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An unsplit, higher order Godunov method for scalar conservation laws in multiple dimensions. (English) Zbl 0684.65088
The authors develop an unsplit higher order Godunov method for scalar conservation laws in two dimensions. The method represents an extension of methods previously developed by {\it P. Collela} [A multidimensional second order Godunov scheme for conservation laws (to appear)] and {\it B. van Leer} [Computing methods in applied sciences and engineering VI, Proc. 6th Int. Symp., Versailles 1983, 493-497 (1984; Zbl 0565.65052)]. The resulting method is shown to satisfy a maximum principle for constant coefficient linear advection. Tests of the method on a variety of linear advection problems indicate that the method is more accurate than existing methods of this type. Although the improvement for the propagation of a pure discontinuity is rather modest, the improvement for smooth structure is more substantial. In particular, the method does a better job of preserving shape of the profile as it is propagated than other methods. The major difficulty with the scheme is its complexity. This renders the method costly for general application. For applications to e.g. porous media flow the computational cost is dominated by the solution of the elliptic pressure equation. For this type of equation where a conservation law is solved as a part of a larger computational tast, the complexity of the scheme does not present a problem.
Reviewer: Ph.Brenner

65M06Finite difference methods (IVP of PDE)
76S05Flows in porous media; filtration; seepage
35L65Conservation laws
Full Text: DOI
[1] Shubin, G. R.; Bell, J. B.: Comput. methods appl. Mech. eng.. 47, 47 (1984)
[2] Bell, J. B.; Shubin, G. R.: Eighth SPE symposium on reservoir simulation. SPE 13514 (February 10--13, 1985)
[3] Colella, P.: SIAM J. Sci. stat. Comput.. 6, 104 (1985)
[4] Colella, P.; Conus, P.; Sethian, J.: Some numerical methods for discontinuous flows in porous media. The mathematics of reservoir simulation, 161 (1983)
[5] P. Colella,A multidimensional second order Godunov scheme for conservation laws, to appear.
[6] Van Leer, B.: Multidimensional explicit difference schemes for hyperbolic conservation laws. Computing methods in applied sciences and engineering, VI, 493 (1984)
[7] P. Colella, Private communication, 1985.
[8] Colella, P.; Woodward, P. R.: J. comput. Phys.. 54, 174 (1984)
[9] Zalesak, Steven T.: J. comput. Phys.. 31, 335 (1979)
[10] Russell, T. F.; Wheeler, M. F.: Finite element and finite difference method for continuous flow problems. The mathematics of reservoir simulation, 35 (1983)
[11] A. Weiser and M. Wheeler, On convergence of block-centered finite difference methods for elliptic problems, to appear. · Zbl 0644.65062
[12] Russell, T. F.; Wheeler, M. F.; Chiang, C.: Proceedings, SEG/SIAM/SPE conference of mathematical and computational methods in seismic explorations and reservoir modeling. (1985)