An adaptive multiresolution flux reconstruction method with local time stepping and artificial viscosity for compressible flows simulations

In this paper, we introduce a novel approach that combines multiresolution (MR) techniques with the flux reconstruction (FR) method to accurately and effciently simulate compressible flows. We achieve further enhancements in effciency through the incorporation of local time stepping, and we add artificial viscosity to capture shocks. With the developed MR-FR algorithm, the layer difference of two adjacent elements can exceed 1, and simulation errors can be adjusted by manipulating a single scalar. To ensure conservation, information communication between nodes at different layers is accomplished using L2 projection. Additionally, we propose an innovative indicator based on MR analysis to detect discontinuities, enabling us to take full advantage of the details generated by MR. By indicating smoothness and adding artificial viscosity only to the finest meshes, computational costs can be reduced and errors resulting from artificial diffusion can be locally limited. Numerical tests demonstrate that the adoption of MR preserve the convergence order of the FR method, and the newly proposed indicator performs well in detecting discontinuities. Overall, the MR-FR algorithm can accurately simulate compressible flows with strong shocks and physical dissipation using significantly fewer grids, making it a promising approach for further applications.