Multi-domain hybrid spectral-WENO methods for hyperbolic conservation laws

In this paper, we introduce the multi-domain hybrid Spectral-WENO method aimed at the discontinuous solutions of hyperbolic conservation laws. The main idea is to conjugate the non-oscillatory properties of the high order weighted essentially non-oscillatory (WENO) finite difference schemes with the high computational efficiency and accuracy of spectral methods. Built in a multi-domain framework, subdomain adaptivity in space and time is used in order to maintain the solutions parts exhibiting high gradients and discontinuities always inside WENO subdomains, while the smooth parts of the solution are kept in spectral ones. A high order multi-resolution algorithm by Ami Harten is used to determine the smoothness of the solution in each subdomain. Numerical experiments with the simulation of compressible flow in the presence of shock waves are performed.