Abstract: Central schemes may serve as universal finite-difference methods for solving nonlinear convection-diffusion equations in the sense that they are not tied to the specific eigen-structure of the problem, and hence can be implemented in a straightforward manner as black-box solvers for general conservation laws and related equations governing the spontaneous evolution of large gradient phenomena. The first-order Lax-Friedrichs scheme \cite{Lx}, is the forerunner for such central schemes. The central Nessyahu-Tadmor (NT) scheme \cite{NT}, offers higher resolution while retaining the simplicity of Riemann-solver-free approach. The numerical viscosity present in these central schemes is of order ${\cal O}((\Delta x)^{2r}/\Delta t)$. In the convective regime where $\Delta t \sim \Delta x$, the improved resolution of NT scheme and its generalizations is achieved by lowering 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 eigen-structure of the problem, yet they enjoy a much smaller numerical viscosity ( -- of the corresponding order ${\cal O}(\Delta x)^{2r-1})$). In particular, our new central schemes maintain their high-resolution independent of ${\cal O}(1/\Delta t)$, and letting $\Delta t\downarrow 0$, they admit a particularly simple semi-discrete formulation. The main idea behind the construction of these central schemes is the use of more precise information of the {\em local propagation speed}. Beyond these CFL related speeds, no characteristic information is required. As a second ingredient in their construction, these central scheme realize the (non-smooth part of the) approximate solution in terms of its cell averages integrated over the Riemann fans of {\em varying size}.
The semi-discrete central scheme is then extended to multidimensional problems, with or without degenerate diffusive terms. Fully discrete versions are obtained with Runge-Kutta solvers. We prove that scalar version of our high-resolution central scheme is non-oscillatory in the sense of satisfying the TVD property in the one-dimensional case and the maximum principle in two-space dimensions. We conclude with a series of numerical examples, considering convex and non-convex problems with and without degenerate diffusion, and scalar and systems of equations in one- and two-space dimensions. The time evolution is carried out by the third- and fourth-order explicit embedded integration Runge-Kutta methods recently proposed by Medovikov \cite{Med}. These numerical studies demonstrate the remarkable resolution of our new family of central scheme.
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