zbMATH — the first resource for mathematics

MPI-AMRVAC: a parallel, grid-adaptive PDE toolkit. (English) Zbl 07288716
Summary: We report on the latest additions to our open-source, block-grid adaptive framework MPI-AMRVAC, which is a general toolkit for especially hyperbolic/parabolic partial differential equations (PDEs). Applications traditionally focused on shock-dominated, magnetized plasma dynamics described by either Newtonian or special relativistic (magneto)hydrodynamics, but its versatile design easily extends to different PDE systems. Here, we demonstrate applications covering any-dimensional scalar to system PDEs, with e.g. Korteweg-de Vries solutions generalizing early findings on soliton behavior, shallow water applications in round or square pools, hydrodynamic convergence tests as well as challenging computational fluid and plasma dynamics applications. The recent addition of a parallel multigrid solver opens up new avenues where also elliptic constraints or stiff source terms play a central role. This is illustrated here by solving several multi-dimensional reaction-diffusion-type equations. We document the minimal requirements for adding a new physics module governed by any nonlinear PDE system, such that it can directly benefit from the code flexibility in combining various temporal and spatial discretization schemes. Distributed through GitHub ,MPI-AMRVAC, can be used to perform 1D, 1.5D, 2D, 2.5D or 3D simulations in Cartesian, cylindrical or spherical coordinate systems, using parallel domain-decomposition, or exploiting fully dynamic block quadtree-octree grids.
35-XX Partial differential equations
65-XX Numerical analysis
Full Text: DOI
[1] Mignone, A.; Bodo, G.; Massaglia, S.; Matsakos, T.; Tesileanu, O.; Zanni, C.; Ferrari, A., PLUTO: A numerical code for computational astrophysics, ApJ Suppl. Ser., 170, 1, 228-242 (2007), arXiv:astro-ph/0701854
[2] Mignone, A.; Zanni, C.; Tzeferacos, P.; van Straalen, B.; Colella, P.; Bodo, G., The PLUTO code for adaptive mesh computations in astrophysical fluid dynamics, ApJ Suppl. Ser., 198, 1, 7 (2012), arXiv:1110.0740
[3] Fryxell, B.; Olson, K.; Ricker, P.; Timmes, F. X.; Zingale, M.; Lamb, D. Q.; MacNeice, P.; Rosner, R.; Truran, J. W.; Tufo, H., FLASH: An adaptive mesh hydrodynamics code for modeling astrophysical thermonuclear flashes, ApJ Suppl. Ser., 131, 1, 273-334 (2000)
[4] Ziegler, U., A three-dimensional Cartesian adaptive mesh code for compressible magnetohydrodynamics, Comput. Phys. Comm., 116, 1, 65-77 (1999)
[5] Ziegler, U., The NIRVANA code: Parallel computational MHD with adaptive mesh refinement, Comput. Phys. Comm., 179, 4, 227-244 (2008) · Zbl 1197.76102
[6] Stone, J. M.; Gardiner, T. A.; Teuben, P.; Hawley, J. F.; Simon, J. B., Athena: A new code for astrophysical MHD, ApJ Suppl. Ser., 178, 1, 137-177 (2008), arXiv:0804.0402
[7] Teyssier, R., Cosmological hydrodynamics with adaptive mesh refinement. A new high resolution code called RAMSES, Astron. Astrophys., 385, 337-364 (2002), arXiv:astro-ph/0111367
[8] Xia, C.; Teunissen, J.; El Mellah, I.; Chané, E.; Keppens, R., MPI-AMRVAC 2.0 for solar and astrophysical applications, ApJ Suppl. Ser., 234, 30 (2018), arXiv:1710.06140
[9] Goedbloed, J. P.; Keppens, R.; Poedts, S., Magnetohydrodynamics of Laboratory and Astrophysical Plasmas (2019), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 1407.82002
