Numerical Algorithm With High Spatial Accuracy for the Fractional Diffusion-Wave Equation With Neumann Boundary Conditions

Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation

Higher-order compact finite difference scheme with multigrid algorithm is applied in this paper for solving one-dimensional and two-dimensional inhomogeneous Helmholtz equations. In two-dimensional case, the suggested scheme has the stencil of twenty one points. An efficient solver multigrid method yields eighth-order accurate approximation on both fine and coarse grids. For the Neumann boundary condition, an eighth-order accurate representation is also developed. The accuracy and efficiency of eighth-order compact difference scheme are exhibited through graphical illustrations and computed results are drafted in tabular form to validate the numerical experiments.

This paper is devoted to the numerical treatment of time fractional diffusion equation with Neumann boundary conditions. A compact difference scheme is derived for solving this problem, by combining the classic finite difference method for Caputo derivative in time, the second order central difference method in space and the compact difference treatment for Neumann boundary conditions. The solvability, stability and convergence of this scheme are rigorously discussed. We prove that the convergence order of this proposed scheme is O(τ2 − α + h2), where τ, α and h are the time step size, the index of fractional derivative and space step size respectively. Numerical experiments are carried out to demonstrate the theoretical analysis.

In this paper, we analyze a compact finite difference scheme for computing a coupled nonlinear Schrodinger equation. The proposed scheme not only conserves the total mass and energy in the discrete level but also is decoupled and linearized in practical computa- tion. Due to the difficulty caused by compact difference on the nonlinear term, it is very hard to obtain the optimal error estimate without any restriction on the grid ratio. In order to overcome the difficulty, we transform the compact difference scheme into a special and equivalent vector form, then use the energy method and some important lemmas to obtain the optimal convergent rate, without any restriction on the grid ratio, at the order of O(h 4 +� 2 ) in the discrete L 1 -norm with time stepand mesh size h. Finally, numerical results are reported to test our theoretical results of the proposed scheme. Mathematics subject classification: 65M06, 65M12.

High-order compact finite difference schemes for solving the regularized long-wave equation

A Three-Point Combined Compact Difference Scheme

A sixth order finite difference scheme for the convection diffusion equation

To investigate the potential of the fourth‐order compact difference scheme within the specific context of numerical atmospheric models, a linear baroclinic adjustment system is discretized, using a variety of candidates for practically meaningful staggered 3D grids. A unified method is introduced to derive the dispersion relationship of the baroclinic geostrophic adjustment process. Eight popular 3D grids are obtained by combining contemporary horizontal staggered grids, such as the Arakawa C and Eliassen grids, with optimal vertical grids, such as the Lorenz and Charney‐Phillip (CP) grids, and their time‐staggered versions. The errors produced on the 3D grids in describing the baroclinic geostrophic adjustment process relative to the differential case are compared in terms of frequency and group velocity components with the elimination of implementation error. The results show that by utilizing the fourth‐order compact difference scheme with high precision, instead of the conventional second‐order centered difference scheme, the errors in describing baroclinic geostrophic adjustment process decrease but only when using the combinations of the horizontally staggered Arakawa C grid and the vertically staggered CP or the C/CP grid, the time‐horizontally staggered Eliassen (EL) grid and vertically staggered CP or the EL/CP grid, the C grid and the vertically time‐staggered versions of Lorenz (LTS) grid or C/LTS grid, EL grid and LTS grid or EL/LTS grid. The errors were found to increase for specific waves on the rest grids. It can be concluded that errors produced on the chosen 3D grids do not universally decrease when using the fourth‐order compact difference scheme; hence, care should be taken when implementing the fourth‐order compact difference scheme, otherwise, the expected benefits may be offset by increased errors. Copyright © 2008 John Wiley & Sons, Ltd.

