×

zbMATH — the first resource for mathematics

ADER schemes for scalar non-linear hyperbolic conservation laws with source terms in three-space dimensions. (English) Zbl 1061.65103
Summary: We develop non-linear ADER schemes for time-dependent scalar linear and non-linear conservation laws in one-, two- and three-space dimensions. Numerical results of schemes of up to fifth order of accuracy in both time and space illustrate that the designed order of accuracy is achieved in all space dimensions for a fixed Courant number and essentially non-oscillatory results are obtained for solutions with discontinuities. We also present preliminary results for two-dimensional non-linear systems.

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
Software:
HE-E1GODF
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Balsara, D.S.; Shu, C.W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. comput. phys, 160, 405-452, (2000) · Zbl 0961.65078
[2] Casper, J.; Atkins, H., A finite-volume high order ENO scheme for two dimensional hyperbolic systems, J. comput. phys., 106, 62-76, (1993) · Zbl 0774.65066
[3] Cockburn, B.; Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. sci. comput., 16, 173-261, (2001) · Zbl 1065.76135
[4] Colella, P., Multidimensional upwind methods for hyperbolic conservation laws, J. comput. phys., 87, 171-200, (1990) · Zbl 0694.65041
[5] Godunov, S.K., A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics, Mat. sbornik., 47, 357-393, (1959)
[6] Davies-Jones, R., Comments on ‘A kinematic analysis of frontogenesis associated with a non-divergent vortex’, J. atm. sci., 42, 2073-2075, (1985)
[7] Jiang, G.S.; Shu, C.W., Efficient implementation of weighted ENO schemes, J. comput. phys., 126, 202-212, (1996) · Zbl 0877.65065
[8] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S.R., Uniformly high order accurate essentially non-oscillatory schemes III, J. comput. phys., 71, 231-303, (1987) · Zbl 0652.65067
[9] Hu, C.; Shu, C.-W., Weighted essentially non-oscillatory schemes on triangular meshes, J. comput. phys., 150, 97-127, (1999) · Zbl 0926.65090
[10] Kolgan, V.P., Application of the minimum-derivative principle in the construction of finite-difference schemes for numerical analysis of discontinuous solutions in gas dynamics, Uchenye zapiski tsagi [sci. notes of central inst. of aerodynamics], 3, 6, 68-77, (1972), (in Russian)
[11] Kolgan, V.P., Finite-difference schemes for computation of three dimensional solutions of gas dynamics and calculation of a flow over a body under an angle of attack, Uchenye zapiski tsagi, 6, 2, 1-6, (1975), (in Russian)
[12] Liu, X.D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J. comput. phys., 115, 200-212, (1994) · Zbl 0811.65076
[13] Rusanov, V.V., Calculation of interaction of non-steady shock waves with obstacles, USSR J. comput. math. phys., 1, 267-279, (1961)
[14] Schwartzkopff, T.; Munz, C.D.; Toro, E.F., ADER-2D: a high-order approach for linear hyperbolic systems in 2D, J. sci. comput., 17, 231-240, (2002) · Zbl 1022.76034
[15] Schwartzkopff, T.; Dumbser, M.; Munz, C.D., Fast high order ADER schemes for linear hyperbolic equations, J. comput. phys., 197, 532-539, (2004) · Zbl 1052.65078
[16] Shi, J.; Hu, C.; Shu, C.-W., A technique for treating negative weights in WENO schemes, J. comput. phys., 175, 108-127, (2002) · Zbl 0992.65094
[17] Tilaeva, N.N., A generalization of the modified Godunov scheme to arbitrary unstructured meshes, Uchenye zapiski tsagi [sci. notes of central inst. of aerodynamics], 17, 18-26, (1986), (in Russian)
[18] Takakura, Y.; Toro, E.F., Arbitrarily accurate non-oscillatory schemes for a nonlinear conservation law, Cfd j., 11, N. 1, 7-18, (2002)
[19] Titarev, V.A.; Toro, E.F., ADER: arbitrary high order Godunov approach, J. sci. comput., 17, 609-618, (2002) · Zbl 1024.76028
[20] Titarev, V.A.; Toro, E.F., High order ADER schemes for the scalar advection-reaction-diffusion equations, Cfd j., 12, 1, 1-6, (2003)
[21] V.A. Titarev, E.F. Toro, Finite-volume WENO schemes for three-dimensional conservation laws, J. Comp. Physics (2004), to appear. Also Preprint NI03057-NPA. Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK, 2003, 33 pp · Zbl 1059.65078
[22] Toro, E.F., A weighted average flux method for hyperbolic conservation laws, Proc. roy. soc. London A, 423, 401-418, (1989) · Zbl 0674.76060
[23] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1999), Springer Berlin · Zbl 0923.76004
[24] Toro, E.F., Shock-capturing methods for free-surface shallow flows, (2001), Wiley New York, 314 pages · Zbl 0996.76003
[25] Toro, E.F.; Millington, R.C.; Nejad, L.A.M., Towards very high order Godunov schemes, (), 907-940 · Zbl 0989.65094
[26] Toro, E.F.; Titarev, V.A., Solution of the generalised Riemann problem for advection-reaction equations, Proc. roy. soc. London, 458, 2018, 271-281, (2002) · Zbl 1019.35061
[27] E.F. Toro, V.A. Titarev, TVD fluxes for the high-order ADER schemes, J. Sci. Comput., to appear. Also Preprint NI03011-NPA, Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK, 2003, 37 pp · Zbl 1096.76029
[28] van Leer, B., Towards the ultimate conservative difference scheme I: the quest for monotonicity, Lecture notes phys., 18, 163-168, (1973)
[29] van Leer, B., Towards the ultimate conservative difference scheme V: a second order sequel to godunov’ method, J. comput. phys., 32, 101-136, (1979) · Zbl 1364.65223
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.