A semi-Lagrangian AMR scheme for 2D transport problems in conservation form

In this paper, we construct a semi-Lagrangian (SL) Adaptive-Mesh-Refinement (AMR) solver for 1D and 2D transport problems in conservation form. First, we describe the à-la-Harten AMR framework: the adaptation process selects a hierarchical set of grids with different resolutions depending on the features of the integrand function, using as criteria the point value prediction via interpolation from coarser meshes, and the appearance of large gradients. We integrate in time by reconstructing at the feet of the characteristics through the Point-Value Weighted Essentially Non-Oscillatory (PV-WENO) interpolator. We propose, then, an extension to the 2D setting by making the time integration dimension-by-dimension thanks to a Strang splitting. We discuss the quality of the results and the speedup with respect to a Fixed Mesh (FM) strategy through the following benchmark tests: in 1D, constant and variable-coefficient advections; in 2D, the 1D Vlasov–Poisson system (2D in the phase space) for the case of constant-coefficient advections, and, for the case of variable-coefficient advections, the deformation flow and the guiding-center model.