Discrete transparent boundary conditions for the two-dimensional leap-frog scheme

TL;DR

This paper introduces a method for implementing approximate discrete transparent boundary conditions for 2D transport equations using the leap-frog scheme, analyzing stability and coupling strategies on rectangular domains.

Contribution

It presents a novel approach for explicit boundary conditions in 2D leap-frog schemes, avoiding corner prescriptions and analyzing stability through normal mode analysis.

Findings

Explicit boundary conditions are derived for each side of the rectangle.

Strong instabilities can occur with certain coupling strategies.

Some coupling strategies show promising stability results.

Abstract

We develop a general strategy in order to implement (approximate) discrete transparent boundary conditions for finite difference approximations of the two-dimensional transport equation. The computational domain is a rectangle equipped with a Cartesian grid. For the two-dimensional leapfrog scheme, we explain why our strategy provides with explicit numerical boundary conditions on the four sides of the rectangle and why it does not require prescribing any condition at the four corners of the computational domain. The stability of the numerical boundary condition on each side of the rectangle is analyzed by means of the so-called normal mode analysis. Numerical investigations for the full problem on the rectangle show that strong instabilities may occur when coupling stable strategies on each side of the rectangle. Other coupling strategies yield promising results.