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     

On the numerical solution of the Klein‐Gordon equation

A predictor–corrector (P–C) scheme based on the use of rational approximants of second‐order to the matrix‐exponential term in a three‐time level reccurence relation is applied to the nonlinear Klein‐Gordon equation. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. Both the predictor and the corrector scheme are analyzed for local truncation error and stability. The proposed method is applied to problems possessing periodic, kinks and single, double‐soliton waves. The accuracy as well as the long time behavior of the proposed scheme is discussed. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A predictor–corrector scheme for the improved Boussinesq equation

An implicit finite-difference method based on rational approximants of second order to the matrix-exponential term in a three-time level recurrence relation has been proposed for the numerical solution of the improved Boussinesq equation already known from the bibliography. The method, which is analyzed for local truncation error and stability, leads to the solution of a nonlinear system. To overcome this difficulty a predictor–corrector (P–C) scheme in which the predictor is also a second order implicit one is proposed. The efficiency of the proposed method is tested to various wave packets and the results arising from the experiments are compared with the relevant ones known in the bibliography.     

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%).     

A NEW MULTI-SYMPLECTIC SCHEME FOR NONLINEAR “GOOD” BOUSSINESQ EQUATION

The Hamiltonian formulations of the linear “good“ Boussinesq (L.G.B.) equation and the multi-symplectic formulation of the nonlinear “good“ Boussinesq (N.G.B.) equation are considered. For the multi-symplectic formulation, a new fifteen-point difference scheme which is equivalent to the multi-symplectic Preissmann integrator is derived. We also present numerical experiments, which show that the symplectic and multisymplectic schemes have excellent long-time numerical behavior.     

广义Boussinesq方程的多辛方法

广义Boussinesq方程作为一类重要的非线性方程有着许多有趣的性质,基于Hamilton空间体系的多辛理论研究了广义Boussinesq方程的数值解法,构造了一种等价于多辛Box格式的新隐式多辛格式,该格式满足多辛守恒律、局部能量守恒律和局部动量守恒律．对广义Boussinesq方程孤子解的数值模拟结果表明,该多辛离散格式具有较好的长时间数值稳定性．     

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     

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

The method of lines is used to transform the initial/boundary-value problem associated with the two-dimensional sine-Gordon equation in two space variables into a second-order initial-value problem. The finite-difference methods are developed by replacing the matrix-exponential term in a recurrence relation with rational approximants. The resulting finite-difference methods are analyzed for local truncation error, stability and convergence. To avoid solving the nonlinear system a predictor–corrector scheme using the explicit method as predictor and the implicit as corrector is applied. Numerical solutions for cases involving the most known from the bibliography line and ring solitons are given.     

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.     

A meshless based numerical technique for traveling solitary wave solution of Boussinesq equation

In this paper, we employ the boundary-only meshfree method to find out numerical solution of the classical Boussinesq equation in one dimension. The proposed method in the current paper is a combination of boundary knot method and meshless analog equation method. The boundary knot technique is an integration free, boundary-only, meshless method which is used to avoid the known disadvantages of the method of fundamental solution. Also, we use the meshless analog equation method to replace the nonlinear governing equation with an equivalent nonhomogeneous linear equation. A predictor-corrector scheme is proposed to solve the resulted differential equation of the collocation. The numerical results and conclusions are obtained for both the ‘good’ and the ‘bad’ Boussinesq equations.     

A global extrapolated procedure for the boussinesq equation

In this paper we use the global extrapolation procedure to study the Boussinesq equation in one dimension. The application of a parametric finite-difference method leads to a three-time level nonlinear scheme, which is analyzed for local truncation error, stability and convergence. Then, the nonlinear term of the equation is properly linearized so, that the scheme becomes linear. The results of a number of numerical experiments for the single-soliton wave are given.     

Conservative finite difference methods for fractional Schrödinger–Boussinesq equations and convergence analysis

In this paper, two conservative finite difference schemes for fractional Schrödinger–Boussinesq equations are formulated and investigated. The convergence of the nonlinear fully implicit scheme is established via discrete energy method, while the linear semi‐implicit scheme is analyzed by means of mathematical induction method. Our schemes are proved to preserve the total mass and energy in discrete level. The numerical results are given to confirm the theoretical analysis.     

New conservative schemes with discrete variational derivatives for nonlinear wave equations

New conservative finite difference schemes for certain classes of nonlinear wave equations are proposed. The key tool there is “discrete variational derivative”, by which discrete conservation property is realized. A similar approach for the target equations was recently proposed by Furihata, but in this paper a different approach is explored, where the target equations are first transformed to the equivalent system representations which are more natural forms to see conservation properties. Applications for the nonlinear Klein–Gordon equation and the so-called “good” Boussinesq equation are presented. Numerical examples reveal the good performance of the new schemes.     

