A smoothness indicator for adaptive algorithms for hyperbolic systems. (English) Zbl 0998.65092

Summary: The formation of shock waves in solutions of hyperbolic conservation laws calls for locally adaptive numerical solution algorithms and requires a practical tool for identifying where adaption is needed. In this paper, a new smoothness indicator (SI) is used to identify “rough” solution regions and is implemented in locally adaptive algorithms. The SI is based on the weak local truncation error of the approximate solution. It was recently reported in S. Karni and A. Kurganov [Local error analysis for approximate solutions of hyperbolic conservation laws, Adv. Comput. Math. 22, No. 1, 79–99 (2005; Zbl 1127.65070)], where error analysis and convergence properties were established.
The present paper is concerned with its implementation in scheme adaption and mesh adaption algorithms. The SI provides a general framework for adaption and is not restricted to a particular discretization scheme. The implementation in this paper uses the central-upwind scheme of A. Kurganov, S. Noelle, and G. Petrova [SIAM J. Sci. Comput. 23, No. 3, 707–740 (2001; Zbl 0998.65092)].
The extension of the SI to two space dimensions is given. Numerical results in one and two space dimensions demonstrate the robustness of the proposed SI and its potential in reducing computational costs and improving the resolution of the solution.


65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
76L05 Shock waves and blast waves in fluid mechanics
76M20 Finite difference methods applied to problems in fluid mechanics
Full Text: DOI Link


[1] Aràndiga, F.; Donat, R., Nonlinear multiscale decompositions: the approach of A. harten, Numer. algorithms, 23, 175, (2000) · Zbl 0952.65015
[2] Aràndiga, F.; Donat, R.; Harten, A., Multiresolution based on weighted averages of the hat function. II. nonlinear reconstruction techniques, SIAM J. sci. comput., 20, 1053, (1999) · Zbl 0940.42020
[3] Berger, M.J.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J. comput. phys., 82, 67, (1989) · Zbl 0665.76070
[4] Berger, M.J.; Oliger, J., Adaptive mesh refinement for hyperbolic partial differential equations, J. comput. phys., 53, 482, (1984) · Zbl 0536.65071
[5] de Boor, C., A practical guide to splines, (1978), Springer-Verlag New York · Zbl 0406.41003
[6] Harabetian, E.; Pego, R., Nonconservative hybrid shock capturing schemes, J. comput. phys., 105, 1, (1993) · Zbl 0781.65076
[7] S. Karni, and, A. Kurganov, Local error analysis for approximate solutions of hyperbolic conservation laws, submitted for publication. · Zbl 1127.65070
[8] Kurganov, A.; Levy, D., A third-order semi-discrete central scheme for conservation laws and convection-diffusion equations, SIAM J. sci. comput., 22, 1461, (2000) · Zbl 0979.65077
[9] Kurganov, A.; Noelle, S.; Petrova, G., Semi-discrete central-upwind scheme for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. sci. comput., 23, 707, (2001) · Zbl 0998.65091
[10] Kurganov, A.; Petrova, G., A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems, Numer. math., 88, 683, (2001) · Zbl 0987.65090
[11] Kurganov, A.; Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. comput. phys., 160, 241, (2000) · Zbl 0987.65085
[12] A. Kurganov, and, E. Tadmor, Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers, submitted for publication. · Zbl 1058.76046
[13] 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
[14] Liu, X.-D.; Osher, S., Nonoscillatory high order accurate self similar maximum principle satisfying shock capturing schemes, I, SIAM J. numer. anal., 33, 760, (1996) · Zbl 0859.65091
[15] Liu, X.-D.; Tadmor, E., Third order nonoscillatory central scheme for hyperbolic conservation laws, Numer. math., 79, 397, (1998) · Zbl 0906.65093
[16] Minion, M.L., A projection method for locally refined grids, J. comput. phys., 127, 158, (1996) · Zbl 0859.76047
[17] Nessyahu, H.; Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws, J. comput. phys., 87, 408, (1990) · Zbl 0697.65068
[18] Nessyahu, H.; Tadmor, E., The convergence rate of approximate solutions for nonlinear scalar conservation laws, SIAM J. numer. anal., 29, 1505, (1992) · Zbl 0765.65092
[19] Nessyahu, H.; Tadmor, E.; Tassa, T., The convergence rate of Godunov type schemes, SIAM J. numer. anal., 31, 1, (1994) · Zbl 0799.65096
[20] Quirk, J.J., An adaptive mesh refinement algorithm for computational shock hydrodynamics, (1991), Cranfield Institute of Technology
[21] Schulz-Rinne, C.W., Classification of the Riemann problem for two-dimensional gas dynamics, SIAM J. math. anal., 24, 76, (1993) · Zbl 0811.35082
[22] Schulz-Rinne, C.W.; Collins, J.P.; Glaz, H.M., Numerical solution of the Riemann problem for two-dimensional gas dynamics, SIAM J. sci. comput., 14, 1394, (1993) · Zbl 0785.76050
[23] Tadmor, E., Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. numer. anal., 28, 891, (1991) · Zbl 0732.65084
[24] Tadmor, E.; Tang, T., Pointwise error estimates for scalar conservation laws with piecewise smooth solutions, SIAM J. numer. anal., 36, 1739, (1999) · Zbl 0934.35088
[25] E. F. Toro, Primitive, conservative and adaptive schemes for hyperbolic conservation laws, in, Numerical Methods for Wave Propagation, edited by, E. F. Toro and J. F. Clarke, Kluwer Academic, Dordrecht/Norwell, MA, 1998. · Zbl 0958.76062
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.