Executive Summary : | This proposal aims to develop hybrid numerical schemes for a system of hyperbolic conservation laws up to dimension three, a crucial partial differential equation model. The classical solution of these equations may cease to exist in finite time due to the presence of shocks, contact discontinuities, and rarefaction waves in the solution profile. Higher-order numerical methods may develop spurious oscillation or blow-up of the solution. Resolving these segments while computing the solution with sufficient accuracy and keeping computational cost within sustainable limits has been a significant research problem for decades. The project proposes hybrid numerical schemes of arbitrary order for hyperbolic conservation laws with convex flux, which converge to the solution with the least computational cost. For non-convex flux, a hybridization between first-order monotone scheme and WENO reconstruction can ensure scheme convergence to correct entropy solution. The proposed work includes developing and analyzing hybrid schemes for two model problems: hybrid numerical schemes for conservation laws with non-convex flux and hybrid numerical for Ideal MHD equations. Troubled-cell indicators, measuring the regularity of the solution, will be developed for an arbitrary higher-order hybrid scheme. The hybrid algorithms developed in this project will be rigorously validated by extensive numerical experiments. Efficient implementations of the hybrid algorithms will be developed using thread-based parallelism via OpenMP or MPI-based parallelism. |