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Scalar hyperbolic PDE simulations and coupling strategies. (English) Zbl 1293.65113
Summary: We report on grid-adaptive, multi-dimensional simulations for hyperbolic PDEs, with a deliberate focus on the analytically tractable scalar case. Motivated by recent efforts towards multi-physics simulation strategies, we investigate a variety of coupling strategies for numerically solving hyperbolic partial differential equations (PDEs). We use adaptive mesh refinement in combination with shock-capturing spatio-temporal discretizations, and first present accuracy and validation tests on both smooth and shock-dominated evolutions. To investigate the feasibility of coupling means for multi-physics simulations, we then introduce new reference tests where spatially different flux prescriptions require coupling strategies across the domains where local advection, Burgers or nonconvex behavior is imposed. For these nonlinear single scalar equations, we can illustrate and analytically explain the evolutions obtained when handling cases where fluxes differ in spatially non-overlapping regions. We discuss both conservative and non-conservative ways of coupling across the region boundaries. When coupling scalar conservation laws where the characteristic speeds change discontinuously across the regional boundaries, these two strategies yield differing, but mathematically sound, solution behaviors. Their relevance for multi-physics simulations where one couples systems of (hyperbolic) PDEs is discussed.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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[1] Goedbloed, J. P.; Poedts, S., Principles of MHD. with application to laboratory and astrophysical plasmas, (2004), Cambridge University Press Cambridge
[2] Goedbloed, J. P.; Keppens, R.; Poedts, S., (Advanced MHD. With Application to Laboratory and Astrophysical Plasmas, (2010), Cambridge University Press Cambridge)
[3] Leveque, R. J., The dynamics of pressureless dust clouds and delta waves, J. Hyperbolic Differ. Equ., 1, 315, (2004) · Zbl 1079.76074
[4] Goodrich, C. C.; Sussman, A. L.; Lyon, J. G.; Shay, M. A.; Cassak, P. A., The CISM code coupling strategy, J. Atmos. Solar-Terr. Phys., 66, 1469, (2004)
[5] Lee, Y. J.; Sussman, A., Efficient communication between parallel programs with intercomm, university of maryland, technical report CS-TR-4557 and UMIACS-TR-2004-04, (2004)
[6] Brown, D. L.; Chesshire, G. S.; Henshaw, W. D.; Quinlan, D. J., Overture: an object oriented software system for solving partial differential equations in serial and parallel environments, (Eight SIAM Conference on Parallel Processing for Scientific Computing, Minneapolis, Minnesota, March 14-17 1997, (1997), SIAM Philadelphia, PA UCRL-JC-132017)
[7] Berger, M. J.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys., 82, 64, (1989) · Zbl 0665.76070
[8] B. Boutin, F. Coquel, Ph.G. LeFloch, Coupling techniques for nonlinear hyperbolic equations, IV. Multi-component coupling and multidimensional well-balanced schemes, 1 Jun 2012. arXiv:1206.0248v1 [math.AP]. · Zbl 1310.65097
[9] Leveque, R. J., Finite volume methods for hyperbolic problems, (2002), Cambridge University Press Cambridge · Zbl 1010.65040
[10] Gottlieb, S.; Shu, C.-W., Total variation diminishing Runge-Kutta schemes, Math. Comp., 67, 221, 73, (1998) · Zbl 0897.65058
[11] Balsara, D. S., Second-order acurate schemes for magnetohydrodynamics with divergence-free reconstruction, Astrophys. J. Suppl. Ser., 151, 149, (2004)
[12] Tóth, G.; Odstrčil, D., Comparison of some flux corrected transport and total variation diminishing numerical schemes for hydrodynamic and magnetohydrodynamic problems, J. Comput. Phys., 128, 82, (1996) · Zbl 0860.76061
[13] Rusanov, V. V., The calculation of the interaction of non-stationary shock waves and obstacles, USSR Comput. Math. Math. Phys., 1, 304, (1961)
[14] van Leer, B., Towards the ultimate conservative difference scheme, V. A second-order sequel to godunov’s method, J. Comput. Phys., 32, 101, (1979) · Zbl 1364.65223
[15] Koren, B., A robust upwind discretization method for advection, diffusion and source terms, (Vreugdenhil, C. B.; Koren, B., Numerical Methods for Advection-Diffusion Problems, Notes on Numerical Fluid Mechanics, vol. 45, (1993), Vieweg Braunschweig), 117 · Zbl 0805.76051
[16] Cada, M.; Torrilhon, M., Compact third-order limiter functions for finite volume methods, J. Comput. Phys., 228, 4118, (2009) · Zbl 1273.76286
[17] Keppens, R.; Meliani, Z.; van Marle, A. J.; Delmont, P.; Vlasis, A.; van der Holst, B., Parallel, grid-adaptive approaches for relativistic hydro and magnetohydrodynamics, J. Comput. Phys., 231, 718, (2012) · Zbl 1426.76385
[18] Löhner, R., An adaptive finite element scheme for transient problems in CFD, Comput. Methods Appl. Mech. Engrg, 61, 323, (1987) · Zbl 0611.73079
[19] Keppens, R.; Nool, M.; Tóth, G.; Goedbloed, J. P., Adaptive mesh refinement for conservative systems: multi-dimensional efficiency evaluation, Comput. Phys. Comm., 153, 317, (2003) · Zbl 1196.76055
[20] van der Holst, B.; Keppens, R., Hybrid block-AMR in Cartesian and curvilinear coordinates: MHD applications, J. Comput. Phys., 226, 925, (2007) · Zbl 1310.76133
[21] Tóth, G., A general code for modeling MHD flows on parallel computers: versatile advection code, Astrophys. Lett. Commun., 34, 245, (1996), See http://www.phys.uu.nl/ toth
[22] Garcia, A. L.; Bell, J. B.; Crutchfield, W. Y.; Alder, B. J., Adaptive mesh and algorithm refinement using direct simulation Monte Carlo, J. Comput. Phys., 154, 134, (1999) · Zbl 0954.76075
[23] Tóth, G., Space weather modeling framework: a new tool for the space science community, J. Geophys. Res., 110, A12226, (2005)
[24] Innocenti, M. E.; Lapenta, G.; Markidis, S.; Beck, A.; Vapirev, A., A multi level multi domain method for particle in cell plasma simulations, J. Comput. Phys., 238, 115, (2013)
[25] Efendiev, Y.; Ginting, V.; Hou, T.; Ewing, R., Accurate multiscale finite element methods for two-phase flow simulations, J. Comput. Phys., 220, 155, (2006) · Zbl 1158.76349
[26] Durlofsky, L. J.; Efendiev, Y.; Ginting, V., An adaptive local-global multiscale finite volume element method for two-phase flow simulations, Adv. Water Res., 30, 576, (2007)
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