A Modified High-Resolution Non-staggered Central Scheme with Adjustable Numerical Dissipation

Abstract

In this article, we present a low numerical dissipation non-staggered central scheme for the Riemann problems in hyperbolic conservation laws and convection–diffusion equations. The main disadvantage of classical non-staggered central schemes is their relatively large numerical dissipation, which will cause numerical solutions to be excessively smooth at discontinuities and large gradients, especially when small time steps are forced by CFL numbers. The presented scheme reduces the numerical dissipation through introducing a time–space dependent parameter $\alpha _{j+\frac{1}{2}}^n$ defined by restriction functions to control the information of previous time level in numerical solutions. The proof of Total Variation Diminishing (TVD) of the presented scheme indicates that the smaller the value of $\alpha _{j+\frac{1}{2}}^n$ is, the narrower the CFL limitation is and the lower the dissipation of solutions is. Motivated by the indication, three restriction functions about the local propagation speed $a\big (u_{j+\frac{1}{2}}^n\big )$ are given, so that $\alpha _{j+\frac{1}{2}}^n$ approaches 0 at the large gradients and 1 in the flat regions. The restriction functions ensure that the presented scheme not only achieves high accuracy, but also retains stability. For ensuring the overall second order accuracy of the presented scheme, the second-order central Runge–Kutta method is utilized for the time discretization. In addition, we supplement the TVD proof of the backward projection in non-staggered central schemes, which is not considered in Jiang et al. (SIAM J Numer Anal 35(6):2147–2168, 1998). Finally, we conclude this article with numerical examples to show the remarkable resolution of the presented scheme at shock waves, contact discontinuities and large gradients.