Abstract

In this article, we present approximations with finite elements of high order using stabilized variational multiscale methods to approximate the equations of motion of a fluid in shallow waters. We write these equations as a system of non-linear and transient convection-diffusion-reaction (CDR) equations and we perform our developments in this general framework. Variational multiscale methods (VMS) are based on the decomposition of the unknowns of the continuous problem in a resolved component in the finite element space and another component that cannot be captured by the finite element mesh, and that we call subscale. The subscale is approximated in terms of the finite element solution, obtaining a robust numerical scheme, which in particular allows one to use the same interpolation for all unknowns and the possibility to deal with convection dominated flows (we will not consider the possibility of dealing with shocks). The two VMS methodologies that we will consider are called algebraic subscales (ASGS) and orthogonal subscales (OSS).