ADER Schemes on Adaptive Triangular Meshes for Scalar Conservation Laws

ADER schemes are recent finite volume methods for hyperbolic conservation laws, which can be viewed as generalizations of the classical first order Godunov method to arbitrary high orders. In the ADER approach, high order polynomial reconstruction from cell averages is combined with high order flux evaluation, where the latter is done by solving generalized Riemann problems across cell interfaces. Currently available nonlinear ADER schemes are restricted to Cartesian meshes. This paper proposes an adaptive nonlinear finite volume ADER method on unstructured triangular meshes for scalar conservation laws, which works with WENO reconstruction. To this end, a customized stencil selection scheme is developed, and the flux evaluation of previous ADER schemes is extended to triangular meshes. Moreover, an a posteriori error indicator is used to design the required adaption rules for the dynamic modification of the triangular mesh during the simulation. The expected convergence orders of the proposed ADER method are confirmed by numerical experiments for linear and nonlinear scalar conservation laws. Finally, the good performance of the adaptive ADER method, in particular its robustness and its enhanced flexibility, is further supported by numerical results concerning Burgers equation.