A numerical method for the one‐dimensional sine‐Gordon equation

A numerical method based on a predictor–corrector (P‐C) scheme arising from the use of rational approximants of order 3 to the matrix‐exponential term in a three‐time level recurrence relation is applied successfully to the one‐dimensional sine‐Gordon equation, already known from the bibliography. In this P‐C scheme a modification in the corrector (MPC) has been proposed according to which the already evaluated corrected values are considered. The method, which uses as predictor an explicit finite‐difference scheme arising from the second order rational approximant and as corrector an implicit one, has been tested numerically on the single and the soliton doublets. Both the predictor and the corrector schemes are analyzed for local truncation error and stability. From the investigation of the numerical results and the comparison of them with other ones known from the bibliography it has been derived that the proposed P‐C/MPC schemes at least coincide in terms of accuracy with them. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

A predictor‐‐corrector scheme for the sine‐Gordon equation

A predictor–corrector scheme is developed for the numerical solution of the sine‐Gordon equation using the method of lines approach. The solution of the approximating differential system satisfies a recurrence relation, which involves the cosine function. Pade' approximants are used to replace the cosine function in the recurrence relation. The resulting schemes are analyzed for order, stability, and convergence. Numerical results demonstrate the efficiency and accuracy of the predictor–corrector scheme over some well‐known existing methods. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 133–146, 2000     

A second‐order accurate difference scheme for the two‐dimensional Burgers' system

In this article, a Crank‐Nicolson‐type finite difference scheme for the two‐dimensional Burgers' system is presented. The existence of the difference solution is shown by Brouwer fixed‐point theorem. The uniqueness of the difference solution and the stability and L₂ convergence of the difference scheme are proved by energy method. An iterative algorithm for the difference scheme is given in detail. Furthermore, a linear predictor–corrector method is presented. The numerical results show that the predictor–corrector method is also convergent with the convergence order of two in both time and space. At last, some comments are provided for the backward Euler scheme. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

The boundary integral equation approach for numerical solution of the one‐dimensional Sine‐Gordon equation

This article describes a numerical method based on the boundary integral equation and dual reciprocity method for solving the one‐dimensional Sine‐Gordon (SG) equation. The time derivative is approximated by the time‐stepping method and a predictor–corrector scheme is employed to deal with the nonlinearity which appears in the problem. Numerical results are presented for some problems to demonstrate the usefulness and accuracy of this approach. In addition, the conservation of energy in SG equation is investigated. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

An exponential time differencing method of lines for the Burgers and the modified Burgers equations

A second–order exponential time differencing scheme using the method of lines is developed in this article for the numerical solution of the Burgers and the modified Burgers equations. For each case, the resulting nonlinear system is solved explicitly using a modified predictor‐corrector method. The efficiency of the method introduced is tested by comparing experimental results with others selected from the available literature.     

A finite element approach for analysis and computational modelling of coupled reaction diffusion models

In this article, we establish the existence and uniqueness of solutions to the coupled reaction–diffusion models using Banach fixed point theorem. The Galerkin finite element method is used for the approximation of solutions, and an a priori error estimate is derived for such approximations. A scheme is proposed by combining the Crank–Nicolson and the predictor–corrector methods for the time discretization. Some numerical examples are considered to illustrate the accuracy and efficiency of the proposed scheme. It is found that the scheme is second‐order convergent. In addition, nonuniform grids are used in some cases to enhance the accuracy of the scheme.     

New predictor‐corrector scheme for solving nonlinear differential equations with Caputo‐Fabrizio operator

In this paper, we develop a new, simple, and accurate scheme to obtain approximate solution for nonlinear differential equation in the sense of Caputo‐Fabrizio operator. To derive this new predictor‐corrector scheme, which suits on Caputo‐Fabrizio operator, firstly, we obtain the corresponding initial value problem for the differential equation in the Caputo‐Fabrizio sense. Hence, by fractional Euler method and fractional trapeziodal rule, we obtain the predictor formula as well as corrector formula. Error analysis for this new method is derived. To test the validity and simplicity of this method, some illustrative examples for nonlinear differential equations are solved.     

