An improved numerical scheme for the sine‐Gordon equation in 2+1 dimensions

Abstract

SUMMARYA rational approximant of order 4, which is applied to a three-time-level recurrence relation, is used to transform the initial/boundary-value problem associated with the two-dimensional sine-Gordon (SG) equation arising in the Josephson junctions problem. The resulting non-linear system, which is analyzed for stability, is solved using an appropriate predictor-corrector (P-C) scheme, in which an explicit scheme of order 2 is used as predictor. For the implementation of the corrector, in order to avoid extended matrix evaluations, an auxiliary vector was successfully introduced. In this P-C scheme, a modification in the corrector has been proposed according to which the already evaluated corrected values are considered. The behavior of this P-C scheme is tested numerically on line and ring solitons known from the bibliography regarding the SG equation and conclusions for both undamped and damped problems are derived.

Meshless local Petrov–Galerkin (MLPG) approximation to the two dimensional sine-Gordon equation

“…The examples are chosen for comparison with the results of [23,34,35,24,36,37,26,39] for undamped and damped equations. We need to iterate between finding an estimation for u and solving the system (4.3) for a new u that described in previous section.…”

“…Numerical solutions for the SG equation have given in [32] using two difference schemes, Christiansen and Lomdahl [26] using a generalized leapfrog method, Argyris et al [23] by the finite element approach, Sheng et al [33] by a split cosine scheme, Djidjeli et al [25] using a two-step one-parameter leapfrog scheme, Bratsos [34] using a three time level fourth order explicit finite-difference scheme, Bratsos [35] using a modified predictor-corrector scheme, Bratsos [24] using the method of lines, Bratsos [36] by a third order numerical scheme, Bratsos [37] using a rational approximant of order 4, which is applied to a three time level recurrence relation, Dehghan and Mirzaei [38] by the dual reciprocity boundary element approximation, Mirzaei and Dehghan [39] using continuous linear boundary elements, Dehghan and Shokri [40] using the radial basis functions, and etc. Also the one-dimensional SG equation has been considered widely in numerical and exact solutions.…”

Application of meshless local radial point interpolation (MLRPI) on a one-dimensional inverse heat conduction problem

“…As it will be described, corresponding to each node We obtain one equation. For nodes which are located in the interior of the domain, i.e., for x i 2 interior X, to obtain the discrete equations from the locally weak forms (37), substituting approximation formulas (25) and (26) into local integral Eq. (37) yields…”

“…Hence the domain and boundary integrals in the weak form methods can easily be evaluated over the regularly shaped sub-domains (spheres in 3D or circles in 2D) and their boundaries. In the literature, several meshless weak form methods have been reported such as diffuse element method (DEM) [24], smooth particle hydrodynamic (SPH) [25,26], the reproducing kernel particle method (RKPM) [27], boundary node method (BNM) [28], partition of unity finite element method (PUFEM) [29], finite sphere method (FSM) [30], boundary point interpolation method (BPIM) [31] and boundary radial point interpolation method (BRPIM) [32]. Liu applied the concept of MLPG and developed meshless local radial point interpolation (MLRPI) method [33][34][35].…”

Numerical Simulation of Dam Break Flows Using a Radial Basis Function Meshless Method with Artificial Viscosity

“…In the literature, several meshless weak form techniques have been reported; among others, the smooth particle hydrodynamic method [6] and boundary point interpolation method are worth noticing [7]. e weak forms are used to derive a set of algebraic equations through a numerical integration process using a set of quadrature domain that might be constructed globally or locally in the domain of the problem.…”