Numerical investigations on the resonance errors of multiscale discontinuous Galerkin methods for one-dimensional stationary Schr\"{o}dinger equation

TL;DR

This paper investigates how penalty parameters influence resonance errors in high-order multiscale discontinuous Galerkin methods for the 1D stationary Schrödinger equation, revealing that positive penalties reduce resonance errors and improve system conditioning.

Contribution

The study provides extensive numerical evidence that positive penalty parameters mitigate resonance errors and enhance the stability of multiscale DG methods for Schrödinger equations.

Findings

Zero penalty parameters cause resonance errors.

Positive penalty parameters reduce resonance errors.

Positive penalties improve matrix condition numbers.

Abstract

In this paper, numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high order multiscale discontinuous Galerkin (DG) methods [6, 7] for one-dimensional stationary Schr\"{o}dinger equation. Previous work showed that penalty parameters were required to be positive in error analysis, but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes. In this work, by performing extensive numerical experiments, we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods, and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers.