013.Google Scholar
[10] Boscheri, W. and Dumbser, M.. A direct Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume scheme on unstructured tetrahedral meshes for conservative and non-conservative hyperbolic systems in 3D. Journal of Computational Physics, 275(0):484–523, 2014.CrossRefGoogle Scholar
[11] Boscheri, W. and Dumbser, M.. An Efficient Quadrature-Free Formulation for High Order Arbitrary-Lagrangian-Eulerian ADER-WENO Finite Volume Schemes on Unstructured Meshes. Journal of Scientific Computing, 66:240–274, 2016.Google Scholar
[12] Boscheri, W., Dumbser, M., and Balsara, D.S.. High Order Lagrangian ADER-WENO Schemes on Unstructured Meshes – Application of Several Node Solvers to Hydrodynamics and Magnetohydrodynamics. International Journal for Numerical Methods in Fluids, 76:737–778, 2014.CrossRefGoogle Scholar
[13] Boscheri, W., Loubère, R., and Dumbser, M.. Direct Arbitrary-Lagrangian-Eulerian ADER-MOOD finite volume schemes for multidimensional hyperbolic conservation laws. Journal of Computational Physics, 292:56–87, 2015.Google Scholar
[14] Castro, M.J., Gallardo, J.M., López, J.A., and Parés, C.. Well-balanced high order extensions of godunov's method for semilinear balance laws. SIAM Journal of Numerical Analysis, 46:1012–1039, 2008.CrossRefGoogle Scholar
[15] Castro, M.J., Gallardo, J.M., and Parés, C.. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. applications to shallow-water systems. Mathematics of Computation, 75:1103–1134, 2006.Google Scholar
[16] Clain, S., Diot, S., and Loubère, R.. A high-order finite volume method for systems of conservation lawsmulti-dimensional optimal order detection (MOOD). Journal of Computational Physics, 230(10):4028–4050, 2011.Google Scholar
[17] Clain, S., Diot, S., and Loubère, R.. Multi-dimensional optimal order detection (mood) a very high-order finite volume scheme for conservation laws on unstructured meshes. In Fort Fürst Halama Herbin Hubert (Eds.), editor, FVCA 6, International Symposium, Prague, June 6-10, volume 4 of Series: Springer Proceedings in Mathematics, 2011. 1st Edition. XVII, 1065 p. 106 illus. in color.Google Scholar
[18] Clain, S. and Machado, G.. A very high-order finite volume method for the one-dimensional time-dependent convection-diffusion problem. Computers and Mathematics with Applications, 2014. in press.Google Scholar
[19] Clain, Stéphane, Machado, Gaspar J, Nóbrega, JM, and Pereira, RMS. A sixth-order finite volume method for multidomain convection–diffusion problem with discontinuous coefficients. Computer Methods in Applied Mechanics and Engineering, 267:43–64, 2013.Google Scholar
[20] Cockburn, B., Karniadakis, G. E., and Shu, C.W.. Discontinuous Galerkin Methods. Lecture Notes in Computational Science and Engineering. Springer, 2000.Google Scholar
[21] Colella, P. and Sekora, M.D.. A limiter for PPM that preserves accuracy at smooth extrema. Journal of Computational Physics, 227:7069–7076, 2008.Google Scholar
[22] Deledicque, V. and Papalexandris, M.V.. An exact riemann solver for compressible two-phase flow models containing non-conservative products. Journal of Computational Physics, 222:217–245, 2007.Google Scholar
[23] B. Després and Mazeran, C.. Lagrangian gas dynamics in two-dimensions and Lagrangian systems. Archive for Rational Mechanics and Analysis, 178:327–372, 2005.Google Scholar
[24] Diot, S., Clain, S., and Loubère, R.. Improved detection criteria for the multi-dimensional optimal order detection (MOOD) on unstructured meshes with very high-order polynomials. Computers and Fluids, 64:43–63, 2012.CrossRefGoogle Scholar
