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High order accurate conservative remapping scheme on polygonal meshes using a posteriori MOOD limiting. (English) Zbl 1390.65101
Summary: In this article, we present a high order accurate 2D conservative remapping method for a general polygonal mesh. This method conservatively projects piecewise constant data from an old mesh onto a possibly uncorrelated new one. First an arbitrary (high) accuracy polynomial reconstruction operator is built. Then, the exact intersection between the old and new mesh is constructed, leading to a submesh of old subcells paving any new cell. Last, a high order accurate quadrature rule is designed to integrate the arbitrary high order accurate polynomials on the subcells to get a final new piecewise constant cell average on any new cell. The technique is limited a posteriori in such a way that effective high accuracy is maintained on smooth solutions while an essentially non-oscillatory behavior is observed on an irregular solution. Moreover, intrinsic physical properties of the system of variables such as positivity can be ensured. Numerical results assess that such a method is effective on problems for scalar remapping (smooth and discontinuous). We have also considered the remapping of multiple coupled quantities, such as mass, momentum and energy. The results confirm that this remap method is efficient in situations containing irregular/discontinuous profiles and smooth parts, for which some physical admissible constraints such as positivity must be ensured.

MSC:
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
Software:
ReALE; r3d; SLIC
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[1] Barth, T.; Jespersen, C., The design and application of upwind schemes on unstructured meshes, AIAA Pap, 89, 89-0366, 1-12, (1989)
[2] Benson, D. J., An efficient, accurate, simple ALE method for nonlinear finite element programs, Comput Methods Appl Mech Eng, 72, 305-350, (1989) · Zbl 0675.73037
[3] Benson, D., Volume of fluid interface reconstruction methods for multi-material problems, Appl Mech Rev, 55, 2, 151-165, (2002)
[4] Boris, J.; Book, D., Flux-corrected transport, J Comput Phys, 135, 2, 172-186, (1997) · Zbl 0939.76074
[5] Boscheri, W.; Loubère, R.; Dumbser, M., Direct arbitrary-Lagrangian-Eulerian ADER-MOOD finite volume schemes for multidimensional hyperbolic conservation laws, J Comput Phys, 292, 0, 56-87, (2015) · Zbl 1349.76312
[6] Clain, S.; Diot, S.; Loubère, R., A high-order finite volume method for hyperbolic systems: multi-dimensional optimal order detection (MOOD), J Comput Phys, 230, 10, 4028-4050, (2011) · Zbl 1218.65091
[7] Cockburn, B.; Johnson, C.; Shu, C.; Tadmor, E., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, (Quarteroni, A., Advanced numerical approximation of nonlinear hyperbolic equations, Lecture notes in mathematics, vol. 1697, (1998), Springer), 325-432 · Zbl 0927.65111
[8] Després, B.; Loubère, R., Convergence and sensitivity analysis of repair algorithms in 1D, Int J Finite Vols, 3, 1, (2006)
[9] Diot, S.; Clain, S.; Loubère, R., Improved detection criteria for the multi-dimensional optimal order detection (MOOD) on unstructured meshes with very high-order polynomials, Comput Fluids, 64, 43-63, (2012) · Zbl 1365.76149
[10] Diot, S.; Loubère, R.; Clain, S., The multidimensional optimal order detection method in the three-dimensional case: very high-order finite volume method for hyperbolic systems, Int J Numer Methods Fluids, 73, 4, 362-392, (2013)
[11] Du, Q.; Faber, V.; Gunzburger, M., Centroidal Voronoi tessellations: applications and algorithms, SIAM Rev, 41, 637-676, (1999) · Zbl 0983.65021
[12] Dukowicz, J.; Baumgardner, J., Incremental remapping as a transport/advection algorithm, J Comput Phys, 160, 318-335, (2000) · Zbl 0972.76079
[13] Dukowicz, J.; Kodis, J., Accurate conservative remapping (rezoning) for arbitrary-Lagrangian-Eulerian computations, SIAM J Sci Stat Comput, 8, 305-321, (1987) · Zbl 0644.76085
[14] Dumbser, M., Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations, Comput Fluids, 39, 1, 60-76, (2010) · Zbl 1242.76161
[15] Dumbser, M.; Balsara, D.; Toro, E.; Munz, C., A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes, J Comput Phys, 227, 8209-8253, (2008) · Zbl 1147.65075
