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AbstractThe study aims at the development and theoretical analyses of a box method used to solve a kind of hyperbolic equations with shifts. By using the discrete energy method, it is shown that numerical solutions converge to exact solutions with an order of O(τ2 + h2) in L∞-norm as τ = O(h). Here, τ and h are temporal and spatial meshsizes, respectively. According to local truncation error, the asymptotic expansion formula of the numerical solutions is derived by introducing two auxiliary problems. By applying this asymptotic expansion formula, a class of Richardson extrapolations methods have been derived to achieve extrapolation solutions with an order of O(τ4 + h4) in L∞-norm. Numerical results show the high performance of the proposed methods and the exactness of theoretical findings. [ABSTRACT FROM AUTHOR]

Du Fort-Frankel (DFF) finite difference method (FDM) was proposed for linear diffusion equations with periodic boundary conditions by Du Fort and Frankel in 1953. It is an explicit and unconditionally von Neumann stable scheme. Thus, it is very easy to be implemented and suitable for long-term simulations. However, there has been no research work on numerical solutions of sine-Gordon equations (SGE) and nonlinear coupled sine-Gordon equations (CSGEs) by using energy-preserving Du Fort-Frankel finite difference methods (EP-DFF-FDMs). In this study, two classes of weighted EP-DFF-FDMs, which are devised by combining DFF FDMs with invariant energy quadratization methods (IEQMs), are suggested for numerical simulations of SGE and CSGEs, respectively. By using the discrete energy method, it is shown that their solutions satisfy the discrete energy conservative laws, and converge to exact solutions with an order of O (τ 2 + h x 2 + h y 2 + (τ h x ) 2 + (τ h y ) 2) in H1-norm. Here, τ denotes time increment, while hx and hy represent spacing grids in x- and y-dimensions, respectively. What is more, our methods with parameter 휃 ≥ 1/4 are unconditionally stable in L2-norm though they are explicit schemes. Finally, numerical results confirm the exactness of theoretical findings, and the superiorities of our algorithms over some existent algorithms in terms of computational efficiency and the ability to conserve the discrete energy. [ABSTRACT FROM AUTHOR]

In this paper, a three‐level finite difference method (FDM), which preserves energy and mass conservative laws, is first derived for one‐dimensional (1D) nonlinear coupled Schrödinger–Boussinesq equations (NCSBEs). Using the discrete energy analysis method, error estimations have been proven to be 𝒪(τ2+hx2) in L2‐, H1‐, and L∞‐norms, respectively. Secondly, this energy‐ and mass‐preserving FDM (EM‐FDM) is generalized to solve NCSBEs in two dimensions. Also, by the discrete energy method, it is shown that numerical solutions provided by the generalized EM‐FDM method for two‐dimensional (2D) NCSBEs converge to exact solutions with a convergent rate of 𝒪(τ2+hx2+hy2) in L2‐ and H2‐norms. By the embedding relations H2(Ω)⊂C(Ω‾), we further derive that the numerical solutions are convergent with an order of 𝒪(τ2+hx2+hy2) in H1‐ and L∞‐norms. Finally, numerical results confirm the exactness of theoretical results, and the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

As we know, second-order backward differentiation formula (BDF2) is L-stable scheme, which can damp unwanted finite oscillations, while Crank-Nicolson (C-N) method is only A-stable scheme and usually causes numerical oscillations. Thus, BDF2 may be more popular than C-N. However, up to now, most of studies focus on the error analysis of the compact alternating direction implicit (ADI) method on the basis of C-N method for temporal discretization in L ∞ -norm. This paper is devoted to the convergence analysis of a compact multi-step ADI method used to solve a two-dimensional (2D) parabolic equation in L 2 -, H 1 - and L ∞ -norms. Besides, the existence of the ADI solution is also analyzed in detail. A class of Richardson extrapolation algorithms are established to improve computational efficiency. By introducing a transformation, our algorithms are easily generalized to the solution of convection-diffusion problems with constant coefficients. Numerical results confirm the performance of our algorithms and support theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, by applying order reduction approach, a second-order accurate box scheme is established to solve a nonlinear delayed convection-diffusion equations with Neumann boundary conditions. By the discrete energy method, it is shown that the difference scheme is uniquely solvable, and has a convergence rate of with respect to - norm in constrained and non-constrained temporal grids. Besides, for constrained temporal step, a Richardson extrapolation method (REM) used along with the box scheme, which makes final solution third-order accurate in both time and space, is developed in detail. Finally, numerical results confirm the accuracy and efficiency of our solvers. [ABSTRACT FROM AUTHOR]

In this article, a high-order compact alternating direction implicit method combined with a Richardson extrapolation technique is developed to solve a class of two-dimensional nonlinear delay hyperbolic differential equations. The solvability, stability and convergence of the method are analysed simultaneously inL2- andH1-norms by the discrete energy method. Numerical experiments are provided to demonstrate the accuracy and efficiency of the schemes. [ABSTRACT FROM AUTHOR]

In this article, a new compact alternating direction implicit finite difference scheme is derived for solving a class of 3-D nonlinear evolution equations. By the discrete energy method, it is shown that the new difference scheme has good stability and can attain second-order accuracy in time and fourth-order accuracy in space with respect to the discrete H1 -norm. A Richardson extrapolation algorithm is applied to achieve fourth-order accuracy in temporal dimension. Numerical experiments illustrate the accuracy and efficiency of the extrapolation algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 [ABSTRACT FROM AUTHOR]