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

Central schemes may serve as universal finite-difference methods for solvin g nonlinear convection-diffusion equations in the sense that they are not t ied to the specific eigenstructure of the problem, and hence can be impleme nted in a straightforward manner as black-box solvers for general conservat ion laws and related equations governing the spontaneous evolution of large gradient phenomena. The first-order Lax-Friedrichs scheme (P. D. Lax, 1954 ) is the forerunner for such central schemes. The central Nessyahu-Tadmor ( NT) scheme (H. Nessyahu and E. Tadmor, 1990) offers higher resolution while retaining the simplicity of the Riemann-solver-free approach. The numerica l viscosity present in these central schemes is of order O((Delta x)(2r)/De lta t). In the convective regime where Delta t similar to Delta x, the impr oved resolution of the NT scheme and its generalizations is achieved by low ering the amount of numerical viscosity with increasing r. At the same time , this family of central schemes suffers from excessive numerical viscosity when a sufficiently small time step is enforced, e.g., due to the presence of degenerate diffusion terms. In this paper we introduce a new family of central schemes which retain the simplicity of being independent of the eigenstructure of the problem, yet which enjoy a much smaller numerical viscosity (of the corresponding order O((Delta x)(2r-1))). In particular, our new central schemes maintain their high-resolution independent of O(1/Delta t), and letting Delta t down arrow 0, they admit a particularly simple semi-discrete formulation. The main id ea behind the construction of these central schemes is the use of more prec ise information of the local propagation speeds. Beyond these CFL related s peeds, no characteristic information is required. As a second ingredient in their construction, these central schemes realize the (nonsmooth part of t he) approximate solution in terms of its cell averages integrated over the Riemann fans of varying size. The semi-discrete central scheme is then extended to multidimensional probl ems, with or without degenerate diffusive terms. Fully discrete versions ar e obtained with Runge-Kutta solvers. We prove that a scalar version of our high-resolution central scheme is nonoscillatory in the sense of satisfying the total-variation diminishing property in the one-dimensional case and t he maximum principle in two-space dimensions. We conclude with a series of numerical examples, considering convex and nonconvex problems with and with out degenerate diffusion, and scalar and systems of equations in one- and t wo-space dimensions. Time evolution is carried out by the: third- and fourt h-order explicit embedded integration Runge-Kutta methods recently proposed by A. Medovikov (1998). These numerical studies demonstrate the remarkable resolution of our new family of central scheme. (C) 2000 Academic Press.