[25] Diot, S., Loubère, R., and Clain, S.. The MOOD method in the three-dimensional case: Very-high-order finite volume method for hyperbolic systems. International Journal of Numerical Methods in Fluids, 73:362–392, 2013.CrossRefGoogle Scholar
[26] Dubiner, M.. Spectral methods on triangles and other domains. Journal of Scientific Computing, 6:345–390, 1991.Google Scholar
[27] 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 on unstructured meshes. Journal of Computational Physics, 227:8209–8253, 2008.Google Scholar
[28] Dumbser, M. and Boscheri, W.. High-order unstructured Lagrangian one–step WENO finite volume schemes for non–conservative hyperbolic systems: Applications to compressible multi–phase flows. Computers and Fluids, 86:405–432, 2013.Google Scholar
[29] Dumbser, M., Castro, M., Parés, C., and Toro, E.F.. ADER schemes on unstructured meshes for non-conservative hyperbolic systems: Applications to geophysical flows. Computers and Fluids, 38:1731–1748, 2009.CrossRefGoogle Scholar
[30] 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, 227:3971–4001, 2008.Google Scholar
[31] Dumbser, M., Hidalgo, A., Castro, M., Parés, C., and Toro, E.F.. FORCE schemes on unstructured meshes II: Non–conservative hyperbolic systems. Computer Methods in Applied Mechanics and Engineering, 199:625–647, 2010.Google Scholar
[32] Dumbser, M., Kaeser, 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
[33] Dumbser, M. and Käser, 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
[34] Dumbser, M. and Toro, E. F.. A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems. Journal of Scientific Computing, 48:70–88, 2011.Google Scholar
[35] Dumbser, M., Uuriintsetseg, A., and Zanotti, O.. On Arbitrary–Lagrangian–Eulerian One–Step WENO Schemes for Stiff Hyperbolic Balance Laws. Communications in Computational Physics, 14:301–327, 2013.Google Scholar
[36] 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
[37] Galera, S., Maire, P.H., and Breil, J.. A two-dimensional unstructured cell-centered multi-material ale scheme using vof interface reconstruction. Journal of Computational Physics, 229:5755–5787, 2010.CrossRefGoogle Scholar
[38] Hidalgo, A. and Dumbser, M.. ADER schemes for nonlinear systems of stiff advection-diffusion-reaction equations. Journal of Scientific Computing, 48:173–189, 2011.Google Scholar
[39] Hu, C. and Shu, C.W.. Weighted essentially non-oscillatory schemes on triangular meshes. Journal of Computational Physics, 150:97–127, 1999.Google Scholar
[40] Jiang, G.-S. and Shu, C.W.. Efficient implementation of weighted ENO schemes. Journal of Computational Physics, 126:202–228, 1996.Google Scholar
[41] Kapila, A.K., Menikoff, R., Bdzil, J.B., Son, S.F., and Stewart, D.S.. Two-phase modelling of DDT in granular materials: reduced equations. Physics of Fluids, 13:3002–3024, 2001.Google Scholar
[42] Karniadakis, G. E. and Sherwin, S. J.. Spectral/hp Element Methods in CFD. Oxford University Press, 1999.Google Scholar
[43] Käser, M. and Iske, A.. ADER schemes on adaptive triangular meshes for scalar conservation laws. Journal of Computational Physics, 205:486–508, 2005.Google Scholar
[44] Knupp, P.M.. Achieving finite element mesh quality via optimization of the jacobian matrix normand associated quantities. part ii – a framework for volume mesh optimization and the condition number of the jacobian matrix. Int. J. Numer. Meth. Engng., 48:1165–1185, 2000.Google Scholar
[45] Loubère, R., Dumbser, M., and Diot, S.. A new family of high order unstructured mood and ader finite volume schemes for multidimensional systems of hyperbolic conservation laws. Communication in Computational Physics, 16:718–763, 2014.Google Scholar
