Transparent Boundary Conditions for Evolution Equations in Infinite Periodic Strips

Abstract

We consider the solution of a generic equation $\gamma\rho(\mathbf{x})\partial^p_tu(\mathbf{x},t)-\Delta u(\mathbf{x},t) +V(\mathbf{x})u(\mathbf{x},t) = f(\mathbf{x},t)$, $\mathbf{x} = (x,y)$, for $t>0$, $p=1,2$ in a domain $\Omega$ which is infinite in $x$ and bounded in $y$. We assume that $f(\cdot,t)$ is supported for all $t>0$ in $\Omega_0 = \{\mathbf{x} \in \Omega \; | \; -a_- < x < a_+\}$ and that $\rho(\mathbf{x})$ and $V(\mathbf{x})$ are x-periodic in $\Omega \setminus \Omega_0$. We consider the associated $\theta$-scheme in time, to obtain a semidiscretized problem. We then show how to obtain for each time step exact boundary conditions on the vertical segments, $\Gamma_0^- = \{\mathbf{x}\in \Omega\; | \; x=-a_-\}$ and $\Gamma_0^+ = \{\mathbf{x}\in \Omega \;| \; x=a_+\}$, that will enable us to find the solution on $\Omega_0 \cup \Gamma_0^+ \cup \Gamma_0^-$. Then the solution can be extended in $\Omega$ in a straightforward manner from the values on $\Gamma_0^-$ and $\Gamma_0^+$. The method is ba...