Error estimates for second order finite volume schemes using a TVD-Runge-Kutta time discretization for a nonlinear scalar hyperbolic conservation law (Q2716798)

The paper examines higher order finite volume schemes for scalar hyperbolic conservation laws. In space, the higher order discretization relies on reconstruction techniques on general unstructured grids in any dimensions. In time, the higher order discretization is realized using an explicit second order total variation diminishing (TVD) Runge-Kutta discretization. The assumptions on the reconstruction and limitation procedure for getting the higher discretization in space are clarified. The author proves a bound for the approximate solution and a weak bounded variation stability result. Subsequently a continuous entropy estimate for the approximate solution is obtained. Finally an a-priori \(L^1\) error estimate in space and time of order \(h^{1/4}\), where \(h\) denotes the mesh-size parameter is obtained.