A New ADER Method Inspired by the Active Flux Method

Abstract

We study numerical methods that are inspired by the active flux method of Eymann and Roe and present several new results for one and two-dimensional hyperbolic problems. For one-dimensional linear problems we show that the unlimited active flux method can be interpreted as an ADER method. This interpretation motivates the construction of new third order accurate methods for nonlinear hyperbolic conservation laws. In the two-dimensional case, equivalent methods are only obtained for scalar linear problems. For two-dimensional linear systems the methods are no longer equivalent. For the two-dimensional acoustic equations we compare the accuracy of the two resulting approaches. While commonly used methods for hyperbolic problems are based on discontinuous reconstructions, the active flux method uses a continuous, piecewise quadratic reconstruction. For nonlinear problems we identify a situation in which the continuous reconstruction leads to an unstable approximation. We propose a limiting strategy which overcomes this problem. Our limited version of the active flux method uses the same local stencil as the original method.