Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. (English) Zbl 0961.65078

This paper concerns a finite difference scheme for numerical integration of the hyperbolic conservation law problems \[ U_t+\text{div } F(U)=0, \] where \(U\in {\mathbb R^m}\), \(F(U)\in {\mathbb R^d}\), and \(F\) represents the flux. The right-hand-side of this equation is transformed by introduction of the numerical flux on the space grid, and the problem is treated as a system of ordinary differential equations. To integrate this system numerically a special kind of explicite Runge-Kutta method is used. In order to insure the possibly high order of accuracy a special technique of interpolation with delimiters is used. This technique allows also to estimate (locally) the regularity of the solution and to preserve the monotonicity. The paper contains tables of coefficients of interpolation formulas. The results of numerical tests are presented.


65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
Full Text: DOI HAL


[1] Balsara, D.S., Linearized formulation of the Riemann problem for adiabatic and isothermal magnetohydrodynamics, Ap. J. supp., 116, 119, (1998)
[2] Balsara, D.S., Total variation diminishing algorithim for adiabatic and isothermal magnetohydrodynamics, Ap. J. supp., 116, 133, (1998)
[3] D. S. Balsara, TVD scheme for relativistic MHD, submitted for publication.
[4] Balsara, D.S.; Spicer, D., A staggered mesh algorithm using higher order Godunov fluxes to ensure solenoidal magnetic fields in MHD simulations, J. comput. phys., 149, 270, (1997) · Zbl 0936.76051
[5] D. S. Balsara, A. Pouquet, D. Ward-Thompson, and, R. M. Crutcher, Numerical MHD studies of turbulence and star formation, in, Interstellar Turbulence, edited by, J. Franco and A. Caraminana, Cambridge Univ. Press, Cambridge, UK, 1998.
[6] Brio, M.; Wu, C.C., An upwind differencing scheme for the equations of MHD, J. comput. phys., 75, 400, (1988) · Zbl 0637.76125
[7] Browning, G.L.; Kreiss, H.-O., Comparison of numerical methods for the calculation of two dimensional turbulence, Math. comput., 52, 369, (1989) · Zbl 0678.76048
[8] Berger, M.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J. comput. phys., 82, 64, (1990) · Zbl 0665.76070
[9] Casper, J., Finite-volume implementation of high-order essentially nonoscillatory schemes in two dimensions, Aiaa j., 30, 2829, (1992) · Zbl 0769.76043
[10] Casper, J.; Shu, C.-W.; Atkins, H., Comparison of two formulations of high-order accurate essentially nonoscillatory schemes, Aiaa j., 32, 1970, (1994) · Zbl 0827.76049
[11] S. R. Chakravarthy, and, S. Osher, Very High Order Accurate TVD Schemes, AIAA Paper # 85-0363. · Zbl 0608.65057
[12] Cockburn, B.; Shu, C.-W., The runge – kutta discontinuous Galerkin method for conservation laws. V. multidimensional systems, J. comput. phys., 141, 199, (1998) · Zbl 0920.65059
[13] Colella, P.; Woodward, P.R., The piecewise parabolic method (PPM) for gas-dynamical simulations, J. comput. phys., 54, 174, (1984) · Zbl 0531.76082
[14] R. Fedkiw, R. Aslam, T. Merriman, and, S. Osher, A non-oscillatory Eulerian approach to interfaces in multimaterial flows, submitted for publication. · Zbl 0957.76052
[15] Ghosal, S., An analysis of the numerical errors in large-eddy simulations of turbulence, J. comput. phys., 125, 187, (1996) · Zbl 0848.76043
[16] Godunov, S.K., Mat. sb., 47, 357, (1959)
[17] Harten, A., The artificial compression method for computation of shocks and contact discontinuities. I. single conservation laws, Comm. pure appl. math., 30, 611, (1977) · Zbl 0343.76023
[18] Harten, A., High resolution schemes for hyperbolic conservation laws, J. comput. phys., 49, 357, (1983) · Zbl 0565.65050
[19] Harten, A., ENO schemes with subcell resolution, J. comput. phys., 83, 148, (1989) · Zbl 0696.65078
[20] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S.J., Uniformly high order essentially non-oscillatory schemes, III, J. comput. phys., 71, 231, (1987) · Zbl 0652.65067
[21] W. D. Henshaw, H.-O. Kreiss, and, L. G. Reyna, ICASE Report #88-8, 1988.
