A High Order Numerical Method for Computing Physical Observables in the Semiclassical Limit of the One-Dimensional Linear Schrödinger Equation with Discontinuous Potentials

We develop a fourth order numerical method for the computation of multivalued physical observables (density, momentum, etc.) in the semiclassical limit of the one dimensional linear Schrödinger equation in the case of discontinuous potentials. We adopt the level set framework developed in [21] which allows one to compute the multivalued physical observables via solving the classical Liouville equation with bounded initial data and approximating delta function integrals. We achieve high order accuracy for our method by studying two issues. The first is to highly accurately compute the solution and its derivatives of the Liouville equation with bounded initial data and discontinuous potentials. The second is to design high order numerical methods to evaluate one dimensional delta function integrals with discontinuous kernel functions. Numerical examples are presented to verify that our method achieves the fourth order L1-norm accuracy for computing multivalued physical observables of the one dimensional linear Schrödinger equation with general discontinuous potentials.