WKB-Based Schemes for the Oscillatory 1D Schrödinger Equation in the Semiclassical Limit

Abstract

Abstract. An efficient and accurate numerical method is presented for the solution of highly oscillatory differential equations. While standard methods would require a very fine grid to resolve the oscillations, the presented approach uses first an analytic (second order) WKB-type transformation, which filters out the dominant oscillations. The resulting ODE is much smoother and can hence be discretized on a much coarser grid, with significantly reduced numerical costs.In many practically relevant examples, the method is even asymptotically correct w.r.t. the small parameter ε that identifies the oscillation wave length. Indeed, in these cases, the error then vanishes for ε → 0, even on a fixed spatial mesh. Applications to the stationary Schrödinger equation are presented.

Coupling finite volume and nonstandard finite difference schemes for a singularly perturbed Schrödinger equation

“…2(b) and Fig. 3 respectively, one can clearly see the computational power of the nonstandard finite volume scheme and the non-classical finite volume scheme, in these two schemes we do not need the requirement of "at least 10 grid nodes per oscillation" as highlighted in [3]. We also highlight the better performance of NSFV scheme when exact schemes for the boundary conditions are used, see Figs.…”

“…It has been documented that for the singularly perturbed Schrödinger equation, standard schemes such as finite difference methods require several grid points to provide an accurate and reliable approximation [3]. Therefore, such methods consume much CPU memory and also computing time will be long.…”

“…This makes its numerical approximation computationally costly. In particular, in [3] it was pointed out that most numerical schemes require at least 10 grid points per oscillation.…”

“…Therefore, a lot of research on the solution of equation (1.1) has been done based on standard finite difference (both classical, adaptive mesh and finite volume), finite element and WKB methods, see for example [3] and the literature therein. The authors in [9] solved the Schrödinger equation on a non-uniform mesh using a finite difference method.…”

“…They employed the finite element method with the basis elements coming from the WKB approximation. Most recently on this trend, the authors in [3] claimed that the high computational cost of classical methods is due to the requirement that the space width be smaller compared to the wavelength of the solution. This idea forms the basis of our current work.…”