[10] (Vande Wouwer, A.; Saucez, P.; Schiesser, W., Adaptive Method of Lines (2001), Chapman & Hall/CRC Press) · Zbl 0977.68022
[11] 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, 718-744 (2012) · Zbl 1426.76385
[12] Collins, D. C.; Xu, H.; Norman, M. L.; Li, H.; Li, S., Cosmological adaptive mesh refinement magnetohydrodynamics with enzo, ApJ Suppl. Ser., 186, 2, 308-333 (2010), arXiv:0902.2594
[13] Teunissen, J.; Keppens, R., A geometric multigrid library for quadtree/octree AMR grids coupled to MPI-AMRVAC (2019), arXiv e-prints, arXiv:1901.11370
[14] Porth, O.; Xia, C.; Hendrix, T.; Moschou, S. P.; Keppens, R., MPI-AMRVAC for solar and astrophysics, ApJ Suppl. Ser., 214, 4 (2014), arXiv:1407.2052
[15] Keppens, R.; Nool, M.; Tóth, G.; Goedbloed, J. P., Adaptive mesh refinement for conservative systems: multi-dimensional efficiency evaluation, Comput. Phys. Comm., 153, 3, 317-339 (2003), arXiv:astro-ph/0403124 · Zbl 1196.76055
[16] Leroy, M. H.J.; Keppens, R., On the influence of environmental parameters on mixing and reconnection caused by the Kelvin-Helmholtz instability at the magnetopause, Phys. Plasmas, 24, 1, 012906 (2017)
[17] Chané, E.; Saur, J.; Keppens, R.; Poedts, S., How is the Jovian main auroral emission affected by the solar wind?, J. Geophys. Res. (Space Phys.), 122, 2, 1960-1978 (2017)
[18] Ruan, W.; Xia, C.; Keppens, R., Extreme-ultraviolet and X-ray emission of turbulent solar flare loops, ApJ Lett., 877, 1, L11 (2019)
[19] Zhou, Y.-H.; Xia, C.; Keppens, R.; Fang, C.; Chen, P. F., Three-dimensional MHD simulations of solar prominence oscillations in a magnetic flux rope, Astrophys. J., 856, 2, 179 (2018), arXiv:1803.03385
[20] Xia, C.; Keppens, R.; Fang, X., Coronal rain in magnetic bipolar weak fields, Astron. Astrophys., 603, A42 (2017), arXiv:1706.01804
[21] El Mellah, I.; Sander, A. A.C.; Sundqvist, J. O.; Keppens, R., Formation of wind-captured disks in supergiant X-ray binaries. Consequences for Vela X-1 and Cygnus X-1, Astron. Astrophys., 622, A189 (2019)
[22] Tóth, G., A general code for modeling MHD flows on parallel computers: Versatile advection code, Astrophys. Lett. Commun., 34, 245 (1996)
[23] 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, 1, 82-100 (1996) · Zbl 0860.76061
[24] Keppens, R.; Tóth, G.; Botchev, M. A.; van der Ploeg, A., Implicit and semi-implicit schemes: Algorithms, Internat. J. Numer. Methods Fluids, 30, 3, 335-352 (1999) · Zbl 0951.76059
[25] Keppens, R.; Tóth, G., Nonlinear dynamics of Kelvin-Helmholtz unstable magnetized jets: Three-dimensional effects, Phys. Plasmas, 6, 5, 1461-1469 (1999), arXiv:astro-ph/9901383
[26] Keppens, R.; Tóth, G.; Westermann, R. H.J.; Goedbloed, J. P., Growth and saturation of the Kelvin-Helmholtz instability with parallel and antiparallel magnetic fields, J. Plasma Phys., 61, 1, 1-19 (1999), arXiv:astro-ph/9901166
[27] Tóth, G., The \(\nabla \cdot\) b=0 constraint in shock-capturing magnetohydrodynamics codes, J. Comput. Phys., 161, 2, 605-652 (2000) · Zbl 0980.76051
[28] Toro, E., Riemann Solvers and Numerical Methods for Fluid dynamics (1997), Springer-Verlag: Springer-Verlag Berlin
[29] Leveque, R. J., Finite Volume Methods for Hyperbolic Problems (2002), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 1010.65040
