Discrete transparent boundary conditions for the two-dimensional leap-frog scheme: approximation and fast implementation

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.

Perfectly Matched Layers Methods for Mixed Hyperbolic–Dispersive Equations

“…[41] for linearized Green-Naghdi equations and [56] for its application to domain decomposition for water wave propagation. However this kind of technique is hard to extend to a two dimensional problem (see [8] for the 2d linear transport equation). Another interesting approach consists in imitating the shallow water equations with nonlocal flux: see [44] in the case of a Boussinesq system where the problem of generating a wave in the computational domain is treated.…”

“…Exact transparent boundary conditions both continuous and discrete were derived and implemented for the linearized KdV equation in [9], and for linearized BBM equation in [10], the bi-directional dispersive wave propagation for the linearized Green-Naghdi model were considered in [41]. The generalization of this approach to the case of two-dimensional problem was proposed in [8] for the transport equation. However, the design of DTBCs for the linear dispersive two-dimensional waves models such as the KdV, the BBM or the Green-Naghdi equations is hardly extended to real situations.…”

Coupling Dispersive Shallow Water Models by Deriving Asymptotic Interface Operators

“…To test this asymptotic domain decomposition method with an additive iterative scheme we compare the monodomain solution with the DDM solution for 𝑥 ∈ (0, 1), 𝜙 𝐿𝑒 𝑓 𝑡 = 0, 𝜙 𝑅𝑖𝑔ℎ𝑡 = −0.5, 𝜕 3 𝑥 𝜁 𝑛 = −1, 𝜇 = 3, Δ𝑥 = 0.05, Δ𝑧 = Δ𝑥 2 . These parameters are convenient to avoid overflow in intermediate calculations in the formula of the analytical solution.…”

“…Since this coupling is a "divide and conquer" type of problem, domain decomposition methods (DDM) can help to obtain further insights. Usually, coupling conditions have been derived for each equation on a case-by-case basis (e.g., [2,3,4,5] ). Here we explore a different approach, based on the steps of the derivation of the GNE and NSWE [6, ch.…”