A SEMI-LARGANGIAN METHOD ON DYNAMICALLY ADAPTED OCTREE MESHES ∗

Abstract. The paper develops a semi-Lagrangian method for the numerical integration of the transport equation discretized on adaptive Cartesian cubic meshes. We use dynamically adaptive graded Cartesian grids. They allows for a fast grid reconstruction in the course of numerical inte-gration. The suggested semi-Lagrangian method uses a higher order interpolation with a limiting strategy and a back-and-forth correction of the numerical solution. The interpolation operators have compact nodal stencils. In a series of experiments with dynamically adapted meshes, we demonstrate that the method has at least second order convergence and acceptable conservation and monotonicity properties. Key words. semi-Lagrangian method, octree meshes, adaptivity 1. Introduction. The well known higher order Eulerian methods TVD or WENO [12,31] for the numerical integration of the transport equation require CFL time step restrictions. Semi-Lagrangian advection schemes are not limited by the CFL condi-tion and hence are more flexible in adapting time steps to accuracy requirements. For non-uniform dynamically adapted Cartesian grids semi-Lagrangian schemes becom