Error estimates for a class of energy- and Hamiltonian-preserving local discontinuous Galerkin methods for the Klein-Gordon-Schrodinger equations

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Error estimates for a class of energy- and Hamiltonian-preserving local discontinuous Galerkin methods for the Klein-Gordon-Schrodinger equations

机译：一类能源和汉密尔顿局保留的局部不连续的Galerkin方法的估计估计为Klein-Gordon-Schrodinger方程

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The Klein-Gordon-Schrodinger (KGS) equations are classical models to describe the interaction between conservative scalar nucleons and neutral scalar mesons through Yukawa coupling. In this paper, we propose local discontinuous Galerkin (LDG) methods to solve the KGS equations. The methods involve a Crank-Nicholson time discretization for the Schrodinger equation part, a Crank-Nicholson leap frog method in time for the Klein-Gordon equation part, and local discontinuous Galerkin methods in space. Our designed numerical methods have high-order convergence rate, and energy- and Hamiltonian-preserving properties. We present the proofs of such conservation properties for both semi-discrete and fully-discrete schemes. We also establish optimal error estimates of the semi-discrete methods for the linearized KGS equations and the fully discrete methods for the KGS equations. The analysis can be extended to LDG methods for the nonlinear Klein-Gordon or Schrodinger equation, and the KGS equations in higher spatial dimensions. Several numerical tests are presented to verify some of our theoretical findings.

机译：Klein-Gordon-Schrodinger（kgs）方程是古典模型，用于描述通过yukawa偶联的保守标量核磁和中性标量介质之间的相互作用。在本文中，我们提出了局部不连续的Galerkin（LDG）方法来解决KGS方程。该方法涉及Schrodinger方程部分的曲柄-Nicholson时间离散化，曲柄-Nicholson跳跃蛙方法及时用于克莱因戈登公式部分，以及空间中的局部不连续的Galerkin方法。我们设计的数值方法具有高阶收敛速度和能量和汉密尔顿源保存的属性。我们介绍了半离散和完全离散方案的这种保护性能的证据。我们还建立了线性化KGS方程的半离散方法的最佳误差估计和KGS方程的完全离散方法。该分析可以扩展到非线性Klein-Gordon或Schrodinger方程的LDG方法，以及较高空间尺寸的KGS方程。提出了几种数值测试以验证我们的一些理论发现。

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《Journal of Applied Mathematics and Computing》 |2020年第2期|共48页
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正文语种 eng
中图分类数学;
关键词
Error estimates; Local discontinuous Galerkin methods; Energy conservation; Hamiltonian-preserving; Klein-Gordon-Schrodinger equations;

机译：错误估计;局部不连续的Galerkin方法;节能;汉密尔顿保留;Klein-Gordon-Schrodinger方程;

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1. Error estimates for a class of energy- and Hamiltonian-preserving local discontinuous Galerkin methods for the Klein-Gordon-Schrodinger equations [J] . Journal of Applied Mathematics and Computing . 2020,第1a2期

机译：一类能源和汉密尔顿局保留的局部不连续的Galerkin方法的估计估计为Klein-Gordon-Schrodinger方程
2. Stability analysis and error estimates of local discontinuous Galerkin methods for convection-diffusion equations on overlapping meshes [J] . Du Jie, Yang Yang, Chung Eric BIT numerical mathematics . 2019,第4期

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4. Preliminary results in Local Discontinuous Galerkin Methods for Two Classes of 2D Nonlinear Wave Equations [C] . Yan Xu, Chi-Wang Shu Computational Mechanics . 2004

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6. Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws: divided difference estimates and accuracy enhancement [O] . Xiong Meng, Jennifer K. Ryan -1

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7. Local Discontinuous Galerkin Finite Element Method and Error Estimates for One Class of Sobolev Equation [O] . Gao F. Z., Qiu J. X., Zhang Q., 2009

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8. Local Discontinuous Galerkin Methods for the Cahn-Hilliard Type Equations [R] . Xia, Y. , Xu, Y. , Shu, C. 2007

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Error estimates for a class of energy- and Hamiltonian-preserving local discontinuous Galerkin methods for the Klein-Gordon-Schrodinger equations

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