Numerical Investigations on the Resonance Errors of Multiscale Discontinuous Galerkin Methods for One-Dimensional Stationary Schrödinger Equation

doi:10.1007/s42967-022-00248-4

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (1): 311-324.doi: 10.1007/s42967-022-00248-4

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Numerical Investigations on the Resonance Errors of Multiscale Discontinuous Galerkin Methods for One-Dimensional Stationary Schrödinger Equation

Bo Dong¹, Wei Wang²

1. Department of Mathematics, University of Massachusetts Dartmouth, North Dartmouth, MA, 02747, USA;
2. Department of Mathematics and Statistics, Florida International University, Miami, FL, 33199, USA

收稿日期:2022-08-30 修回日期:2022-12-19 发布日期:2024-04-16
通讯作者: Wei Wang,E-mail:weiwang1@fiu.edu;Bo Dong,E-mail:bdong@umassd.edu E-mail:weiwang1@fiu.edu;bdong@umassd.edu
基金资助:
The authors consent to comply with all the Publication Ethical Standards. The research of the first author is supported by the National Science Foundation grant DMS-1818998.

Numerical Investigations on the Resonance Errors of Multiscale Discontinuous Galerkin Methods for One-Dimensional Stationary Schrödinger Equation

Bo Dong¹, Wei Wang²

1. Department of Mathematics, University of Massachusetts Dartmouth, North Dartmouth, MA, 02747, USA;
2. Department of Mathematics and Statistics, Florida International University, Miami, FL, 33199, USA

Received:2022-08-30 Revised:2022-12-19 Published:2024-04-16
Contact: Wei Wang,E-mail:weiwang1@fiu.edu;Bo Dong,E-mail:bdong@umassd.edu E-mail:weiwang1@fiu.edu;bdong@umassd.edu
Supported by:
The authors consent to comply with all the Publication Ethical Standards. The research of the first author is supported by the National Science Foundation grant DMS-1818998.

摘要/Abstract

摘要： 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 (Dong et al. in J Sci Comput 66: 321-345, 2016; Dong and Wang in J Comput Appl Math 380: 1-11, 2020) for a one-dimensional stationary Schrödinger 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.

关键词: Discontinuous Galerkin (DG) method, Multiscale method, Resonance errors, One-dimensional Schrödinger equation

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 (Dong et al. in J Sci Comput 66: 321-345, 2016; Dong and Wang in J Comput Appl Math 380: 1-11, 2020) for a one-dimensional stationary Schrödinger 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.

Key words: Discontinuous Galerkin (DG) method, Multiscale method, Resonance errors, One-dimensional Schrödinger equation

Bo Dong, Wei Wang. Numerical Investigations on the Resonance Errors of Multiscale Discontinuous Galerkin Methods for One-Dimensional Stationary Schrödinger Equation[J]. Communications on Applied Mathematics and Computation, 2024, 6(1): 311-324.

参考文献

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Numerical Investigations on the Resonance Errors of Multiscale Discontinuous Galerkin Methods for One-Dimensional Stationary Schrödinger Equation

Numerical Investigations on the Resonance Errors of Multiscale Discontinuous Galerkin Methods for One-Dimensional Stationary Schrödinger Equation

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