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Conservative Local Discontinuous Galerkin method for the fractional Klein-Gordon-Schrödinger system with generalized Yukawa interaction
Numerical Algorithms ( IF 2.1 ) Pub Date : 2019-06-24 , DOI: 10.1007/s11075-019-00761-3
P. Castillo , S. Gómez

The formulation of the Local Discontinuous Galerkin (LDG) method applied to the space fractional Klein-Gordon-Schrödinger system with generalized interaction is presented. By considering its primal formulation and taking advantage of the symmetry of the bilinear form associated to the discretization of the Riesz differential operator, conservation of discrete analogues of the mass and the energy can be demonstrated for the semi-discrete problem and for the fully discrete problem using, as time marching scheme, a combination of the modified Crank-Nicolson method for the fractional nonlinear Schrödinger equation and the Newmark method for the nonlinear Klein-Gordon equation. Boundedness of the numerical solution in the L2 norm is derived from the conservation properties of the fully discrete method. A series of numerical experiments with high order approximations illustrates our conservation results and shows that optimal rates of convergence can be also achieved.



中文翻译:

广义Yukawa相互作用的分数Klein-Gordon-Schrödinger系统的保守局部不连续Galerkin方法

提出了局部不连续Galerkin(LDG)方法的公式化,该方法应用于具有广义相互作用的空间分数Klein-Gordon-Schrödinger系统。通过考虑其原始公式,并利用与Riesz微分算子离散化相关的双线性形式的对称性,可以证明对于半离散问题和完全离散问题,质量和能量的离散类似物的守恒性使用改进的Crank-Nicolson方法(用于分数阶非线性Schrödinger方程)和Newmark方法(用于非线性Klein-Gordon方程)作为时间行进方案。L 2中数值解的有界性规范是从完全离散方法的守恒性质导出的。一系列具有高阶近似的数值实验说明了我们的守恒结果,并表明还可以实现最佳收敛速度。

更新日期:2020-04-22
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