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 leap-frog 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.
Highlights
We are concerned with the construction and numerical implementation of discrete transparent boundary conditions for linear transport equations
This general strategy may be applied to any finite difference scheme
We focus here on the discretization of the two-dimensional transport equation by means of the leap-frog scheme
Summary
We are concerned with the construction and numerical implementation of discrete transparent boundary conditions for linear transport equations. We shall construct fully discrete, approximate transparent boundary conditions for the two-dimensional leap-frog scheme on a rectangle. Our strategy is to start from the exact transparent boundary conditions for a half-space and, as in [19], to localize them with respect to the tangential variable by means of a suitable “small frequency approximation” This general strategy may be applied to any finite difference scheme. The nice feature of this approximation is that it does not dissipate high frequency signals – and allows for a precise analysis of wave reflections on each side of the rectangle – and, its stencil exhibits some “dimensional splitting” The latter feature will be helpful in our construction since we shall not have to develop a specific numerical treatment for the four corners of the computational domain. For the half-space problem, the stability of our approximate DTBC and report on some numerical simulations on the rectangle
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