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Abstracts

Numerical solution of differential and integral equations

documentstyle[12pt]{article} title{ Unconditionally Stable Explicit High-Order Numerical Method for the Nonlinear Schr"{o}dinger Equation } author{A. V. Shapeev} begin{document} maketitle The initial-boundary-value problem for nonlinear Schr"odinger equation begin{equation} left{ begin{array}{l} displaystyle i frac{partial A}{partial t} + d(t) frac{mathstrut partial^2 A}{mathstrut partial x^2} + c(t) |A|^2 A = i G(t) A, displaystyle A(0,x)=A^0(x), displaystyle A(t,x)=A(t,x+L) end{array} right. end{equation} is considered. One of the most widespread method of solving this problem is split step Fourier method (SSFM) [1]. It is unconditionally stable and is second order accurate in $t$. Direct application of known high-order methods (Runge-Kutta, Adams methods and their generalizations) to this equation gives us at most conditionally stable discretizations. The aim of the present work is to construct unconditionally stable explicit numerical method of increased order with respect to $t$. The approach proposed is based on the following. The linearized problem (i.e. problem (1) with $c(t)=0$) has analytical representation of general solution and therefore transition operator $S_{t_0}^t$ associating the initial data at the point of time $t_0$ with the solution of linear problem at the point of time $t$ can be given by an explicit formula. As it is done in the method of variation of constants, substitution $A(t,x) = S_0^t,u(t,x)$ is made. Finally, applying Runge-Kutta high-order methods to equation for newly introduced function $u(t,x)$ we obtain unconditionally stable explicit discretizations. Numerical experiments show that the proposed numerical method is more efficient than the known methods in majority of cases which are of interest. end{document}

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