Abstract
This paper investigates a first-order and a second-order approximation technique for the shallow water equation with topography using continuous finite elements. Both methods are explicit in time and are shown to be well-balanced. The first-order method is invariant domain preserving and satisfies local entropy inequalities when the bottom is flat. Both methods are positivity preserving. Both techniques are parameter free, work well in the presence of dry states, and can be made high order in time by using strong stability preserving time stepping algorithms. 1. Introduction. The objective of this paper is to develop an invariant domain preserving well-balanced approximation of the shallow water equation with bathymetry using continuous finite elements. There are many finite volume and dis-continuous Galerkin (DG) techniques available in the literature that can solve this problem efficiently up to second and higher order in space. Examples of schemes that are well balanced at rest and robust in the presence of dry states can be found, for example, in Audusse and Bristeau [1], Audusse et al. [2], Bollermann, Noelle, and Lukáčová-Medvidová [6], Gallardo, Parés, and Castro [14], Kurganov and Petrova [23], Perthame and Simeoni [27], Ricchiuto and Bollermann [28]. We refer the reader to the book of Bouchut [7] for a review on this topic, to the paper of Xing and Shu [32] for a survey on finite volume and DG methods, and to the paper [23] for a survey of central-upwind schemes. However, to the best of our knowledge, these types of approximations are not developed in the context of continuous finite elements. Or we should say that no robust continuous finite element technique is yet available in the literature that guarantees second-order accuracy, works properly in every regime (subcritical, transcritical, transcritical with hydraulic jumps, wet, and dry regions) and is well-balanced at rest. We propose such a method in the present paper. Two variants of the method are discussed: one variant is first-order accurate in space, positivity preserving, and preserves every convex invariant domain of the system in the absence of bathymetry; the other variant is second-order accurate in space and positivity preserving. Both variants are explicit in time and use continuous finite elements on unstructured meshes.