Simon Johansson: A Modal PML for the 2D Shallow Water Equations in Conservative Variables

Abstract:

A modal Perfectly Matched Layer (PML) is constructed for the two-dimensional, non-linear, Shallow Water Equations (SWEs) in conservative variables. The result is an analytical continuation of the original equations where absorption is applied to the outgoing wave modes which are damped exponentially fast in the direction of propagation. Numerical tests are performed using a variation of the Diagonally Implicit Runge-Kutta (DIRK) integration scheme in conjunction with Lax-Wendroff’s method for smooth solutions and some variation of Roe’s method for discontinuous solutions. Different absorption functions are used, i.e. the absorption function of the original PML constructed by Berenger for electromagnetic waves, and some variations of the hyperbola functions. The results clearly show that the PML is better than the characteristic boundary condition, but also that improvements through some sort of optimization should lead to better parameter choices, potentially decreasing the reflections further.