Central Runge--Kutta Schemes for Conservation Laws

Abstract

In this work, a new formulation for central schemes based on staggered grids is proposed. It is \nbased on a novel approach, in which first a time discretization is carried out, followed by the space \ndiscretization. The schemes obtained in this fashion have a simpler structure than previous central \nschemes. For high order schemes, this simplification results in higher computational efficiency. In \nthis work, schemes of order 2 to 5 are proposed and tested, although CRK schemes of any order \nof accuracy can be constructed in principle. The application to systems of equations is carefully \nstudied, comparing algorithms based on a componentwise extension of the scalar scheme, and \nalgorithms based on projection along characteristic directions.