an:01452596
Zbl 0987.65085
Kurganov, Alexander; Tadmor, Eitan
New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations
EN
J. Comput. Phys. 160, No. 1, 241-282 (2000).
00065221
2000
j
65M06 35L65 65M20 65M15
hyperbolic conservation laws; multidimensional systems; degenerate diffusion; central difference schemes; non-oscillatory time differencing; total-variation diminishing property; semi-discrete scheme; convection-diffusion equations; Runge-Kutta methods; numerical examples
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.
Song Wang (Nedlands)