On the Construction and Comparison of Difference Schemes (1968)

Previous article Next article On the Construction and Comparison of Difference SchemesGilbert StrangGilbert Stranghttps://doi.org/10.1137/0705041PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] J. Barkley Rosser, A Runge-Kutta for all seasons, SIAM Rev., 9 (1967), 417–452 10.1137/1009069 MR0219242 0243.65041 LinkISIGoogle Scholar[2] Robert D. Richtmyer and , K. W. Morton, Difference methods for initial-value problems, Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967xiv+405 MR0220455 0155.47502 Google Scholar[3] Peter D. Lax and , Burton Wendroff, Difference schemes for hyperbolic equations with high order of accuracy, Comm. Pure Appl. Math., 17 (1964), 381–398 MR0170484 0233.65050 CrossrefISIGoogle Scholar[4] Samuel Z. Burstein, Numerical methods in multidimensional shocked flows, AIAA J., 2 (1964), 2111–2117 MR0180067 CrossrefGoogle Scholar[5] S. Z. Burstein, Finite-difference calculations for hydrodynamic flows containing discontinuities, J. Comp. Phys., 2 (1967), 198–222 0166.22103 Google Scholar[6] C. Berger, On the numerical range of powers of an operator, to appear Google Scholar[7] W. P. Crowley, Second-order numerical advection, J. Comp. Phys., 1 (1967), 471–484 10.1016/0021-9991(67)90053-8 0149.44403 CrossrefGoogle Scholar[8] Gilbert Strang, Accurate partial difference methods. I. Linear Cauchy problems, Arch. Rational Mech. Anal., 12 (1963), 392–402 MR0146970 0113.32303 CrossrefISIGoogle Scholar[9] A. R. Gourlay and , J. Ll. Morris, A multistep formulation of the optimized Lax-Wendroff method for nonlinear hyperbolic systems in two space variables, Math. Comp., 22 (1968), 715–719 MR0251931 0172.19904 CrossrefISIGoogle Scholar[10] Peter Lax and , Burton Wendroff, Systems of conservation laws, Comm. Pure Appl. Math., 13 (1960), 217–237 MR0120774 0152.44802 CrossrefISIGoogle Scholar[11] R. D. Richtmyer, A survey of difference methods for nonsteady fluid dynamics, Tech. Note, 63-2, National Center for Atmospheric Research, Boulder, Colorado, 1962 Google Scholar[12] Gilbert Strang, Accurate partial difference methods. II. Non-linear problems, Numer. Math., 6 (1964), 37–46 10.1007/BF01386051 MR0166942 0143.38204 CrossrefGoogle Scholar[13] Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons Inc., New York, 1962xi+407 MR0135729 0112.34901 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Semiexplicit symplectic integrators for non-separable Hamiltonian systems23 September 2022 | Mathematics of Computation, Vol. 92, No. 339 Cross Ref A combined volume penalization / selective frequency damping approach for immersed boundary methods applied to high-order schemesJournal of Computational Physics, Vol. 472 Cross Ref Numerical computation for rogue waves in the coupled nonlinear Schrödinger equations with the coherent coupling effect29 April 2022 | International Journal of Computer Mathematics, Vol. 99, No. 12 Cross Ref A second-order maximum bound principle preserving operator splitting method for the Allen–Cahn equation with applications in multi-phase systemsMathematics and Computers in Simulation, Vol. 202 Cross Ref Ultrafast laser matter interactions: modeling approaches, challenges, and prospects27 September 2022 | Modelling and Simulation in Materials Science and Engineering, Vol. 30, No. 8 Cross Ref An efficient IMEX-DG solver for the compressible Navier-Stokes equations for non-ideal gasesJournal of Computational Physics, Vol. 471 Cross Ref Implications of CO 2 mass transport dynamics for large-scale CCS in basalt formationsInternational Journal of Greenhouse Gas Control, Vol. 121 Cross Ref A charge-preserving compact splitting method for solving the coupled stochastic nonlinear Schrödinger equationsApplied Numerical Mathematics, Vol. 181 Cross Ref Local effects in vortex-flame interactions: Implications for turbulent premixed flame scaling and observablesCombustion and Flame, Vol. 245 Cross Ref Hamiltonian preserving nonlinear opticsPhysica D: Nonlinear Phenomena, Vol. 439 Cross Ref A fully coupled, fully implicit simulation method for unsteady flames using Jacobian approximation and clusteringCombustion and Flame, Vol. 245 Cross Ref Reactive Transport Modeling with a Coupled OpenFOAM®-PHREEQC Platform7 October 2022 | Transport in Porous Media, Vol. 145, No. 2 Cross Ref Multirate Exponential Rosenbrock MethodsVu Thai Luan, Rujeko Chinomona, and Daniel R. 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Magnetic Vol. Cross Ref Local wave