The dual reciprocity boundary integral equation technique to solve a class of the linear and nonlinear fractional partial differential equations

In this paper, we apply the boundary integral equation technique and the dual reciprocity boundary elements method (DRBEM) for the numerical solution of linear and nonlinear time‐fractional partial differential equations (TFPDEs). The main aim of the present paper is to examine the applicability and efficiency of DRBEM for solving TFPDEs. We employ the time‐stepping scheme to approximate the time derivative, and the method of linear radial basis functions is also used in the DRBEM technique. This method is improved by using a predictor–corrector scheme to overcome the nonlinearity that appears in the nonlinear problems under consideration. To confirm the accuracy of the new approach, several examples are presented. The convergence of the DRBEM is studied numerically by comparing the exact solutions of the problems under investigation. Copyright © 2015 John Wiley & Sons, Ltd.     

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. This research was co-funded by E.U. (75%) and by the Greek Government (25%).     

Fourth‐order compact scheme for the one‐dimensional sine‐Gordon equation

Finite difference scheme to the generalized one‐dimensional sine‐Gordon equation is considered in this paper. After approximating the second order derivative in the space variable by the compact finite difference, we transform the sine‐Gordon equation into an initial‐value problem of a second‐order ordinary differential equation. Then Padé approximant is used to approximate the time derivatives. The resulting fully discrete nonlinear finite‐difference equation is solved by a predictor‐corrector scheme. Both Dirichlet and Neumann boundary conditions are considered in our proposed algorithm. Stability analysis and error estimate are given for homogeneous Dirichlet boundary value problems using energy method. Numerical results are given to verify the condition for stability and convergence and to examine the accuracy and efficiency of the proposed algorithm. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

New approach to the Lax‐Wendroff modified differential equation for linear and nonlinear advection

The paper presents an enhanced analysis of the Lax‐Wendroff difference scheme—up to the eighth‐order with respect to time and space derivatives—of the modified‐partial differential equation (MDE) of the constant‐wind‐speed advection equation. The modified equation has been so far derived mainly as a fourth‐order equation. The Π ‐form of the first differential approximation (differential approximation or equivalent equation) derived by expressing the time derivatives in terms of the space derivatives is used for presenting the MDE. The obtained coefficients at higher order derivatives are analyzed for indications of the character of the dissipative and dispersive errors. The authors included a part of the stencil applied for determining the modified differential equation up to the eighth‐order of the analyzed modified differential equation for the second‐order Lax‐Wendroff scheme. Neither the derived coefficients at the space derivatives of order p ∈ (7 – 8) in the modified differential equation for the Lax‐Wendroff difference scheme nor the results of analyses on the basis of these coefficients of the group velocity, phase shift errors, or dispersive and dissipative features of the scheme have been published. The MDEs for 2 two‐step variants of the Lax‐Wendroff type difference schemes and the MacCormack predictor–corrector scheme (see MacCormack's study) constructed for the scalar hyperbolic conservation laws are also presented in this paper. The analysis of the inviscid Burgers equation solution with the initial condition in a form of a shock wave has been discussed on their basis. The inviscid Burgers equation with the source is also presented. The theory of MDE started to develop after the paper of C. W. Hirt was published in 1968.     

A fourth algebraic order trigonometrically fitted predictor–corrector scheme for IVPs with oscillating solutions

A scheme of trigonometrically fitted predictor–corrector (P–C) Adams–Bashforth–Moulton methods is constructed in this paper. Our new P–C method is based on the third order Adams–Bashforth scheme (as predictor) and on the fourth order Adams–Moulton scheme (as corrector). We tested the efficiency of our newly developed scheme against well known methods, with excellent results. The numerical experimentation showed that our method is considerably more efficient compared to well known methods used for the numerical solution of initial value problems with oscillating solutions.     

