A Compact Difference Scheme for Time Fractional Diffusion Equation with Neumann Boundary Conditions

A fourth-order compact algorithm is discussed for solving the time fractional diffusion-wave equation with Neumann boundary conditions. The $$L1$$ discretization is applied for the time-fractional derivative and the compact difference approach for the spatial discretization. The unconditional stability and the global convergence of the compact difference scheme are proved rigorously, where a new inner product is introduced for the theoretical analysis. The convergence order is $$\mathcal{O }(\tau ^{3-\alpha }+h^4)$$ in the maximum norm, where $$\tau $$ is the temporal grid size and $$h$$ is the spatial grid size, respectively. In addition, a Crank---Nicolson scheme is presented and the corresponding error estimates are also established. Meanwhile, a compact ADI difference scheme for solving two-dimensional case is derived and the global convergence order of $$\mathcal{O }(\tau ^{3-\alpha }+h_1^4+h_2^4)$$ is given. Then extension to the case with Robin boundary conditions is also discussed. Finally, several numerical experiments are included to support the theoretical results, and some comparisons with the Crank---Nicolson scheme are presented to show the effectiveness of the compact scheme.

In this article, a linear block-centred finite difference method for the nonlinear distributed-ordered diffusion differential equation with Neumann boundary condition is constructed and analysed. We derive rigorous error estimates and obtain the optimal order of convergence both for pressure and velocity, where are the step sizes in time, distributed order and space, respectively. Finally, some numerical experiments are presented to show the efficiency and accuracy of our constructed scheme.

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

In this paper, we consider the local discontinuous Galerkin (LDG) finite element method for one-dimensional linear time-fractional Tricomi-type equation (TFTTE), which is obtained from the standard one-dimensional linear Tricomi-type equation by replacing the first-order time derivative with a fractional derivative (of order ?, with 1?<?????2). The proposed LDG is based on LDG finite element method for space and finite difference method for time. We prove that the method is unconditionally stable, and the numerical solution converges to the exact one with order O(h k?+?1?+?? 2), where h, ? and k are the space step size, time step size, polynomial degree, respectively. The comparison of the LDG results with the exact solutions is made, numerical experiments reveal that the LDG is very effective.

In this paper, we consider two types of space-time fractional diffusion equations(STFDE) on a finite domain. The equation can be obtained from the standard diffusion equation by replacing the second order space derivative by a Riemann-Liouville fractional derivative of order s (1 <s ≤ 2), and the first order time derivative by a Caputo fractional derivative of order ? (0 <? ≤ 1). For the 0 <? < 1 case, we present two schemes to approximate the time derivative and finite element methods for the space derivative, the optimal convergence rate can be reached O(?2?? + h2) and O(?2 + h2), respectively, in which ? is the time step size and h is the space step size. And for the case ? = 1, we use the Crank-Nicolson scheme to approximate the time derivative and obtain the optimal convergence rate O(?2 + h2) as well. Some numerical examples are given and the numerical results are in good agreement with the theoretical analysis.

Efficient algorithm and convergence analysis of conservative SAV compact difference scheme for Boussinesq Paradigm equation

In this article, we consider the finite element methods (FEM) for Grwünwald–Letnikov time-fractional diffusion equation, which is obtained from the standard two-dimensional diffusion equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0 < α < 1). The proposed method is based on high-order FEM for spacial discretization and finite difference method for time discretization. We prove that the method is unconditionally stable, and the numerical solution converges to the exact one with order O(h r+1 + τ2-α), where h, τ and r are the space step size, time step size and polynomial degree, respectively. A numerical example is presented to verify the order of convergence.

Two higher-order compact finite difference approaches, the Hermitian and the Lax-Wendroff are examined by applying them to the viscous Burgers' equation. The difference equations obtained by the two methods were integrated in time through the third-order Strong-Stability-Preserving Runge-Kutta scheme. Absolute errors are computed by using an exact solution. The results are also compared with a second order central difference scheme. The Hermitian approach is far easier to implement. On uniform grids the Lax-Wendroff scheme produces smaller errors during the initial stages, but both methods are equally good for larger durations of integration. The convergence rate of the Hermitian scheme is slightly higher than the Lax-Wendroff scheme although both are of fourth order. Both schemes are unstable beyond a certain step size in time and space. When numerical boundary conditions are imposed, second-order conditions produce the best results whereas linear extrapolation proved to be the worst. It was also observed that large domains were required to implement the numerical boundary conditions properly. There was no detrimental effect on the accuracy of the results obtained through either of the two schemes when the size of the domain was greatly increased. Both schemes showed remarkable improvement in accuracy when clustered grids were employed. However much smaller time steps are required for stable solutions.

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.

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 &amp; Sons, Ltd.

A Three-Point Combined Compact Difference Scheme

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.

In this article, two sets of fourth‐order compact finite difference schemes are constructed for solving heat‐conducting problems of two or three dimensions, respectively. Both problems are with Neumann boundary conditions. These works are extensions of our earlier work (Zhao et al., Fourth order compact schemes of a heat conduction problem with Neumann boundary conditions, Numerical Methods Partial Differential Equations, to appear) for the one‐dimensional case. The local one‐dimensional method is employed to construct these two sets of schemes, which are proved to be globally solvable, unconditionally stable, and convergent. Numerical examples are also provided. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

Linearized ADI schemes for two-dimensional space-fractional nonlinear Ginzburg–Landau equation