[30] van der Holst, B.; Keppens, R.; Meliani, Z., A multidimensional grid-adaptive relativistic magnetofluid code, Comput. Phys. Comm., 179, 9, 617-627 (2008), arXiv:0807.0713 · Zbl 1197.76085
[31] Rusanov, V., The calculation of the interaction of non-stationary shock waves and obstacles, USSR Comp. Math. Math. Phys., 1, 304-320 (1961)
[32] Harten, A.; Lax, P.; van Leer, B., On upstream differencing and godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25, 35-61 (1983) · Zbl 0565.65051
[33] Tóth, G., The LASY preprocessor and its application to general multidimensional codes, J. Comput. Phys., 138, 2, 981-990 (1997) · Zbl 0903.76077
[34] Lohner, R., An adaptive finite element scheme for transient problems in CFD, Comput. Methods Appl. Mech. Engrg., 61, 3, 323-338 (1987) · Zbl 0611.73079
[35] Keppens, R.; Porth, O., Scalar hyperbolic PDE simulations and coupling strategies, J. Comput. Appl. Math., 266, 87-101 (2014) · Zbl 1293.65113
[36] Zabusky, N. J.; Kruskal, M. D., Interaction of “Solitons” in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15, 6, 240-243 (1965) · Zbl 1201.35174
[37] Lee, C.-T.; Lin, J.-E.; Lee, C.-C.; Liu, M.-L., Some remarks on the stability condition of numerical scheme of the KdV-type equation, J. Math. Res., 9 (2017)
[38] Delis, A. I.; Katsaounis, T., Numerical solution of the two-dimensional shallow water equations by the application of relaxation methods, Appl. Math. Model., 29, 754-783 (2005) · Zbl 1067.76586
[39] Zoppou, C.; Roberts, S., Numerical solution of the two-dimensional unsteady dam break, Appl. Math. Model., 24, 457-475 (2000) · Zbl 1004.76064
[40] Hendrix, T.; Keppens, R.; van Marle, A. J.; Camps, P.; Baes, M.; Meliani, Z., Pinwheels in the sky, with dust: 3D modelling of the wolf-rayet 98a environment, Mon. Not. R. Astron. Soc., 460, 4, 3975-3991 (2016), arXiv:1605.09239
[41] Toro, E., Shock-Capturing Methods for Free-Surface Shallow Flows (2001), Wiley: Wiley New York · Zbl 0996.76003
[42] van der Holst, B.; Keppens, R., Hybrid block-AMR in cartesian and curvilinear coordinates: MHD applications, J. Comput. Phys., 226, 1, 925-946 (2007) · Zbl 1310.76133
[43] Koren, B., Numerical Methods for Advection-Diffusion Problems (1993), Vieweg: Vieweg Braunschweig · Zbl 0788.00033
[44] Chang, P.; Wadsley, J.; Quinn, T. R., A moving-mesh hydrodynamic solver for ChaNGa, Mon. Not. R. Astron. Soc., 471, 3, 3577-3589 (2017), arXiv:1707.05333
[45] Wadsley, J. W.; Keller, B. W.; Quinn, T. R., Gasoline2: a modern smoothed particle hydrodynamics code, Mon. Not. R. Astron. Soc., 471, 2, 2357-2369 (2017), arXiv:1707.03824
[46] Mocz, P.; Vogelsberger, M.; Sijacki, D.; Pakmor, R.; Hernquist, L., A discontinuous Galerkin method for solving the fluid and magnetohydrodynamic equations in astrophysical simulations, Mon. Not. R. Astron. Soc., 437, 1, 397-414 (2014), arXiv:1305.5536
[47] Hopkins, P. F., A new class of accurate, mesh-free hydrodynamic simulation methods, Mon. Not. R. Astron. Soc., 450, 1, 53-110 (2015), arXiv:1409.7395
[48] Gresho, P. M.; Chan, S. T., On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. II - Implementation, Internat. J. Numer. Methods Fluids, 11, 621-659 (1990) · Zbl 0712.76036
[49] Frisch, U., Turbulence (1995), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 0727.76064
