Construction, Structure and Asymptotic Approximations of a Microdifferential Transparent Boundary Condition for the Linear Schrödinger Equation
Résumé
A transparent boundary condition for the two-dimensional linear Schrödinger equation is constructed through a microlocal approximation of the operator associating the Dirichlet data to the Neumann one in a M-quasi hyperbolic region. Several quasi-analytic characterization results concerning the asymptotic expansion of the total symbol of this operator in a subclass of inhomogeneous symbols with a quasi-polynomial-like structure are stated. In particular, a high-frequency control giving the behavior of these symbols is precised. It highlights the way of how to derive some consistent asymptotic artificial boundary conditions involving fractional derivatives with respect to the time variable by approximating the micro-transparent condition in the high-frequency regime. These approximate conditions are local according to the space variable and should lead to some efficient and accurate numerical simulations if they are used to truncate the unbounded domain of propagation.