Research

Nonlinear conservation laws govern the behavior of fluid phenomena such as compressible or shallow water flows. Numerical simulations of such phenomena are sensitive to problems of instability. For low order methods, this instability can be offset by the presence of numerical dissipation, which decreases accuracy but has a stabilizing effect on simulations. However, the low numerical dissipation of high order methods becomes a double-edged sword, reducing robustness and rendering many high order discretizations unstable without additional regularization such as artificial viscosity, slope limiting, or filtering. This regularization can, in turn, reduce accuracy to first or second order.

Discretely entropy stable methods restore stability while maintaining high order accuracy by ensuring that numerical solutions satisfy a physically consistent entropy inequality. This entropy inequality serves as a generalization of the conservation or dissipation of energy to nonlinear PDEs, and ensures that the numerical solution does not blow up so long as the solution is physical (e.g. positive density, pressure).

Like standard finite element methods, discontinuous Galerkin (DG) methods are numerical methods designed to achieve arbitrarily high orders of accuracy on unstructured meshes (which are typically necessary for complex geometries). Unlike standard finite element methods, DG methods enforce a "weak" continuity between neighboring elements by means of an interface flux. We can also naturally add stabilization terms through the interface flux to improve solution quality and accuracy. 

In our group, the primary reason we use DG methods is that, when paired with explicit time-stepping methods, matrix-free implementations of DG methods become fairly simple to implement while achieving relative computational efficiency. We are also interested in high order continuous finite element methods which are well-suited to explicit time-stepping, such as the spectral element method.