On the construction of compact high-order semi-implicit DG schemes and applications

Abstract:
In this talk we present a class of semi-implicit methods on unstructured staggered meshes in 2D and 3D. The numerical solution is computed using a DG approach, but on staggered meshes. We will consider several PDE systems such as the incompressible/compressible Navier Stokes and linear elasticity. The proper implicit discretization of the terms associated to the fast scale, i.e. the pressure, allows to obtain a weak CFL stability restriction. The resulting maximum time step is then limited by the local velocity and not by the celerity, allowing the algorithm to perform well in the low Mach number regime. The resulting linear system is symmetric and at least semi-positive definite for most of the applications, so an efficient matrix-free implementation of the conjugate gradient method can be used for the solution of the linear part. Thanks to the chosen discretization, the stencil results very compact and involves only the direct neighbors. Besides, high order in time can also be obtained using a genuine space-time DG or a simple IMEX approach. Several extensions to ph-adaptivity and GPU computing will be discussed, as well as a possible application to networks of compliant pipes, useful to simulate physiological flows.