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Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes

Published online by Cambridge University Press:  03 June 2015

Walter Boscheri  and
Michael Dumbser
Walter Boscheri*
Affiliation:
Laboratory of Applied Mathematics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, I-38123 Trento, Italy
Michael Dumbser*
Affiliation:
Laboratory of Applied Mathematics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, I-38123 Trento, Italy
*
Email:[email protected]
Corresponding author.Email:[email protected]
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Abstract

In this article we present a new class of high order accurate Arbitrary-Eulerian-Lagrangian (ALE) one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimensional unstructured triangular meshes. A WENO reconstruction algorithm is used to achieve high order accuracy in space and a high order one-step time discretization is achieved by using the local space-time Galerkin predictor proposed in. For that purpose, a new element-local weak formulation of the governing PDE is adopted on moving space-time elements. The space-time basis and test functions are obtained considering Lagrange interpolation polynomials passing through a predefined set of nodes. Moreover, a polynomial mapping defined by the same local space-time basis functions as the weak solution of the PDE is used to map the moving physical space-time element onto a space-time reference element. To maintain algorithmic simplicity, the final ALE one-step finite volume scheme uses moving triangular meshes with straight edges. This is possible in the ALE framework, which allows a local mesh velocity that is different from the local fluid velocity. We present numerical convergence rates for the schemes presented in this paper up to sixth order of accuracy in space and time and show some classical numerical test problems for the two-dimensional Euler equations of compressible gas dynamics.

