An Adaptive Multiresolution Ultra-weak Discontinuous Galerkin Method for Nonlinear Schrödinger Equations

doi:10.1007/s42967-020-00096-0

Communications on Applied Mathematics and Computation ›› 2022, Vol. 4 ›› Issue (1): 60-83.doi: 10.1007/s42967-020-00096-0

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An Adaptive Multiresolution Ultra-weak Discontinuous Galerkin Method for Nonlinear Schrödinger Equations

Zhanjing Tao¹, Juntao Huang², Yuan Liu³, Wei Guo⁴, Yingda Cheng⁵

1 School of Mathematics, Jilin University, Jilin 130012, China;
2 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA;
3 Department of Mathematics, Statistics and Physics, Wichita State University, Wichita, KS 67260, USA;
4 Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 70409, USA;
5 Department of Mathematics, Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI 48824, USA

收稿日期:2020-06-30 修回日期:2020-09-11 出版日期:2022-03-20 发布日期:2022-03-01
通讯作者: Juntao Huang, Zhanjing Tao, Yuan Liu, Wei Guo, Yingda Cheng E-mail:huangj75@msu.edu;zjtao@jlu.edu.cn;liu@math.wichita.edu;weimath.guo@ttu.edu;ycheng@msu.edu

An Adaptive Multiresolution Ultra-weak Discontinuous Galerkin Method for Nonlinear Schrödinger Equations

Zhanjing Tao¹, Juntao Huang², Yuan Liu³, Wei Guo⁴, Yingda Cheng⁵

1 School of Mathematics, Jilin University, Jilin 130012, China;
2 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA;
3 Department of Mathematics, Statistics and Physics, Wichita State University, Wichita, KS 67260, USA;
4 Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 70409, USA;
5 Department of Mathematics, Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI 48824, USA

Received:2020-06-30 Revised:2020-09-11 Online:2022-03-20 Published:2022-03-01
Contact: Juntao Huang, Zhanjing Tao, Yuan Liu, Wei Guo, Yingda Cheng E-mail:huangj75@msu.edu;zjtao@jlu.edu.cn;liu@math.wichita.edu;weimath.guo@ttu.edu;ycheng@msu.edu

摘要/Abstract

摘要： This paper develops a high-order adaptive scheme for solving nonlinear Schrödinger equations. The solutions to such equations often exhibit solitary wave and local structures, which make adaptivity essential in improving the simulation efciency. Our scheme uses the ultra-weak discontinuous Galerkin (DG) formulation and belongs to the framework of adaptive multiresolution schemes. Various numerical experiments are presented to demonstrate the excellent capability of capturing the soliton waves and the blow-up phenomenon.

关键词: Multiresolution, Sparse grid, Ultra-weak discontinuous Galerkin method, Schrödinger equation, Adaptivity

Abstract: This paper develops a high-order adaptive scheme for solving nonlinear Schrödinger equations. The solutions to such equations often exhibit solitary wave and local structures, which make adaptivity essential in improving the simulation efciency. Our scheme uses the ultra-weak discontinuous Galerkin (DG) formulation and belongs to the framework of adaptive multiresolution schemes. Various numerical experiments are presented to demonstrate the excellent capability of capturing the soliton waves and the blow-up phenomenon.

Key words: Multiresolution, Sparse grid, Ultra-weak discontinuous Galerkin method, Schrödinger equation, Adaptivity

中图分类号:

65M60

Zhanjing Tao, Juntao Huang, Yuan Liu, Wei Guo, Yingda Cheng. An Adaptive Multiresolution Ultra-weak Discontinuous Galerkin Method for Nonlinear Schrödinger Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 60-83.

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An Adaptive Multiresolution Ultra-weak Discontinuous Galerkin Method for Nonlinear Schrödinger Equations

An Adaptive Multiresolution Ultra-weak Discontinuous Galerkin Method for Nonlinear Schrödinger Equations

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