A class of implicit, second-order accurate, dissipative schemes for solving systems of conservation laws

All explicit difference schemes for solving systems of conservation laws are subject to the Courant-Friedrichs-Lewy [1] convergence condition. This condition manifests itself as a restrictive condition on the size of the time steps which can be used in the schemes. Implicit schemes, on the other hand, automatically satisfy the convergence condition. However, most implicit schemes used in the past have either only been first-order accurate or second-order accurate but nondissipative (Gary [2], Zwas and Abarbanel [17]). This paper develops a class of second-order-accurate implicit schemes which are dissipative. Some numerical results are presented which show their usefulness in solving problems involving discontinuities. These results appear promising for the case of a single equation. However, there appears to be some computational difficulties in the case of systems of equations which require further investigation. © 1974.