A new reconstruction procedure in central schemes for hyperbolic conservation laws

Abstract

This paper presents a new point value reconstruction algorithm based on average values or flux values for central Runge-Kutta schemes in the resolution of hyperbolic conservation laws. This reconstruction employs a fourth-order accurate approximation of point values of the solution at the two extrema and at the mid-point of each cell. These point values are modified in order to enforce monotonicity and shape preserving properties. This correction has been applied essentially in the cells close to the maxima and minima of the solution and in these cases, it has been proven that the reconstruction is fourth-order accurate. In the cells with a maximum or minimum of the solution, a correction has also been applied to such point values with the aim of ensuring that the resulting numerical solution has a non-oscillatory behavior. Several standard one- and two-dimensional test cases are used to verify high-order accuracy, non-oscillatory behavior and high-resolution properties for smooth and discontinuous solutions, and also in their componentwise extension to the Euler gas dynamics equations. © 2011 John Wiley & Sons, Ltd.