Keywords

65Mxx35LxxArbitrary Lagrangian-Eulerianhigh order reconstructionWENOfinite volumelocal space-time Galerkin predictormoving unstructured meshesEuler equations
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 5 , November 2013 , pp. 1174 - 1206
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Abgrall, R.On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. Journal of Computational Physics, 144:45–58, 1994.CrossRefGoogle Scholar
[2]Aboiyar, T., Georgoulis, E. H., and Iske, A.Adaptive ADER Methods Using KernelBased Polyharmonic Spline WENO Reconstruction. SIAM Journal on Scientific Computing, 32:3251–3277, 2010.Google Scholar
[3]Balsara, D. S. and Shu, C. W.Monotonicity perserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys., 160:405–452, 2000.Google Scholar
[4]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
[5]Ben-Artzi, M. and Falcovitz, J.A second-order godunov-type scheme for compressible fluid dynamics. Journal of Computational Physics, 55:1–32, 1984.Google Scholar
[6]Benson, D. J.Computational methods in Lagrangian and Eulerian hydrocodes. Computer Methods in Applied Mechanics and Engineering, 99:235–394, 1992.Google Scholar
[7]Boscheri, W., Dumbser, M., and Righetti, M.A semi-implicit scheme for 3d free surface flows with high order velocity reconstruction on unstructured voronoi meshes. International Journal for Numerical Methods in Fluids. DOI: 10.1002/fld.3753, in press.Google Scholar
[8]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.CrossRefGoogle Scholar
[9]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.CrossRefGoogle Scholar
[10]Carré, G., Del Pino, S., Després, 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.Google Scholar
[11]Casulli, V.Semi-implicit finite difference methods for the two-dimensional shallow water equations. Journal of Computational Physics, 86:56–74, 1990.Google Scholar
[12]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
[13]Cesenek, J., Feistauer, M., Horacek, J., Kucera, V., and Prokopova, J.Simulation of compressible viscous flow in time-dependent domains. Applied Mathematics and Computation. DOI: 10.1016/j.amc.2011.08.077, in press.Google Scholar
[14]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.Google Scholar
[15]Cheng, J. and Shu, C. W.A cell-centered Lagrangian scheme with the preservationof symmetry and conservation properties for compressible fluid flows in two-dimensional cylindrical geometry. Journal of Computational Physics, 229:7191–7206, 2010.Google Scholar
[16]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
[17]Clain, S., Diot, S., and Loubere, R.A high-order finite volume method for systems of conservation laws - Multi-dimensional Optimal Order Detection (MOOD). Journal of Computational Physics, 230:4028–4050, 2011.Google Scholar
[18]Claisse, A., Després, B., Labourasse, E., and Ledoux, F.A new exceptional points method with application to cell-centered Lagrangian schemes and curved meshes. Journal of Computational Physics, 231:4324–4354, 2012.Google Scholar
[19]Cockburn, B., Karniadakis, G. E., and Shu, C. W.Discontinuous Galerkin Methods. Lecture Notes in Computational Science and Engineering. Springer, 2000.Google Scholar
[20]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
[21]Després, B. and Mazeran, C.Symmetrization of Lagrangian gas dynamic in dimension two and multimdimensional solvers. C.R. Mecanique, 331:475–480, 2003.Google Scholar
[22]Després, B. and Mazeran, C.Lagrangian gas dynamics in two-dimensions and Lagrangian systems. Archive for Rational Mechanics and Analysis, 178:327–372, 2005.CrossRefGoogle Scholar
[23]Dubcova, L., Feistauer, M., Horacek, J., and Svacek, P.Numerical simulation of interaction between turbulent flow and a vibrating airfoil. Computing and Visualization in Science, 12:207–225, 2009.Google Scholar
[24]Dubiner, M.Spectral methods on triangles and other domains. Journal of Scientific Computing, 6:345–390, 1991.Google Scholar
[25]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
[26]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
[27]Dumbser, M. and Kaeser, M.Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. Journal of Computational Physics, 221:693–723, 2007.CrossRefGoogle Scholar
[28]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
[29]Dumbser, M. and Toro, E. F.On universal oshertype schemes for general nonlinear hyperbolic conservation laws. Communications in Computational Physics, 10:635–671, 2011.Google Scholar
[30]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
[31]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
[32]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.CrossRefGoogle Scholar
[33]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.Google Scholar
[34]Feistauer, M., Horacek, J., Ruzicka, M., and Svacek, P.Numerical analysis of flow-induced nonlinear vibrations of an airfoil with three degrees of freedom. Computers and Fluids, 49:110–127, 2011.Google Scholar
[35]Feistauer, M., Kucera, V., Prokopova, J., and Horacek, J.The ALE discontinuous Galerkin method for the simulatio of air flow through pulsating human vocal folds. AIP Conference Proceedings, 1281:83–86, 2010.Google Scholar
[36]Ferrari, A.SPH simulation of free surface flow over a sharp-crested weir. Advances in Water Resources, 33:270–276, 2010.Google Scholar
[37]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
[38]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.Google Scholar
[39]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.CrossRefGoogle Scholar
[40]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
[41]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
