New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations (Q1979136)

The authors first review some central difference schemes and then propose a new fully discrete second order central difference scheme for systems of one dimensional hyperbolic conservation laws. This scheme has the merit that its numerical viscosity is of order \({\mathcal O}(\Delta x)^{2r-1})\) which is independent of \(1\Delta t\). (The parameter \(r\) has never been defined in the paper, but it is presumably the order of the scheme.) Furthermore, this scheme becomes semi-discrete when \(\Delta t \rightarrow 0\). In this case the scheme satisfies the scalar total-variation diminishing property. Extensions of the semi-discrete scheme to convection-diffusion equations and to multi-dimensions are also discussed. Fully discrete schemes can also be obtained by applying Runge-Kutta methods to the semi-discrete formula. A number of numerical examples is solved by the method to justify the theoretical results obtained.