Implicit–explicit predictor–corrector schemes for nonlinear parabolic differential equations

In the present paper, a family of predictor–corrector (PC) schemes are developed for the numerical solution of nonlinear parabolic differential equations. Iterative processes are avoided by use of the implicit–explicit (IMEX) methods. Moreover, compared to the predictor schemes, the proposed methods usually have superior accuracy and stability properties. Some confirmation of these are illustrated by using the schemes on the well-known Fisher’s equation.     

A non‐standard finite difference scheme for an advection‐diffusion‐reaction equation

In this note, a non‐standard finite difference (NSFD) scheme is proposed for an advection‐diffusion‐reaction equation with nonlinear reaction term. We first study the diffusion‐free case of this equation, that is, an advection‐reaction equation. Two exact finite difference schemes are constructed for the advection‐reaction equation by the method of characteristics. As these exact schemes are complicated and are not convenient to use, an NSFD scheme is derived from the exact scheme. Then, the NSFD scheme for the advection‐reaction equation is combined with a finite difference space‐approximation of the diffusion term to provide a NSFD scheme for the advection‐diffusion‐reaction equation. This new scheme could preserve the fixed points, the positivity, and the boundedness of the solution of the original equation. Numerical experiments verify the validity of our analytical results. Copyright © 2014 JohnWiley & Sons, Ltd.     

A parametric scheme for the numerical solution of the Boussinesq equation

A parametric scheme is proposed for the numerical solution of the nonlinear Boussinesq equation. The numerical method is developed by approximating the time and the space partial derivatives by finite-difference re placements and the nonlinear term by an appropriate linearized scheme. The resulting finite-difference method is analyzed for local truncation error and stability. The results of a number of numerical experiments are given for both the single and the double-soliton wave.     

A staggered-grid numerical algorithm for the extended Boussinesq equations

A second-order accurate numerical scheme is developed to solve Nwogu’s extended Boussinesq equations. A staggered-grid system is introduced with the first-order spatial derivatives being discretized by the fourth-order accurate finite-difference scheme. For the time derivatives, the fourth-order accurate Adams predictor–corrector method is used. The numerical method is validated against available analytical solutions, other numerical results of Navier–Stokes equations, and experimental data for both 1D and 2D nonlinear wave transformation problems. It is shown that the new algorithm has very good conservative characteristics for mass calculation. As a result, the model can provide accurate and stable results for long-term simulation. The model has proven to be a useful modeling tool for a wide range of water wave problems.     

Error estimate for the upwind finite volume method for the nonlinear scalar conservation law

In this paper we estimate the error of upwind first order finite volume schemes applied to scalar conservation laws. As a first step, we consider standard upwind and flux finite volume scheme discretization of a linear equation with space variable coefficients in conservation form. We prove that, in spite of their lack of consistency, both schemes lead to a first order error estimate. As a final step, we prove a similar estimate for the nonlinear case. Our proofs rely on the notion of geometric corrector, introduced in our previous paper by Bouche et al. (2005) [24] in the context of constant coefficient linear advection equations.     

An unstructured FEM model based on Boussinesq equations and its application to the calculation of multidirectional wave run-up in a cylinder group

In this paper, a numerical model based on the improved Boussinesq equations derived by Beji and Nadaoka [5] is first developed using unstructured finite element technique. A locally rotated coordinate system is introduced to improve the treatment for the fully reflective boundaries whose orientation does not coincide with the coordinate system. The Adams–Bashforth–Moulton predictor–corrector scheme is used for time integration. Typical examples are employed to validate the numerical model. Based on the developed model, multidirectional wave propagation through a cylinder group is numerically calculated and the effects of the wave directionality on the waves in the group and the wave run-up on the cylinders are investigated. Numerical results show that the wave directionality has considerable effect on the wave run-up in the cylinder group.     

Substructuring tools for probabilistic analysis of instrumented nonlinear moving oscillator–beam systems

The study considers the application of finite element modeling, combined with numerical solutions of governing stochastic differential equations, to analyze instrumented nonlinear moving vehicle–structure systems. The focus of the study is on achieving computational efficiency by deploying, within a single modeling framework, three substructuring schemes with different methodological moorings. The schemes considered include spatial substructuring schemes (involving free-interface coupling methods), a spatial mesh partitioning scheme for governing stochastic differential equations (involving the use of a predictor corrector method with implicit integration schemes for linear regions and explicit schemes for local nonlinear regions), and application of the Rao–Blackwellization scheme (which permits the use of Kalman's filtering for linear substructures and Monte Carlo filters for nonlinear substructures). The main effort in this work is expended on combining these schemes with provisions for interfacing of the substructures by taking into account the relative motion of the vehicle and the supporting structure. The problem is formulated with reference to an archetypal beam and multi-degrees of freedom moving oscillator with spatially localized nonlinear characteristics. The study takes into account imperfections in mathematical modeling, guide way unevenness, and measurement noise. The numerical results demonstrate notable reduction in computational effort achieved on account of introduction of the substructuring schemes.