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New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. (English) Zbl 0987.65085
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.

##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35L65 Hyperbolic conservation laws 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
HLLE
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