[50] Balsara, D. S.; Dumbser, M., Divergence-free MHD on unstructured meshes using high order finite volume schemes based on multidimensional Riemann solvers, J. Comput. Phys., 299, 687-715 (2015) · Zbl 1351.76092
[51] Felker, K. G.; Stone, J. M., A fourth-order accurate finite volume method for ideal MHD via upwind constrained transport, J. Comput. Phys., 375, 1365-1400 (2018), arXiv:1711.07439 · Zbl 1416.76147
[52] Yang, Y.; Feng, X.-S.; Jiang, C.-W., An upwind CESE scheme for 2D and 3D MHD numerical simulation in general curvilinear coordinates, J. Comput. Phys., 371, 850-869 (2018) · Zbl 1415.76509
[53] Gardiner, T. A.; Stone, J. M., An unsplit Godunov method for ideal MHD via constrained transport in three dimensions, J. Comput. Phys., 227, 8, 4123-4141 (2008), arXiv:0712.2634 · Zbl 1317.76057
[54] Jiang, C.; Feng, X.; Zhang, J.; Zhong, D., AMR simulations of magnetohydrodynamic problems by the CESE method in curvilinear coordinates, Sol. Phys., 267, 2, 463-491 (2010)
[55] Li, S., An HLLC Riemann solver for magneto-hydrodynamics, J. Comput. Phys., 203, 1, 344-357 (2005) · Zbl 1299.76302
[56] Čada, M.; Torrilhon, M., Compact third-order limiter functions for finite volume methods, J. Comput. Phys., 228, 11, 4118-4145 (2009) · Zbl 1273.76286
[57] Kondo, S.; Miura, T., Reaction-diffusion model as a framework for understanding biological pattern formation, Science, 329, 5999, 1616-1620 (2010) · Zbl 1226.35077
[58] Turing, A. M., The chemical basis of morphogenesis, Bull. Math. Biol., 52, 1-2, 153-197 (1990)
[59] Gray, P.; Scott, S., Autocatalytic reactions in the isothermal, continuous stirred tank reactor, Chem. Eng. Sci., 38, 1, 29-43 (1983)
[60] Pearson, J. E., Complex patterns in a simple system, Science, 261, 5118, 189-192 (1993)
[61] Schnakenberg, J., Simple chemical reaction systems with limit cycle behaviour, J. Theoret. Biol., 81, 3, 389-400 (1979)
[62] Hundsdorfer, W.; Verwer, J. G., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Vol. 33 (2013), Springer Science & Business Media
[63] Ruuth, S. J., Implicit-explicit methods for reaction-diffusion problems in pattern formation, J. Math. Biol., 34, 2, 148-176 (1995) · Zbl 0835.92006
[64] Porth, O.; Komissarov, S. S.; Keppens, R., Three-dimensional magnetohydrodynamic simulations of the Crab nebula, Mon. Not. R. Astron. Soc., 438, 1, 278-306 (2014), arXiv:1310.2531
[65] Porth, O.; Komissarov, S. S.; Keppens, R., Rayleigh-Taylor instability in magnetohydrodynamic simulations of the Crab nebula, Mon. Not. R. Astron. Soc., 443, 1, 547-558 (2014), arXiv:1405.4029
[66] Meliani, Z.; Grandclément, P.; Casse, F.; Vincent, F. H.; Straub, O.; Dauvergne, F., GR-AMRVAC code applications: accretion onto compact objects, boson stars versus black holes, Classical Quantum Gravity, 33, 15, 155010 (2016) · Zbl 1346.83004
[67] Porth, O.; Olivares, H.; Mizuno, Y.; Younsi, Z.; Rezzolla, L.; Moscibrodzka, M.; Falcke, H.; Kramer, M., The black hole accretion code, Comput. Astrophys. Cosmol., 4, 1, 1 (2017), arXiv:1611.09720
