The discontinuous Galerkin (DG) method is now an established method for computing approximate solutions of partial differential equations in many applications. The power of the DG method lies in the local nature of the spatial operators, with high order accuracy, and the flexibility of the method for resolving complex geometries using unstructured and/or boundary conforming curvilinear meshes. Because of the spatial locality of the operators, DG easily lends itself to efficient parallel numerical algorithms on modern heterogeneous high performance computing platforms. However, a crucial component of the DG method is the numerical flux, inherited from finite volume and finite difference methods for hyperbolic PDEs, based on approximate or exact solutions of the Riemann problem.
The choice of the numerical flux is critical for accuracy, stability and robustness of the underlying DG method.
In this project, we will extend the newly developed physics based numerical flux to the
nonlinear shallow water wave equations. In particular, we will explore the benefits of the physics based numerical flux as opposed to classical Rusanov and Godunov fluxes, by comparing numerical solutions.
Further comments: Programing in Python/Matlab will be required. No previous knowledge of DG or finite element methods is required.