Athanassios G. Bratsos

A finite-difference method for solving the cubic Schro¨dinger equation

A family of finite-difference methods is used to transform the initial/boundary-value problem associated with the nonlinear Schrodinger equation into a first-order, linear, initial-value problem. Numerical methods are developed by replacing the time and space derivatives by central-difference replacements. The resulting finite-difference methods are analysed for local truncation, errors, stability and convergence. The results of a number of numerical experiments are given.

The solution of the Boussinesq equation using the method of lines

Abstract The method of lines is used to transform the initial/boundary-value problem associated with the nonlinear hyperbolic Boussinesq equation, into a first-order, nonlinear, initial-value problem. Numerical methods are developed by replacing the matrix-exponential term in a recurrence relation by rational approximants. The resulting finite-difference methods are analysed for local truncation errors, stability and convergence. The results of a number of numerical experiments are given.

A modified predictor–corrector scheme for the two-dimensional sine-Gordon equation

A three-time level finite-difference scheme based on a fourth order in time and second order in space approximation has been proposed for the numerical solution of the nonlinear two-dimensional sine-Gordon equation. The method, which is analysed for local truncation error and stability, leads to the solution of a nonlinear system. To avoid solving it, a predictor–corrector scheme using as predictor a second-order explicit scheme is proposed. The procedure of the corrector has been modified by considering as known the already evaluated corrected values instead of the predictor ones. This modified scheme has been tested on the line and circular ring soliton and the numerical experiments have proved that there is an improvement in the accuracy over the standard predictor–corrector implementation.

A third order numerical scheme for the two-dimensional sine-Gordon equation

A rational approximant of third order, which is applied to a three-time level recurrence relation, is used to transform the two-dimensional sine-Gordon (SG) equation into a second-order initial-value problem. The resulting nonlinear finite-difference scheme, which is analyzed for stability, is solved by an appropriate predictor-corrector (P-C) scheme, in which the predictor is an explicit one of second order. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behavior of the proposed P-C/MPC schemes is tested numerically on the line and ring solitons known from the bibliography, regarding SG equation and conclusions for both the mentioned schemes regarding the undamped and the damped problem are derived.

A discrete Adomian decomposition method for discrete nonlinear Schrodinger equations

We present a new discrete Adomian decomposition method to approximate the theoretical solution of discrete nonlinear Schrodinger equations. The method is examined for plane waves and for single soliton waves in case of continuous, semi-discrete and fully discrete Schrodinger equations. Several illustrative examples and Mathematica program codes are presented.

The solution of the sine-Gordon equation using the method of lines

The method of lines is used to transform the initial/boundary-value problem assckiated with the nonlinear hyperbolic sine-Gordon equation, into a first-order, nonlinear, initial-vjalue problem. Numerical methods are developed by replacing the matrix-exponential term \n a recurrence relation by rational approximants. The resulting finite-difference methods ar|e analysed for local truncation errors, stability and convergence. The results of a number of numerical experiments are given.

A fourth-order numerical scheme for solving the modified Burgers equation

A finite-difference scheme based on fourth-order rational approximants to the matrix-exponential term in a two-time level recurrence relation is proposed for the numerical solution of the modified Burgers equation. The resulting nonlinear system, which is analyzed for stability, is solved using an already known modified predictor-corrector scheme. The results arising from the experiments are compared with the corresponding ones known from the available literature.

A family of parametric finite-difference methods for the solution of the sine-Gordon equation

A family of finite difference methods is used to transform the initial/boundary-value problem associated with the nonlinear hyperbolic sine-Gordon equation, into a linear algebraic system. Numerical methods are developed by replacing the time and space derivatives by finite-difference approximants. The resulting finite-difference methods are analysed for local truncation errors, stability and convergence. The results of a number of numerical experiments are given.

A fourth order numerical scheme for the one-dimensional sine-Gordon equation

A numerical scheme arising from the use of a fourth order rational approximants to the matrix-exponential term in a three-time level recurrence relation is proposed for the numerical solution of the one-dimensional sine-Gordon (SG) equation already known from the bibliography. The method for its implementation uses a predictor–corrector scheme in which the corrector is accelerated by using the already evaluated corrected values modified predictor–corrector scheme. For the implementation of the corrector, in order to avoid extended matrix evaluations, an auxiliary vector was successfully introduced. Both the predictor and the corrector schemes are analysed for stability. The predictor–corrector/modified predictor–corrector (P-C/MPC) schemes are tested on single and soliton doublets as well as on the collision of breathers and a comparison of the numerical results with the corresponding ones in the bibliography is made. Finally, conclusions for the behaviour of the introduced MPC over the standard P-C scheme are derived.

A second order numerical scheme for the solution of the one-dimensional Boussinesq equation

A predictor–corrector (P-C) scheme is applied successfully to a nonlinear method arising from the use of rational approximants to the matrix-exponential term in a three-time level recurrence relation. The resulting nonlinear finite-difference scheme, which is analyzed for local truncation error and stability, is solved using a P-C scheme, in which the predictor and the corrector are explicit schemes of order 2. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behaviour of the P-C/MPC schemes is tested numerically on the Boussinesq equation already known from the bibliography free of boundary conditions. The numerical results are derived for both the bad and the good Boussinesq equation and conclusions from the relevant known results are derived.