Numerical Solution of the Viscous Surface Wave with Discontinuous Galerkin Method∗

We consider an incompressible viscous flow without surface tension in a finite-depth domain of two dimensions, with free top boundary and fixed bottom boundary. This system is governed by the Navier–Stokes equations in this moving domain and the transport equation on the moving boundary. In this paper, we construct a stable numerical scheme to simulate the evolution of this system by discontinuous Galerkin method, and discuss the error analysis of the fluid under certain assumptions. Our formulation is mainly based on the geometric structure introduced in [Y. Guo and Ian Tice, Anal. PDE 6 (2013) 287–369; Y. Guo and Ian Tice, Arch. Ration. Mech. Anal. 207 (2013) 459–531; L. Wu, SIAM J. Math. Anal. 46 (2014) 2084–2135], and the natural energy estimate, which is rarely used in the numerical study of this system before.