Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes

Several numerical schemes for the solution of hyperbolic conservation laws are based on exploiting the information obtained by considering a sequence of Riemann problems. It is argued that in existing schemes much of this information is degraded and that only certain features of the exact solution are worth striving for. It is shown that these features can be obtained by constructing a matrix with a certain “Property U.” Matrices having this property are exhibited for the equations of steady and unsteady gasdynamics. In order to construct them, it is found helpful to introduce “parameter vectors” which notably simplify the structure of the conservation laws.