[22] S. Jamme, F. Torres, and, J. B. Cazalbou, AIAA Paper 97-2070, 1997.
[23] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. comput. phys., 126, 202, (1996) · Zbl 0877.65065
[24] Lax, P.D., Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. pure appl. math., 7, 159, (1954) · Zbl 0055.19404
[25] Leonard, B.P.; Lock, A.P.; MacVean, M.K., The NIRVANA scheme applied to one-dimensional advection, Int. J. num. meth. heat fluid flow, 5, 341, (1995) · Zbl 0849.76063
[26] Lesieur, M.; Comte, P., Large eddy simulations of compressible turbulent flows, Cours AGARD-VKI, turbulence in compressible flows, (1997) · Zbl 0910.76062
[27] Liu, X.-D.; Osher, S.; Chan, T., Weighted essentially nonoscillatory schemes, J. comput. phys., 115, 200, (1994)
[28] Meadows, K.R.; Kumar, A.; Hussaini, M.Y., A computational study of the interaction between a vortex and a shock wave, (1989)
[29] S. P. Pao, and, M. D. Salas, AIAA Paper 81-1205, 1981.
[30] Porter, D.H.; Woodward, P.R.; Pouquet, A., Inertial range structures in decaying compressible trubulence flows, Phys. fluids, 10, 237, (1998) · Zbl 1185.76783
[31] Quirk, J., A contribution of the great Riemann solver debate, Int. J. numer. meth. fluids, 18, 555, (1994) · Zbl 0794.76061
[32] Rogerson, A.; Meiberg, E., A numerical study of the convergence properties of ENO schemes, J. sci. comput., 5, 151, (1990) · Zbl 0732.65086
[33] Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. comput. phys., 43, 357, (1981) · Zbl 0474.65066
[34] P. L. Roe, Some contributions to the modelling of discontinuous flows, in, Lect. in Appl. Math. edited by, B. Engquist, S. Osher, R. J. Somerville, Amer. Math. Soc, 1985, Vol, 22.
[35] Roe, P.R., Discrete models for the numerical analysis of time-dependent multidimensional gas dynamics, J. comput. phys., 63, 458, (1988)
[36] Roe, P.L.; Balsara, D.S., Notes on the eigensystem of magnetohydrodynamics, SIAM J. num. anal., 56, 57, (1996) · Zbl 0845.35092
[37] Ryu, D.; Jones, T., Numerical MHD in astrophysics: algorithm and tests for one-dimensional flow, Ap. j., 442, 228, (1995)
[38] Shu, C.-W.; Osher, S.J., Efficient implementation of essentially non-oscillatory shock capturing schemes, J. comput. phys., 77, 439, (1988) · Zbl 0653.65072
[39] Shu, C.-W.; Osher, S.J., Efficient implementation of essentially non-oscillatory shock capturing schemes II, J. comput. phys., 83, 32, (1989) · Zbl 0674.65061
[40] Shu, C.-W., Numerical experiments on the accuracy of ENO and modified ENO schemes, J. sci. comput., 5, 127, (1990) · Zbl 0732.65085
[41] C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, edited by, B. Cockburn, C. Johnson, C.-W. Shu, and E. Tadmor (Editor-in-Chief: A. Quarteroni, ), Springer, 1998, p, 325. [Lecture Notes in Mathematics, Vol, 1697.]
[42] Shu, C.-W.; Zang, T.A.; Erlebacher, G.; Whitaker, D.; Osher, S., High order ENO schemes applied to two- and three-dimensional compressible flow, Appl. numer. math., 9, 45, (1992) · Zbl 0741.76052
[43] Sod, G.A., A survey of finite difference methods for systems of nonlinear hyperbolic conservation laws, J. comput. phys., 27, 1, (1978) · Zbl 0387.76063
[44] Strang, G., On the construction and comparison of difference schemes, SIAM J. numer. anal., 5, 506, (1968) · Zbl 0184.38503
[45] Suresh, A.; Huynh, H.T., Accurate monotonicity preserving scheme with runge – kutta time-stepping, J. comput. phys., 136, 83, (1997) · Zbl 0886.65099
[46] Sweby, P.K., High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. num. anal., 21, 995, (1984) · Zbl 0565.65048
[47] Tadmor, E., Numerical viscosity and the entropy condition for conservative difference schemes, Math. comput., 43, 369, (1984) · Zbl 0587.65058
[48] vanLeer, B., Towards the ultimate conservative difference scheme. V. A second order sequel to Godunov’s method, J. comput. phys., 32, 101, (1979)
[49] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. comput. phys., 54, 115, (1984) · Zbl 0573.76057
[50] Yang, H., An artificial compression method for ENO schemes, the slope modification method, J. comput. phys., 89, 125, (1990) · Zbl 0705.65062
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.