A moving Kriging‐based MLPG method for nonlinear Klein–Gordon equation

In this paper, the meshless local Petrov–Galerkin approximation is proposed to solve the 2‐D nonlinear Klein–Gordon equation. We used the moving Kriging interpolation instead of the MLS approximation to construct the meshless local Petrov–Galerkin shape functions. These shape functions possess the Kronecker delta function property. The Heaviside step function is used as a test function over the local sub‐domains. Here, no mesh is needed neither for integration of the local weak form nor for construction of the shape functions. So the present method is a truly meshless method. We employ a time‐stepping method to deal with the time derivative and a predictor–corrector scheme to eliminate the nonlinearity. Several examples are performed and compared with analytical solutions and with the results reported in the extant literature to illustrate the accuracy and efficiency of the presented method. Copyright © 2016 John Wiley & Sons, Ltd.     

Trigonometrically fitted predictor–corrector methods for IVPs with oscillating solutions

In this paper we develop a trigonometrically fitted predictor–corrector (P–C) scheme, which is based on the well-known two-step second-order Adams–Bashforth method (as predictor) and on the third-order Adams–Moulton method (as corrector). Numerical experiments show that the new trigonometrically fitted P–C method is substantially more efficient than widely used methods for the numerical solution of initial-value problems (IVPs) with oscillating solutions.     

Runge–Kutta–Nyström‐type parallel block predictor–corrector methods

This paper describes the construction of block predictor–corrector methods based on Runge–Kutta–Nyström correctors. Our approach is to apply the predictor–corrector method not only with stepsize h, but, in addition (and simultaneously) with stepsizes a _i h, i = 1 ...,r. In this way, at each step, a whole block of approximations to the exact solution at off‐step points is computed. In the next step, these approximations are used to obtain a high‐order predictor formula using Lagrange or Hermite interpolation. Since the block approximations at the off‐step points can be computed in parallel, the sequential costs of these block predictor–corrector methods are comparable with those of a conventional predictor–corrector method. Furthermore, by using Runge–Kutta–Nyström corrector methods, the computation of the approximation at each off‐step point is also highly parallel. Numerical comparisons on a shared memory computer show the efficiency of the methods for problems with expensive function evaluations.     

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.      

The arbitrary order implicit multistep schemes of exponential fitting and their applications

We develop the arbitrary order implicit multistep schemes of exponential fitting (EF) for systems of ordinary differential equations. We use an explicit EF scheme to predict an approximation, and then use an implicit EF scheme to correct this prediction. This combination is called a predictor–corrector EF method. We demonstrate the accuracy and efficiency of the new predictor–corrector methods via application to a variety of test cases and comparison with other analytical and numerical results. The numerical results show that the schemes are highly accurate and computationally efficient.     

A Flux Predictor–Corrector Scheme for Solving a 3D Heat Transfer Problem

In this paper, a flux predictor–corrector scheme in the three-dimensional case is proposed and studied. This scheme has no shortcomings of a scheme constructed on the basis of a Douglas–Gunn prototype-scheme. Numerical experiments with the scheme proposed in this paper demonstrate second-order accuracy.     

On operator split technique for the time integration within finite strain viscoplasticity in explicit FEM

In this contribution, the operator split technique is applied to the time integration within viscoplasticity for explicit FEM. As an example, the finite strain viscoplastic material model of Shutov and Kreißig is analyzed. In the new solution scheme, some evolution equations are solved using an explicit update formula for implicit time stepping. The solution procedure is split into three steps: an elastic predictor and two viscoplastic corrector steps. Aspects of accuracy and stability of the algorithm are discussed. As shown, the proposed method is superior compared to a fully explicit integration of evolution equations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A meshless method using the radial basis functions for numerical solution of the regularized long wave equation

This article discusses on the solution of the regularized long wave (RLW) equation, which is introduced to describe the development of the undular bore, has been used for modeling in many branches of science and engineering. A numerical method is presented to solve the RLW equation. The main idea behind this numerical simulation is to use the collocation and approximating the solution by radial basis functions (RBFs). To avoid solving the nonlinear system, a predictor‐corrector scheme is proposed. Several test problems are given to validate the new technique. The numerical simulation, includes the propagation of a solitary wave, interaction of two positive solitary waves, interaction of a positive and a negative solitary wave, the evaluation of Maxwellian pulse into stable solitary waves and the development of an undular bore. The three invariants of the motion are calculated to determine the conservation properties of the algorithm. The results of numerical experiments are compared with analytical solution and with those of other recently published methods to confirm the accuracy and efficiency of the presented scheme.© 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010