Central Runge - Kutta schemes for conservation laws

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