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Compact implicit integration factor methods for some complex-valued nonlinear equations 被引量:1
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作者 张荣培 《Chinese Physics B》 SCIE EI CAS CSCD 2012年第4期49-53,共5页
The compact implicit integration factor (cIIF) method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF me... The compact implicit integration factor (cIIF) method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF method to some complex-valued nonlinear evolutionary equations such as the nonlinear SchrSdinger (NLS) equation and the complex Ginzburg-Landau (GL) equation. Detailed algorithm formulation and practical implementation of cIIF method are performed. The numerical results indicate that this method is very accurate and efficient. 展开更多
关键词 compact implicit integration factor method finite difference nonlinear Schrodinger equa-tion complex Ginzburg Landau equation
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A conservative numerical method for the fractional nonlinear Schrodinger equation in two dimensions
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作者 Rongpei Zhang Yong-Tao Zhang +2 位作者 Zhen Wang Bo Chen Yi Zhang 《Science China Mathematics》 SCIE CSCD 2019年第10期1997-2014,共18页
This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schr?dinger(NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grü... This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schr?dinger(NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grünwald-Letnikov difference(WSGD) operator for the spatial fractional Laplacian. We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations. By introducing the differentiation matrices, the semi-discrete fractional nonlinear Schr?dinger(FNLS) equation can be rewritten as a system of nonlinear ordinary differential equations(ODEs) in matrix formulations. Two kinds of time discretization methods are proposed for the semi-discrete formulation. One is based on the Crank-Nicolson(CN) method which can be proved to preserve the fully discrete mass and energy conservation. The other one is the compact implicit integration factor(c IIF) method which demands much less computational effort. It can be shown that the cIIF scheme can approximate CN scheme with the error O(τ~2). Finally numerical results are presented to demonstrate the method’s conservation, accuracy, efficiency and the capability of capturing blow-up. 展开更多
关键词 fractional nonlinear Schrodinger equation weighted and shifted Grünwald-Letnikov difference compact integration factor method CONSERVATION
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