A new reconstruction procedure in central schemes for hyperbolic conservation laws

Abstract

SUMMARYThis 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.

A new well-balanced non-oscillatory central scheme for the shallow water equations on rectangular meshes

“…In this paper, the numerical scheme of [8] is extended to two spatial dimensions using a uniform rectangular grid and following the methodology developed in [2] and [14]. In the next sections we develop the numerical model adapting it to the resolution of the two dimensional shallow water system (2). This involves designing a new source term treatment to verify the exact C-property.…”

“…where f [2] (û i,j ) is the second component of the flux function f (u) at x = x i and y = y j . If q 1 = q 2 = 0 and η = η * , then…”

“…It can be seen from Equation (2), that the first component of the term in square brackets in (25) (similarly in (26)) is identically zero. We have to prove that the second and third components of this term are zero.…”

“…Central schemes of Bianco et al [5], Levy et al [14], Pareschi et al [19] or Balaguer-Beser [2] are particularly useful when the eigensystem of the equations is very complex or even unknown. Balaguer and Conde [1] compare the accuracy of upwind and central schemes to solve scalar conservation laws.…”