zbMATH — the first resource for mathematics

Adaptive mesh refinement for conservative systems: multi-dimensional efficiency evaluation. (English) Zbl 1196.76055
Summary: Obtainable computational efficiency is evaluated when using an Adaptive Mesh Refinement (AMR) strategy in time accurate simulations governed by sets of conservation laws. For a variety of 1D, 2D, and 3D hydro- and magnetohydrodynamic simulations, AMR is used in combination with several shock-capturing, conservative discretization schemes. Solution accuracy and execution times are compared with static grid simulations at the corresponding high resolution and time spent on AMR overhead is reported. Our examples reach corresponding efficiencies of 5 to 20 in multi-dimensional calculations and only 1.5-8% overhead is observed. For AMR calculations of multi-dimensional magnetohydrodynamic problems, several strategies for controlling the backward difference \(\dot B=0\) constraint are examined. Three source term approaches suitable for cell-centered B representations are shown to be effective. For 2D and 3D calculations where a transition to a more globally turbulent state takes place, it is advocated to use an approximate Riemann solver based discretization at the highest allowed level(s), in combination with the robust Total Variation Diminishing Lax-Friedrichs method on the coarser levels. This level-dependent use of the spatial discretization acts as a computationally efficient, hybrid scheme.

76M25 Other numerical methods (fluid mechanics) (MSC2010)
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
Full Text: DOI
[1] Almgren, A.S; Bell, J.B; Colella, P; Howell, L.H; Welcome, M.L, A conservative adaptive projection method for the variable density incompressible navier – stokes equations, J. comput. phys., 142, 1, (1998) · Zbl 0933.76055
[2] Balsara, D.S, Divergence-free adaptive mesh refinement for magnetohydrodynamics, J. comput. phys., 174, 614, (2001) · Zbl 1157.76369
[3] Balsara, D.S; Spicer, D.S, A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations, J. comput. phys., 149, 270, (1999) · Zbl 0936.76051
[4] Bell, J; Berger, M; Saltzman, J; Welcome, M, Three-dimensional adaptive mesh refinement for hyperbolic conservation laws, SIAM J. sci. comput., 15, 127, (1994) · Zbl 0793.65072
[5] Bell, J.B; Colella, P; Glaz, H.M, A second-order projection method for the incompressible navier – stokes equations, J. comput. phys., 85, 257, (1989) · Zbl 0681.76030
[6] Berger, M.J, Data structures for adaptive grid generation, SIAM J. sci. stat. comput., 7, 904, (1986) · Zbl 0625.65116
[7] Berger, M.J; Colella, P, Local adaptive mesh refinement for shock hydrodynamics, J. comput. phys., 82, 64, (1989) · Zbl 0665.76070
[8] Berger, M.J; LeVeque, R.J, Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems, SIAM J. numer. anal., 35, 2298, (1998) · Zbl 0921.65070
[9] Berger, M.J; Rigoutsos, I, An algorithm for point clustering and grid generation, IEEE trans. on systems, man cybernetics, 21, 1278, (1991)
[10] Berkeley Lab AMR homepage at http://seesar.lbl.gov/
[11] Boris, J.P; Book, D.L, Flux-corrected transport. I. SHASTA, A fluid transport algorithm that works, J. comput. phys., 11, 38, (1973) · Zbl 0251.76004
[12] Brackbill, J.U; Barnes, D.C, The effect of nonzero \(∇· B\) on the numerical solution of the magnetohydrodynamic equations, J. comput. phys., 35, 426, (1980) · Zbl 0429.76079
[13] Brio, M; Wu, C.C, An upwind differencing scheme for the equations of ideal magnetohydrodynamics, J. comput. phys., 75, 400, (1988) · Zbl 0637.76125
[14] Chiang, Y.-L; van Leer, B; Powell, K.G, Simulation of unsteady inviscid flow on an adaptively refined Cartesian grid, Aiaa, 92-0443, (1992)
[15] Dellar, P.J, A note on magnetic monopolies and the one-dimensional MHD Riemann problem, J. comput. phys., 172, 392, (2001) · Zbl 1065.35523
[16] Dedner, A; Kemm, F; Kröner, D; Munz, C.-D; Schnitzer, T; Wesenberg, M, Hyperbolic divergence cleaning for the MHD equations, J. comput. phys., 175, 645, (2002) · Zbl 1059.76040
[17] Friedel, H; Grauer, R; Marliani, C, Adaptive mesh refinement for singular current sheets in incompressible magnetohydrodynamic flows, J. comput. phys., 134, 190, (1997) · Zbl 0879.76079
[18] 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
[19] Harten, A, High resolution schemes for hyperbolic conservation laws, J. comput. phys., 49, 357, (1983) · Zbl 0565.65050
