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 P. Collela [A multidimensional second order Godunov scheme for conservation laws (to appear)] and 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.