A Uniformly Third Order Accurate Scheme for Genuinely Nonlinear Conservation Laws

The entropy condition and the total variation boundedness of weak solutions of convex scalar conservation laws are enforced by Lip+-stability which the physical solutions satisfy. The first order Godunov and LaxFriedrichs schemes, and the second order Maxmod scheme are consistent with this Lip+-stability and are, therefore, entropy convergent. In this paper, a uniformly third order accurate scheme is introduced, which is consistent with the Lip+-stability and, hence, is entropy convergent. Error estimates, both global and local, are obtained. Numerical experiments on systems of conservation laws demonstrate excellent results and sharp resolution of shock discontinuities. A Fortran subroutine of the reconstruction procedure is spelled out in the end of the paper. ∗Courant Institute of Mathematical Sciences, 251 Mercer Street, NY, NY 10012; [email protected]. This author was supported by NSF Grant DMS-9112654 and ONR Grant #N00014-91-J-1034. †School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978 Israel; [email protected]. This author was supported by ONR Grant #N0014-91-J-1343. ‡Department of Mathematics, University of California, Los Angeles, CA 90024; [email protected]. This author was supported by ONR Grant #N00014-92-J-1890.