Improved Accuracy of High-Order WENO Finite Volume Methods on Cartesian Grids

We will present our recent result on the construction of high order WENO finite volume methods for the approximation of hyperbolic partial differential equations on Cartesian grids. The simplest way to use WENO methods on multidimensional Cartesian grids consists in applying a one-dimensional WENO scheme in each direction. This spatial discretization is typically combined with a Runge-Kutta method in time. In the finite difference context, such WENO methods lead to a high order accurate approximation. In contrast to this, finite volume WENO methods based on a dimension-by-dimension approach are only second order accurate for nonlinear problems, see [2].