A conservative local discontinuous Galerkin method for the solution of nonlinear Schrodinger equation in two dimensions
A conservative local discontinuous Galerkin method for the solution of nonlinear Schrodinger equation in two dimensions
复制标题
求解二维非线性薛定谔方程的保守局部间断伽辽金法
DOI：
10.1007/s11425-016-9118-x
复制
发表时间：
2017
期刊：
影响因子：
--
通讯作者：
XiangGui Li
中科院分区：
文献类型：
--
作者：
Rongpei Zhang;Xijun Yu;Mingjun Li;XiangGui Li
在本研究中，我们提出了一个守恒的局部间断Galerkin（LDG）方法来数值求解二维非线性薛定谔（NLS）方程。将NLS方程改写为一阶方程组，并构造了具有适当数值通量的LDG方程。半离散格式的质量守恒和能量守恒定律可以根据不同的数值通量的选择来证明，如中心通量，交替通量和迎风通量。我们将提出两种时间离散化方法的半离散配方。一种是基于Crank-Nicolson方法，可以证明保持离散的质量和能量守恒。另一种方法是Krylov隐式积分因子法（IIF），该方法计算量小.各种数值实验，以证明质量和能量守恒定律，最佳的收敛速度，和爆破现象。
In this study, we present a conservative local discontinuous Galerkin (LDG) method for numerically solving the two-dimensional nonlinear Schr¨odinger (NLS) equation. The NLS equation is rewritten as a firstorder system and then we construct the LDG formulation with appropriate numerical flux. The mass and energy conserving laws for the semi-discrete formulation can be proved based on different choices of numerical fluxes such as the central, alternative and upwind-based flux. We will propose two kinds of time discretization methods for the semi-discrete formulation. One is based on Crank-Nicolson method and can be proved to preserve the discrete mass and energy conservation. The other one is Krylov implicit integration factor (IIF) method which demands much less computational effort. Various numerical experiments are presented to demonstrate the conservation law of mass and energy, the optimal rates of convergence, and the blow-up phenomenon.