Modified equation and error analyses on adaptive meshes for the resolution of evolutionary PDEs with Finite Volume schemes

Adaptive mesh techniques, such as Adaptive Mesh Refinement (AMR) and adaptive multiresolution, enable computational efficiency by dynamically refining or coarsening meshes in the resolution of evolutionary PDEs, which are strongly multiscale in space and time. In this work, we introduce the modified equation analysis for Finite Volume schemes on adaptive meshes, with a focus on adaptive multiresolution. This approach allows us to quantify how the reference scheme’s modified equation is perturbed according to how fluxes are approximated in the presence of coarse cells in the mesh. By leveraging symbolic computations, we offer a systematic means of assessing these perturbations. Our findings highlight a crucial point: when high-order schemes are employed, inadequate flux reconstruction can degrade accuracy, making high-order methods ineffective in smooth regions.