Discontinuous Galerkin Methods for Phase Transitional Flows
Jaap van der Vegt
Numerical techniques for simulating seismic waves is an important aspect of predicting earth quakes, but also plays a central role in exploration geophysics. The propagation of these waves is modeled by the linear elastic wave equations, for which several numerical methods have already been developed, including finite difference methods, finite element methods and spectral methods. However, for many practical applications there is still a need for more efficient and flexible methods. Some of the challenges in practical applications are dealing with large domains, complex boundaries and material interfaces and detailed local structures. Large domains result in high computational costs and long running times, while complex interfaces require a flexible mesh. Small local structures require local refinement of the mesh and time step size. A Discontinuous Galerkin (DG) finite element method, combined with a suitable time integration method, should meet these challenges. Discontinuous Galerkin methods can deal with unstructured nonconforming-meshes and allow local refinement of the approximation order (p) as well as mesh size (h), known as hp-refinement. This makes the method suitable for dealing with complex domains, while this gets troublesome for standard finite difference methods. Since the method is based on discontinuous functions with support on only a single element, the mass matrix is block-diagonal and therefore solving the mass matrix equations for explicit time integration methods is significantly faster than with conforming finite element methods. The DG method can also deal with arbitrary high order polynomial basis functions on any standard element shape, while this is more challenging for mass-lumped finite element methods. In this research I will develop and analyze a suitable DG method for the anisotropic elastic wave equations and construct a suitable time integration method for dealing with detailed local structures. The code for this method will be implemented in our software package hpGEM and should be able to run parallel, such that it can run on large supercomputers.