Two sixth-order, [formula omitted] convergent, and stable compact difference schemes for the generalized Rosenau-KdV-RLW equation

A novel fourth-order three-point compact operator for the nonlinear convection term uux is provided in this paper. The operator makes the numerical analysis of higher-order difference schemes become possible for a wide class of nonlinear evolutionary equations under the unified framework. We take the classical viscous Burgers’ equation as an example and establish a new conservative fourth-order implicit compact difference scheme based on the method of order reduction. A detailed theoretical analysis is carried out by the discrete energy argument and mathematical induction. It is rigorously proved that the difference scheme is conservative, uniquely solvable, stable, and unconditionally convergent in discrete $L^{\infty }$ -norm. The convergence order is two in time and four in space, respectively. Furthermore, we derive a three-level linearized compact difference scheme for viscous Burgers’ equation based on the proposed operator. All numerically theoretical results similar to that of the nonlinear numerical scheme are inherited completely; meanwhile, it is more time saving. Applying the compact operator to other more complex and higher-order nonlinear evolutionary equations is feasible, including Benjamin-Bona-Mahony-Burgers’ equation, Korteweg-de Vries equation, Kuramoto-Sivashinsky equation, and classification to name a few. Numerical results demonstrate that the presented schemes for Burgers’ equation can achieve second-order accuracy in time and fourth-order accuracy in space in $L^{\infty }$ -norm.

Fast high-order compact finite difference schemes are investigated for solving the two-dimensional nonlinear Schrödinger equation with periodic boundary conditions. These schemes are convergent of order s in space, where () with second-order temporal accuracy. The discrete conservation laws and convergence of the finite difference schemes are rigorously demonstrated. Thanks to the circulant matrices resulting from spatial discretization, we significantly reduce the computation complexity and storage requirement of the proposed schemes via fast Fourier transform. Numerical examples are presented to show the accuracy and efficiency of these schemes and verify the theoretical analysis. Moreover, we extend the tenth-order scheme to solve the generalized nonlinear Schrödinger equation.

In this paper, we consider the compact difference approximation of the fourth and second-order schemes on a three-point stencil for Klein–Gordon equations with variable coefficients. Despite the linearity of the differential and difference problems, it is not possible in this case to apply the well-known results on the theory of stability of three-layer operator-difference schemes by A. A. Samarskii. The main purpose is to prove the stability with respect to the initial data and the right-hand side of compact difference schemes in the grid norms L 2 (W h ), W 1 2 (W h ), C (W h ). Using the method of energy inequalities, the corresponding a priori estimates, expressing the stability and convergence of the solution to the difference problem with the assumption h ≤ = h 0, h 0 = const, τ≥h is obtained. The conducted numerical experiment shows how Runge rule is used to determine the different orders of the convergence rate of the difference scheme in the case of two independent variables.

In this article, the high-order upwinding combined compact difference scheme developed in a three-point grid stencil is applied to solve the incompressible Navier-Stokes (NS) and energy equations in three dimensions. The time integrator with symplectic property is employed to approximate the temporal derivative term in inviscid Euler equation so as to numerically retain the embedded Hamiltions and Casimir to get long-time accurate solutions. For the sake of computational efficiency in solving the three-dimensional NS equations, all the calculations will be accelerated using the hybrid CUDA and OpenAcc GPU programing models. The parallel speedup performance compared to the multicore of an Intel Xeon E5-2690V5 CPU is reported.

A combined compact finite difference scheme for predicting the evolution of a mean curvature driven interface

In this paper, we propose a compact difference scheme of second order temporal convergence for the analysis of sub-diffusion fourth-order neutral fractional delay differential equations. In this regard, a difference scheme combining the compact difference operator for spatial discretization along with $$L2-1_{\sigma }$$ formula for Caputo fractional derivative is constructed and analyzed. Unique solvability, stability, and convergence of the proposed scheme are proved using the discrete energy method in $$L_2$$ norm. Established scheme is of second-order convergence in time and fourth-order convergence in spatial dimension, i.e., $$O(\tau ^{3-\alpha }+h^4)$$, where $$\tau$$ and h are time and space mesh sizes respectively and $$\alpha \in (0,1)$$. Finally, some numerical experiments are given to show the authenticity, efficiency, and accuracy of our theoretical results.