[46] Loubère, Raphaël, Maire, Pierre-Henri, Shashkov, Mikhail, Breil, Jérôme, and Galera, Stéphane. Reale: A reconnection-based Arbitrary-Lagrangian-Eulerian method. J. Comput. Phys., 229(12):4724–4761, 2010.Google Scholar
[47] 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
[48] Maire, P.H. and Nkonga, B.. Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics. Journal of Computational Physics, 228:799–821, 2009.Google Scholar
[49] Olliver-Gooch, C. and Van Altena, M.. A high-order–accurate unstructured mesh finite–volume scheme for the advection–diffusion equation. Journal of Computational Physics, 181:729–752, 2002.Google Scholar
[50] Parés, C.. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM Journal on Numerical Analysis, 44:300–321, 2006.Google Scholar
[51] Rhebergen, S., Bokhove, O., and van der Vegt, J.J.W.. Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations. Journal of Computational Physics, 227:1887–1922, 2008.Google Scholar
[52] Costa, Stéphane Clain Ricardo, Machado, Gaspar J.. Sixth-order finite volume method for the 1d biharmonic operator: application to the intramedullary nail simulation. International Journal of Applied Mathematics and Computer Science (AMCS), 2014.Google Scholar
[53] Saurel, R. and Abgrall, R.. A multiphase godunov method for compressible multifluid and multiphase flows. Journal of Computational Physics, 150:425–467, 1999.Google Scholar
[54] Saurel, R., Gavrilyuk, S., and Renaud, F.. A multiphase model with internal degrees of freedom: Application to shock-bubble interaction. Journal of Fluid Mechanics, 495:283–321, 2003.Google Scholar
[55] Schwendeman, D.W., Wahle, C.W., and Kapila, A.K.. The riemann problem and a high-resolution godunov method for a model of compressible two-phase flow. Journal of Computational Physics, 212:490–526, 2006.CrossRefGoogle Scholar
[56] Stroud, A.H.. Approximate Calculation of Multiple Integrals. Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1971.Google Scholar
[57] Suresh, A. and Huynh, H.T.. Accurate monotonicity-preserving schemes with runge-kutta time stepping. Journal of Computational Physics, 136:83–99, 1997.Google Scholar
[58] Toro, E.F.. Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, second edition, 1999.Google Scholar
[59] Toumi, I.. A weak formulation of roes approximate riemann solver. Journal of Computational Physics, 102:360–373, 1992.Google Scholar
[60] Desveaux, V.. Contribution à l’approximation numérique des systèmes hyperboliques. PhD thesis, Université de Nantes, 2013.Google Scholar
[61] Desveaux, V. and Berthon, C.. An entropic mood scheme for the euler equations. International Journal of Finite Volumes, 2013.Google Scholar
[62] Winslow, Alan M.. Numerical solution of the quasilinear poisson equation in a nonuniform triangle mesh. J. Comput. Phys., 135(2):128–138, August 1997.Google Scholar
[63] Woodward, P. and Colella, P.. The numerical simulation of two-dimensional fluid flow with strong shocks. Journal of Computational Physics, 54:115–173, 1984.Google Scholar
[64] Zanotti, O., Dumbser, M., Loubère, R., and Diot, S.. A posteriori subcell limiting for discontinuous galerkin finite element method for hyperbolic system of conservation laws. J. Comput. Phys., 278:47–75, 2014.Google Scholar
[65] Zanotti, O., Fambri, F., and Dumbser, M.. Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement. Mon. Not. R. Astron. Soc., 452:3010–3029, 2015.CrossRefGoogle Scholar
[66] Zanotti, Olindo, Fambri, Francesco, Dumbser, Michael, and Hidalgo, Arturo. Space-time adaptive ader discontinuous galerkin finite element schemes with a posteriori sub-cell finite volume limiting. Computers & Fluids, 118(0):204–224, 2015.Google Scholar