[16] Dumbser, M.; Zanotti, O.; Loubère, R.; Diot, S., A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws, J Comput Phys, 278, 47-75, (2014) · Zbl 1349.65448
[17] Dyadechko, V.; Shashkov, M., Reconstruction of multi-material interfaces from moment data, J Comput Phys, 227, 11, 5361-5384, (2008) · Zbl 1220.76048
[18] Farrell, P.; Piggott, M.; Pain, C.; Gorman, G.; Wilson, C., Conservative interpolation between unstructured meshes via supermesh construction, Comput Methods Appl Mech Eng, 198, 2632-2642, (2009) · Zbl 1228.76105
[19] Friedrich, O., Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids, J Comput Phys, 144, 194-212, (1998) · Zbl 1392.76048
[20] Galera, S.; Maire, P.-H.; Breil, J., A two-dimensional unstructured cell-centered multi-material ALE scheme using VOF interface reconstruction, J Comput Phys, 229, 5755-5787, (2010) · Zbl 1346.76105
[21] Grandy, J., Conservative remapping and region overlays by intersecting arbitrary polyhedra, J Comput Phys, 148, 2, 133-466, (1999) · Zbl 0932.76073
[22] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order accurate essentially non-oscillatory schemes III, J Comput Phys, 71, 231-303, (1987) · Zbl 0652.65067
[23] Harten, A.; Osher, S., Uniformly high-order accurate nonoscillatory schemes I, SIAM J Numer Anal, 24, 279-309, (1987) · Zbl 0627.65102
[24] Hirt, C.; Amsden, A.; Cook, J., An arbitrary Lagrangian-Eulerian computing method for all flow speeds, J Comput Phys, 14, (1974) · Zbl 0292.76018
[25] Hirt, C.; Nichols, B., Volume of fluid (VOF) method for the dynamics of free boundaries, J Comput Phys, 39, 201-225, (1981) · Zbl 0462.76020
[26] Hoch P.. An Arbitrary Lagrangian-Eulerian strategy to solve compressible fluid flows. HAL, http://hal.archives-ouvertes.fr/hal-00366858. Private communication; 2009.
[27] Huang, Y.; Quin, H.; Wang, D., Centroidal Voronoi tessellation-based finite element superconvergence, Int J Numer Methods Eng, 76, 1819-1839, (2008) · Zbl 1195.65171
[28] Jiang, G.; Shu, C., Efficient implementation of weighted ENO schemes, J Comput Phys, 126, 202-228, (1996) · Zbl 0877.65065
[29] Kucharík, M.; Breil, J.; Galera, S.; Maire, P.-H.; Berndt, M.; Shashkov, M., Hybrid remap for multi-material ALE, Comput Fluids, 46, 1, 293-297, (2011) · Zbl 1433.76133
[30] Kucharík, M.; Garimella, R.; Schofield, S.; Shashkov, M., A comparative study of interface reconstruction methods for multi-material ALE simulations, J Comput Phys, 229, 2432-2452, (2009) · Zbl 1423.76343
[31] Kucharík, M.; Shashkov, M., Extension of efficient, swept-integration-based conservative remapping method for meshes with changing connectivity, Int J Numer Methods Fluids, 56, 8, 1359-1365, (2007) · Zbl 1384.65018
[32] Kucharík, M.; Shashkov, M., One-step hybrid remapping algorithm for multi-material arbitrary Lagrangian-Eulerian methods, J Comput Phys, 231, 7, 2851-2864, (2012) · Zbl 1323.74108
[33] Kuzmin, D., On the design of general-purpose flux limiters for finite element schemes. I. scalar convection, J Comput Phys, 219, 513-531, (2006) · Zbl 1189.76342
[34] Kuzmin, D., A vertex-based hierarchical slope limiter for p-adaptive discontinuous Galerkin methods, J Comput Appl Math, 233, 12, 3077-3085, (2010) · Zbl 1252.76045
[35] Kuzmin, D., Hierarchical slope limiting in explicit and implicit discontinuous Galerkin methods, J Comput Phys, 257, Part B, 0, 1140-1162, (2014) · Zbl 1352.65355
[36] (Kuzmin, D.; Löhner, R.; Turek, S., Flux-corrected transport. principles, algorithms and applications, (2005), Springer)
[37] Liska, R.; Shashkov, M.; Váchal, P.; Wendroff, B., Optimization-based synchronized flux-corrected conservative interpolation (remapping) of mass and momentum for arbitrary Lagrangian-Eulerian methods., J Comput Phys, 229, 5, 1467-1497, (2010) · Zbl 1329.76269
[38] Liu, X.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J Comput Phys, 115, 200-212, (1994) · Zbl 0811.65076
[39] Lloyd, S. P., Least squares quantization in PCM, IEEE Trans Inf Theor, 28, 2, 129-137, (1982) · Zbl 0504.94015
[40] Loubère, R.; Dumbser, M.; Diot, S., A new family of high order unstructured MOOD and ADER finite volume schemes for multidimensional systems of hyperbolic conservation laws, Commun Comput Phys, 16, 3, 718-763, (2014) · Zbl 1373.76137
[41] Loubère, R.; Maire, P.-H.; Shashkov, M., Reale: a reconnection arbitrary-Lagrangian-Eulerian method in cylindrical geometry, Comput Fluids, 46, 59-69, (2011) · Zbl 1305.76066
