Asymptotically Correct Finite Difference Schemes for Highly Oscillatory ODEs Available to Purchase

We are concerned with the numerical integration of ODE‐initial value problems of the form $ε2φxx+a(x)φ = 0$ with given $a(x)≥a0>0$ in the highly oscillatory regime $0<ε⋘1$ (appearing as a stationary Schrödinger equation, e.g.). In two steps we derive an accurate finite difference scheme that does not need to resolve each oscillation: With a WKB‐ansatz the dominant oscillations are “transformed out”, yielding a much smoother ODE. For the resulting oscillatory integrals we devise an asymptotic expansion both in ε and h. The resulting scheme typically has a step size restriction of $h = o(ε).$ If the phase of the WKB‐transformation can be computed explicitly, then the scheme is asymptotically correct with an error bound of the order $o(ε3h2).$ As an application we present simulations of a 1D‐model for ballistic quantum transport in a MOSFET (metal oxide semiconductor field‐effect transistor).