[20] Janhunen, P, A positive conservative method for magnetohydrodynamics based on HLL and roe methods, J. comput. phys., 160, 649, (2000) · Zbl 0967.76061
[21] Keppens, R; Nool, M; Zegeling, P.A; Goedbloed, J.P, Dynamic grid adaptation for computational magnetohydrodynamics, (), 61
[22] Keppens, R; Tóth, G, Simulating magnetized plasma with the versatile advection code, (), 680
[23] Keppens, R; Tóth, G, Using high performance Fortran for magnetohydrodynamic simulations, Parallel comput., 26, 705, (2000) · Zbl 0947.68572
[24] Keppens, R; Tóth, G; Botchev, M.A; van der Ploeg, A, Implicit and semi-implicit schemes: algorithms, Int. J. numer. meth. fluids, 30, 335, (1999) · Zbl 0951.76059
[25] 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, (1999)
[26] Keppens, R; Nool, M; Goedbloed, J.P, Zooming in on 3D magnetized plasmas with grid-adaptive simulations, (), 215
[27] Keppens, R; Tóth, G, parallelism for multi-dimensional grid-adaptive magnetohydrodynamic simulations, (), 940 · Zbl 1053.76526
[28] Langseth, J.O; LeVeque, R.J, A wave propagation method for three-dimensional hyperbolic conservation laws, J. comput. phys., 165, 126, (2000) · Zbl 0967.65095
[29] LeVeque, R.J, CLAWPACK 4.0 website
[30] LeVeque, R.J, Wave propagation algorithms for multi-dimensional hyperbolic systems, J. comput. phys., 131, 327, (1997) · Zbl 0872.76075
[31] MacNeice, P; Olson, K.M; Mobarry, C; de Fainchtein, R; Packer, C, PARAMESH: A parallel adaptive mesh refinement community toolkit, Comput. phys. commun., 126, 330, (2000) · Zbl 0953.65088
[32] Marder, B, A method for incorporating Gauss’ law into electromagnetic PIC codes, J. comput. phys., 68, 48, (1987) · Zbl 0603.65079
[33] Nool, M; Keppens, R, AMRVAC: a multi-dimensional grid-adaptive magnetofluid dynamics code, Comput. methods appl. math., 2, 92, (2002) · Zbl 1142.65483
[34] Orszag, A; Tang, C.M, Small-scale structure of two-dimensional magnetohydrodynamic turbulence, J. fluid mech., 90, 129, (1979)
[35] Picone, J.M; Dahlburg, R.B, Evolution of the orszag – tang vortex system in a compressible medium. II. supersonic flow, Phys. fluids B, 3, 29, (1991)
[36] K.G. Powell, An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension), ICASE Report No 94-24, Langley, VA, 1994
[37] Powell, K.G; Roe, P.L; Linde, T.J; Gombosi, T.I; De Zeeuw, D.L, A solution-adaptive upwind scheme for ideal magnetohydrodynamics, J. comput. phys., 154, 284, (1999) · Zbl 0952.76045
[38] Roe, P.L, Approximate Riemann solvers, parameter vectors, and difference schemes, J. comput. phys., 43, 357, (1981) · Zbl 0474.65066
[39] Roe, P.L; Balsara, D.S, Notes on the eigensystem of magnetohydrodynamics, SIAM J. appl. math., 56, 57, (1996) · Zbl 0845.35092
[40] Steiner, O; Knölker, M; Schüssler, M, Dynamic interaction of convection with magnetic flux sheets: first results of a new MHD code, ()
[41] Strang, G, On the construction and comparison of difference schemes, SIAM J. numer. anal., 5, 506, (1968) · Zbl 0184.38503
[42] Tóth, G, The \(∇· B=0\) constraint in shock-capturing magnetohydrodynamic codes, J. comput. phys., 161, 605, (2000) · Zbl 0980.76051
[43] Tóth, G, The LASY preprocessor and its application to general multi-dimensional codes, J. comput. phys., 138, 981, (1997) · Zbl 0903.76077
[44] Tóth, G, A general code for modeling MHD flows on parallel computers: versatile advection code, Astrophys. lett. comm., 34, 245, (1996)
[45] Tóth, G; Keppens, R; Botchev, M.A, Implicit and semi-implicit schemes in the versatile advection code: numerical tests, Astron. astrophys., 332, 1159, (1998)
[46] 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
[47] Tóth, G; Roe, P.L, Divergence- and curl-preserving prolongation and restriction formulas, J. comput. phys., 180, 736, (2002) · Zbl 1143.65322
[48] 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
[49] Walder, R; Folini, D, A-MAZE: A code package to compute 3D magnetic flows, 3D NLTE radiative transfer, and synthetic spectra, (), 281
[50] Walder, R; Folini, D; Motamen, S, Colliding winds in WR binaries: further developments within a complicated story, (), 298
[51] Woodward, P.R; Colella, P, The numerical simulation of two-dimensional fluid flow with strong shocks, J. comput. phys., 54, 115, (1984) · Zbl 0573.76057
[52] H.C. Yee, A class of high-resolution explicit and implicit shock-capturing methods, NASA TM-101088, 1989
[53] Ziegler, U, A three-dimensional Cartesian adaptive mesh code for compressible magnetohydrodynamics, Comput. phys. commun., 116, 65, (1999)
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.