[42] Loubère, R.; Maire, P.-H.; Shashkov, M.; Breil, J.; Galera, S., Reale: a reconnection-based arbitrary-Lagrangian-Eulerian method, J Comput Phys, 229, 4724-4761, (2010) · Zbl 1305.76067
[43] Loubère, R.; Staley, M.; Wendroff, B., The repair paradigm: new algorithms and applications to compressible flow, J Comput Phys, 1, 2, 385-404, (2006) · Zbl 1138.76406
[44] Murman, S. M.; Berger, M.; Aftosmis, M. J., Analysis of slope limiters on irregular grids, Technical Report NAS-05-007, (2005), NAS
[45] M. Kucharik, M. S.; Wendroff, B., An efficient linearity-and-bound-preserving remapping method, J Comput Phys, 188, 2, 462-471, (2003) · Zbl 1022.65009
[46] Margolin, L., Introduction to “an arbitrary Lagrangian-Eulerian computing method for all flow speeds”, J Comput Phys, 135, 198-202, (1997) · Zbl 0938.76067
[47] Margolin, L.; Shashkov, M., Second-order sign-preserving conservative interpolation (remapping) on general grids, J Comput Phys, 184, 1, 266-298, (2003) · Zbl 1016.65004
[48] Margolin, L.; Shashkov, M., Remapping, recovery and repair on staggered grid, Comput Methods Appl Mech Eng, 193, 4139-4155, (2004) · Zbl 1068.76058
[49] Menon, S.; Schmidt, D., Conservative interpolation on unstructured polyhedral meshes: an extension of the supermesh approach to cell-centered finite-volume variables, Comput Methods Appl Mech Eng, 200, 41-44, 2797-2804, (2011) · Zbl 1230.76034
[50] Noh, W.; Woodward, P., SLIC (simple line interface calculation), 5th International conference on numerical methods in fluid dynamics, (1976) · Zbl 0382.76084
[51] Okabe, A.; Boots, B.; Sugihara, K.; Chiu, S., Spatial tessellations: concepts and applications of Voronoi diagrams, (2000), John Wiley & Sons Chichester · Zbl 0946.68144
[52] Ollivier-Gooch, C., High-order ENO schemes for unstructured meshes based on least-squares reconstruction, Tech. Rep, (1996), Argonne National Laboratory
[53] Powell, D.; Abel, T., An exact general remeshing scheme applied to physically conservative voxelization, J Comput Phys, 297, 340-356, (2015) · Zbl 1349.65084
[54] Qiu, J.; Shu, C., Finite difference WENO schemes with Lax-Wendroff type time discretization, SIAM J Sci Comput, 24, 6, 2185-2198, (2003) · Zbl 1034.65073
[55] Rider, W.; Margolin, L. G., Simple modifications of monotonicity-preserving limiter, J Comput Phys, 174, 1, 473-488, (2001) · Zbl 1009.76067
[56] Schofield, S.; Garimella, R.; Francois, M.; Loubère, R., A second-order accurate material-order-independent interface reconstruction technique for multi-material flow simulations, J Comput Phys, 228, 731-745, (2009) · Zbl 1169.76047
[57] Sedov, L. I., Similarity and dimensional methods in mechanics, (1959), Academic Press New York · Zbl 0121.18504
[58] Shashkov, M.; Wendroff, B., The repair paradigm and application to conservation laws, J Comput Phys, 198, 265-277, (2004) · Zbl 1107.65341
[59] Shashkov, M.; Solovjov, A., Numerical simulation of two-dimensional flows by the free-Lagrangian flow simulations, Report TUM-M9105, (1991), Mathematisches Institut, Technische Universität, München
[60] Sugihara, K., A robust and consistent algorithm for intersecting convex polyhedra, Comput Graph Forum, 3, 45-54, (1994)
[61] Váchal, P.; Liska, R.; Shashkov, M.; Wendroff, B., Synchronized flux-corrected remapping for ALE methods, Comput Fluids, 46, 1, 312-317, (2011) · Zbl 1433.76135
[62] van Leer, B., Towards the ultimate conservative difference scheme. III - upstream-centered finite-difference schemes for ideal compressible flow. IV - A new approach to numerical convection, J Comput Phys, 23, 263-299, (1977) · Zbl 0339.76039
[63] van Leer, B., Towards the ultimate conservative difference scheme, J Comput Phys, 32, 101-136, (1979) · Zbl 1364.65223
[64] Velechovský, J.; Kucharík, M.; Liska, R.; Shashkov, M.; Váchal, P., Symmetry- and essentially-bound-preserving flux-corrected remapping of momentum in staggered ALE hydrodynamics, J Comput Phys, 255, 590-611, (2013) · Zbl 1349.76402
[65] Venkatakrishnan, V., Convergence to steady state solutions of the Euler equations on unstructured grids with limiters, J Comput Phys, 118, 120-130, (1995) · Zbl 0858.76058
[66] Youngs, D., An interface tracking method for a 3D Eulerian hydrodynamics code, Technical Report AWE/44/92/35, (1984), Atomic Weapon Research Establishment, Aldermaston, Berkshire, UK
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