A compact finite difference method for the solution of the generalized Burgers–Fisher equation

In this article, numerical solutions of the generalized Burgers–Fisher equation are obtained using a compact finite difference method with minimal computational effort. To verify this, a combination of a sixth‐order compact finite difference scheme in space and a low‐storage third‐order total variation diminishing Runge–Kutta scheme in time have been used. The computed results with the use of this technique have been compared with the exact solution to show the accuracy of it. The approximate solutions to the equation have been computed without transforming the equation and without using linearization. Comparisons indicate that there is a very good agreement between the numerical solutions and the exact solutions in terms of accuracy. The present method is seen to be a very good alternative to some existing techniques for realistic problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Conservative compact finite difference scheme for the N‐coupled nonlinear Klein–Gordon equations

In this article, a compact finite difference method is developed for the periodic initial value problem of the N‐coupled nonlinear Klein–Gordon equations. The present scheme is proved to preserve the total energy in the discrete sense. Due to the difficulty in obtaining the priori estimate from the discrete energy conservation law, the cut‐off function technique is employed to prove the convergence, which shows the new scheme possesses second order accuracy in time and fourth order accuracy in space, respectively. Additionally, several numerical results are reported to confirm our theoretical analysis. Lastly, we apply the reliable method to simulate and study the collisions of solitary waves numerically.     

Compact finite difference scheme for the solution of a time fractional partial integro‐differential equation with a weakly singular kernel

In this paper, we develop a high‐order finite difference scheme for the solution of a time fractional partial integro‐differential equation with a weakly singular kernel. The fractional derivative is used in the Riemann‐Liouville sense. We prove the unconditional stability and convergence of scheme using energy method and show that the convergence order is . We provide some numerical experiments to confirm the efficiency of suggested scheme. The results of numerical experiments are compared with analytical solutions to show the efficiency of proposed scheme. It is illustrated that the numerical results are in good agreement with theoretical ones.     

High‐order compact finite difference and laplace transform method for the solution of time‐fractional heat equations with dirchlet and neumann boundary conditions

The work presents a novel coupling of the Laplace Transform and the compact fourth‐order finite‐difference discretization scheme for the efficient and accurate solution of linear time‐fractional nonhomogeneous diffusion equations subject to both Dirichlet and Neumann boundary conditions. A translational transformation of the dependent variable ensures the Caputo derivative is aligned with the Riemann‐Louiville fractional derivative. The resulting scheme is computationally efficient and shown to be uniquely solvable in all cases, accurate and convergent to in the spatial domain. The convergence rates in the temporal domain are contour dependent but exhibit geometric convergence. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1184–1199, 2016     

The combination of collocation,finite difference,and multigrid methods for solution of the two‐dimensional wave equation

In this article, we apply compact finite difference approximations of orders two and four for discretizing spatial derivatives of wave equation and collocation method for the time component. The resulting method is unconditionally stable and solves the wave equation with high accuracy. The solution is approximated by a polynomial at each grid point that its coefficients are determined by solving a linear system of equations. We employ the multigrid method for solving the resulted linear system. Multigrid method is an iterative method which has grid independently convergence and solves the linear system of equations in small amount of computer time. Numerical results show that the compact finite difference approximation of fourth order, collocation and multigrid methods produce a very efficient method for solving the wave equation. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

A linearized,decoupled, and energy‐preserving compact finite difference scheme for the coupled nonlinear Schrödinger equations

In this article, a decoupled and linearized compact finite difference scheme is proposed for solving the coupled nonlinear Schrödinger equations. The new scheme is proved to preserve the total mass and energy which are defined by using a recursion relationship. Besides the standard energy method, an induction argument together with an H¹ technique are introduced to establish the optimal point‐wise error estimate of the proposed scheme. Without imposing any constraints on the grid ratios, the convergence order of the numerical solution is proved to be of with mesh size h and time step τ. Numerical results are reported to verify the theoretical analysis, and collision of two solitary waves are also simulated. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 840–867, 2017     

High‐order difference scheme for the solution of linear time fractional klein–gordon equations

In this article, we apply a high‐order difference scheme for the solution of some time fractional partial differential equations (PDEs). The time fractional Cattaneo equation and the linear time fractional Klein–Gordon and dissipative Klein–Gordon equations will be investigated. The time fractional derivative which has been described in the Caputo's sense is approximated by a scheme of order , and the space derivative is discretized with a fourth‐order compact procedure. We will prove the solvability of the proposed method by coefficient matrix property and the unconditional stability and ‐convergence with the energy method. Numerical examples demonstrate the theoretical results and the high accuracy of the proposed scheme. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1234–1253, 2014     

Long time error analysis of the fourth‐order compact finite difference methods for the nonlinear Klein–Gordon equation with weak nonlinearity

We present the fourth‐order compact finite difference (4cFD) discretizations for the long time dynamics of the nonlinear Klein–Gordon equation (NKGE), while the nonlinearity strength is characterized by ?^p with a constant p ∈ ?⁺ and a dimensionless parameter ? ∈ (0, 1] . Based on analytical results of the life‐span of the solution, rigorous error bounds of the 4cFD methods are carried out up to the time at O(?^?p) . We pay particular attention to how error bounds depend explicitly on the mesh size h and time step τ as well as the small parameter ? ∈ (0, 1] , which indicate that, in order to obtain ‘correct’ numerical solutions up to the time at O(?^?p) , the ? ‐scalability (or meshing strategy requirement) of the 4cFD methods should be taken as: h = O(?^p/4) and τ = O(?^p/2) . It has better spatial resolution capacity than the classical second order central difference methods. By a rescaling in time, it is equivalent to an oscillatory NKGE whose solution propagates waves with wavelength at O(1) in space and O(?^p) in time. It is straightforward to get the error bounds of the oscillatory NKGE in the fixed time. Finally, numerical results are provided to confirm our theoretical analysis.