[42]Healy, R. W. and Russel, T. F.Solution of the advection-dispersion equation in two dimensions by a finite-volume Eulerian-Lagrangian localized adjoint method. Advances in Water Resources, 21:11–26, 1998.Google Scholar
[43]Hidalgo, A. and Dumbser, M.ADER schemes for nonlinear systems of stiff advectiondiffusionreaction equations. Journal of Scientific Computing, 48:173–189, 2011.Google Scholar
[44]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.Google Scholar
[45]Hu, C. and Shu, C. W.A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics. Journal of Computational Physics, 150:561–594, 1999.Google Scholar
[46]Huang, C. S., Arbogast, T., and Qiu, J.An Eulerian-Lagrangian WENO finite volume scheme for advection problems. Journal of Computational Physics, 231:4028–4052, 2012.Google Scholar
[47]Jiang, G. S. and Shu, C. W.Efficient implementation of weighted eno schemes. Journal of Computational Physics, pages 202–228, 1996.Google Scholar
[48]Karniadakis, G. E. and Sherwin, S. J.Spectral/hp Element Methods in CFD. Oxford University Press, 1999.Google Scholar
[49]Kaäser, M. and Iske, A.Ader schemes on adaptive triangular meshes for scalar conservation laws. Journal of Computational Physics, 205:486–508, 2005.Google Scholar
[50]Kidder, R. E.Laser-driven compression of hollow shells: power requirements and stability limitations. Nucl. Fus., 1:3–14, 1976.Google Scholar
[51]Kurganov, A. and Tadmor, E.Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers. Numerical Methods for Partial Differential Equations, 18:584–608, 2002.Google Scholar
[52]Lentine, M., Greétarsson, J. T., and Fedkiw, R.An unconditionally stable fully conservative semi-Lagrangian method. Journal of Computational Physics, 230:2857–2879, 2011.Google Scholar
[53]Liu, W., Cheng, J., and Shu, C. W.High order conservative Lagrangian schemes with Lax-Rendroff type time discretization for the compressible Euler equations. Journal of Computational Physics, 228:8872–8891, 2009.Google Scholar
[54]Maire, P.-H.A high-order one-step sub-cell force-based discretization for cell-centered La-grangian hydrodynamics on polygonal grids. Computers and Fluids, 46(1):341–347, 2011.Google Scholar
[55]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.CrossRefGoogle Scholar
[56]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
[57]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
[58]Monaghan, J. J.Simulating free surface flows with SPH. Journal of Computational Physics, 110:399–406, 1994.Google Scholar
[59]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
[60]Munz, C. D.On Godunov-type schemes for Lagrangian gas dynamics. SIAM Journal on Numerical Analysis, 31:17–42, 1994.Google Scholar
[61]Olliver-Gooch, C. and Van, M.Altena. A high-order-accurate unstructured mesh finite-volume scheme for the advectiondiffusion equation. Journal of Computational Physics, 181:729–752, 2002.Google Scholar
[62]Ortega, A. Loópez and Scovazzi, G.A geometrically-conservative, synchronized, flux-corrected remap for arbitrary Lagrangian-Euleriancomputations with nodal finite elements. Journal of Computational Physics, 230:6709–6741,2011.Google Scholar
[63]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
[64]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
[65]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
[66]Riemslagh, K., Vierendeels, J., and Dick, E.An arbitrary Lagrangian-Eulerian finite-volume method for the simulation of rotary displaecment pump flow. Applied Numerical Mathematics, 32:419–433, 2000.Google Scholar
[67]Scovazzi, G.Lagrangianshock hydrodynamicson tetrahedral meshes: A stable and accurate variational multiscale approach. Journal of Computational Physics, 231:8029–8069, 2012.Google Scholar
[68]Smith, R. W.AUSM(ALE): a geometrically conservative arbitrary Lagrangian-Eulerian flux splitting scheme. Journal of Computational Physics, 150:268–286, 1999.Google Scholar
[69]Stroud, A. H.Approximate Calculation of Multiple Integrals. Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1971.Google Scholar
[70]Titarev, V. A. and Toro, E. F.ADER: Arbitrary high order Godunov approach. Journal of Scientific Computing, 17:609–618, 2002.Google Scholar
[71]Titarev, V. A. and Toro, E. F.ADER schemes for three-dimensional nonlinear hyperbolic systems. Isaac Newton Institute for Mathematical Sciences Preprint Series, 2003.Google Scholar
[72]Titarev, V. A., Tsoutsanis, P., and Drikakis, D.WENO schemes for mixed-element unstructured meshes. Communications in Computational Physics, 8:585–609, 2010.Google Scholar
[73]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
[74]Toro, E. F.Anomalies of conservative methods: analysis, numerical evidence and possible cures. International Journal of Computational Fluid Dynamics, 11:128–143, 2002.Google Scholar
[75]Toro, E. F.Riemann Solvers and Numerical Methods for Fluid Dynamics: a Practical Introduction. Springer, 2009.Google Scholar
[76]Toro, E. F., Spruce, M., and Speares, W.Restoration of the contact surface in the Harten-Lax-van Leer Riemann solver. Journal of Shock Waves, 4:25–34, 1994.Google Scholar
[77]Tsoutsanis, P., Titarev, V. A., and Drikakis, D.WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions. Journal of Computational Physics, 230:1585–1601, 2011.Google Scholar
[78]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.Google Scholar
[79]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
[80]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
[81]Zhang, Y. T. and Shu, C. W.Third order WENO scheme on three dimensional tetrahedral meshes. Communications in Computational Physics, 5:836–848, 2009.Google Scholar