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Runge-Kutta convolution quadrature and FEM-BEM coupling for the time-dependent linear Schrödinger equation. (English) Zbl 1361.65076

Summary: We propose a numerical scheme to solve the time-dependent linear Schrödinger equation. The discretization is carried out by combining a Runge-Kutta time stepping scheme with a finite element discretization in space. Since the Schrödinger equation is posed on the whole space \(\mathbb{R}^d\), we combine the interior finite element discretization with a convolution quadrature based boundary element discretization. In this paper, we analyze the resulting fully discrete scheme in terms of stability and convergence rate. Numerical experiments confirm the theoretical findings.

MSC:

65M38 Boundary element methods for initial value and initial-boundary value problems involving PDEs
35Q41 Time-dependent Schrödinger equations and Dirac equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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