[68] Porth, O.; Chatterjee, K.; Narayan, R.; Gammie, C. F.; Mizuno, Y.; Anninos, P.; Baker, J. G.; Bugli, M.; Chan, C.-k.; Davelaar, J.; Del Zanna, L.; Etienne, Z. B.; Fragile, P. C.; Kelly, B. J.; Liska, M.; Markoff, S.; McKinney, J. C.; Mishra, B.; Noble, S. C.; Olivares, H.; Prather, B.; Rezzolla, L.; Ryan, B. R.; Stone, J. M.; Tomei, N.; White, C. J.; Younsi, Z.; Akiyama, K.; Alberdi, A.; Alef, W.; Asada, K.; Azulay, R.; Baczko, A.-K.; Ball, D.; Baloković, M.; Barrett, J.; Bintley, D.; Blackburn, L.; Boland, W.; Bouman, K. L.; Bower, G. C.; Bremer, M.; Brinkerink, C. D.; Brissenden, R.; Britzen, S.; Broderick, A. E.; Broguiere, D.; Bronzwaer, T.; Byun, D.-Y.; Carlstrom, J. E.; Chael, A.; Chatterjee, S.; Chen, M.-T.; Chen, Y.; Cho, I.; Christian, P.; Conway, J. E.; Cordes, J. M.; Geoffrey; Crew, B.; Cui, Y.; De Laurentis, M.; Deane, R.; Dempsey, J.; Desvignes, G.; Doeleman, S. S.; Eatough, R. P.; Falcke, H.; Fish, V. L.; Fomalont, E.; Fraga-Encinas, R.; Freeman, B.; Friberg, P.; Fromm, C. M.; Gómez, J. L.; Galison, P.; García, R.; Gentaz, O.; Georgiev, B.; Goddi, C.; Gold, R.; Gu, M.; Gurwell, M.; Hada, K.; Hecht, M. H.; Hesper, R.; Ho, L. C.; Ho, P.; Honma, M.; Huang, C.-W. L.; Huang, L.; Hughes, D. H.; Ikeda, S.; Inoue, M.; Issaoun, S.; James, D. J.; Jannuzi, B. T.; Janssen, M.; Jeter, B.; Jiang, W.; Johnson, M. D.; Jorstad, S.; Jung, T.; Karami, M.; Karuppusamy, R.; Kawashima, T.; Keating, G. K.; Kettenis, M.; Kim, J.-Y.; Kim, J.; Kim, J.; Kino, M.; Koay, J. Y.; Patrick; Koch, M.; Koyama, S.; Kramer, M.; Kramer, C.; Krichbaum, T. P.; Kuo, C.-Y.; Lauer, T. R.; Lee, S.-S.; Li, Y.-R.; Li, Z.; Lindqvist, M.; Liu, K.; Liuzzo, E.; Lo, W.-P.; Lobanov, A. P.; Loinard, L.; Lonsdale, C.; Lu, R.-S.; MacDonald, N. R.; Mao, J.; Marrone, D. P.; Marscher, A. P.; Martí-Vidal, I.; Matsushita, S.; Matthews, L. D.; Medeiros, L.; Menten, K. M.; Mizuno, I.; Moran, J. M.; Moriyama, K.; Moscibrodzka, M.; Müller, C.; Nagai, H.; Nagar, N. M.; Nakamura, M.; Narayanan, G.; Natarajan, I.; Neri, R.; Ni, C.; Noutsos, A.; Okino, H.; Oyama, T.; Özel, F.; Palumbo, D. C.M.; Patel, N.; Pen, U.-L.; Pesce, D. W.; Piétu, V.; Plambeck, R.; PopStefanija, A.; Preciado-López, J. A.; Psaltis, D.; Pu, H.-Y.; Ramakrishnan, V.; Rao, R.; Rawlings, M. G.; Raymond, A. W.; Ripperda, B.; Roelofs, F.; Rogers, A.; Ros, E.; Rose, M.; Roshanineshat, A.; Rottmann, H.; Roy, A. L.; Ruszczyk, C.; Rygl, K. L.J.; Sánchez, S.; Sánchez-Arguelles, D.; Sasada, M.; Savolainen, T.; Schloerb, F. P.; Schuster, K.-F.; Shao, L.; Shen, Z.; Small, D.; Sohn, B. W.; SooHoo, J.; Tazaki, F.; Tiede, P.; Tilanus, R. P.J.; Titus, M.; Toma, K.; Torne, P.; Trent, T.; Trippe, S.; Tsuda, S.; van Bemmel, I.; van Langevelde, H. J.; van Rossum, D. R.; Wagner, J.; Wardle, J.; Weintroub, J.; Wex, N.; Wharton, R.; Wielgus, M.; Wong, G. N.; Wu, Q.; Young, K.; Young, A.; Yuan, F.; Yuan, Y.-F.; Zensus, J. A.; Zhao, G.; Zhao, S.-S.; Zhu, Z., The event horizon general relativistic magnetohydrodynamic code comparison project, Astrophys. J. Suppl. Ser., 243, 2, 26 (2019), arXiv:1904.04923
[69] Ripperda, B.; Bacchini, F.; Porth, O.; Most, E. R.; Olivares, H.; Nathanail, A.; Rezzolla, L.; Teunissen, J.; Keppens, R., General relativistic resistive magnetohydrodynamics with robust primitive variable recovery for accretion disk simulations (2019), arXiv e-prints, arXiv:1907.07197
[70] Olivares, H.; Porth, O.; Davelaar, J.; Most, E. R.; Fromm, C. M.; Mizuno, Y.; Younsi, Z.; Rezzolla, L., Constrained transport and adaptive mesh refinement in the black hole accretion code (2019), arXiv e-prints, arXiv:1906.10795
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.