A space–time discontinuous Galerkin (DG) finite element method for nonlinear water waves in an inviscid and incompressible fluid is presented. The space–time DG method results in a conservative numerical discretization on time dependent deforming meshes which follow the free surface evolution. The algorithm is higher order accurate, both in space and time, and closely related to an arbitrary Lagrangian Eulerian (ALE) approach. A detailed derivation of the numerical algorithm is given including an efficient procedure to solve the nonlinear algebraic equations resulting from the space–time discretization. Numerical examples are shown on a series of model problems to demonstrate the accuracy and capabilities of the method.
van der Vegt, J. J. W., & Xu, Y. (2007). Space-time discontinuous Galerkin method for nonlinear water waves. Journal of computational physics, 224(2/1), 17-39. [10.1016/j.jcp.2006.11.031]. https://doi.org/10.1016/j.jcp.2006.11.031