Alternating Direction Implicit Difference Scheme Research Articles

Compact ADI Difference Scheme for the 2D Time Fractional Nonlinear Schrödinger Equation

In this paper, we will introduce a compact alternating direction implicit (ADI) difference scheme for solving the two-dimensional (2D) time fractional nonlinear Schrödinger equation. The difference scheme is constructed by using the L1−2−3 formula to approximate the Caputo fractional derivative in time and the fourth-order compact difference scheme is adopted in the space direction. The proposed difference scheme with a convergence accuracy of O(τ1+α+hx4+hy4)(α∈(0,1)) is obtained by adding a small term, where τ, hx, hy are the temporal and spatial step sizes, respectively. The convergence and unconditional stability of the difference scheme are obtained. Moreover, numerical experiments are given to verify the accuracy and efficiency of the difference scheme.

The High-Order ADI Difference Method and Extrapolation Method for Solving the Two-Dimensional Nonlinear Parabolic Evolution Equations

In this paper, the numerical solution for two-dimensional nonlinear parabolic equations is studied using an alternating-direction implicit (ADI) Crank–Nicolson (CN) difference scheme. Firstly, we use the CN format in the time direction, and then use the CN format in the space direction before discretizing the second-order center difference quotient. In addition, we strictly prove that the proposed ADI difference scheme has unique solvability and is unconditionally stable and convergent. The extrapolation method is further applied to improve the numerical solution accuracy. Finally, two numerical examples are given to verify our theoretical results.

Pointwise-in-time $ \alpha $-robust error estimate of the ADI difference scheme for three-dimensional fractional subdiffusion equations with variable coefficients

<abstract><p>In this paper, a fully-discrete alternating direction implicit (ADI) difference method is proposed for solving three-dimensional (3D) fractional subdiffusion equations with variable coefficients, whose solution presents a weak singularity at $ t = 0 $. The proposed method is established via the L1 scheme on graded mesh for the Caputo fractional derivative and central difference method for spatial derivative, and an ADI method is structured to change the 3D problem into three 1D problems. Using the modified Grönwall inequality we prove the stability and $ \alpha $-robust convergence. The results presented in numerical experiments are in accordance with the theoretical analysis.</p></abstract>

An efficient ADI difference scheme for the nonlocal evolution equation with multi-term weakly singular kernels in three dimensions

The paper constructs a fast efficient numerical scheme for the nonlocal evolution equation with three weakly singular kernels in three-dimensional space. In the temporal direction, We apply the backward Euler (BE) alternating direction implicit (ADI) method for the time derivative, simultaneously the first-order convolution quadrature formula is employed to deal with Riemann-Liouville (R-L) fractional integral term. In order to obtain a completely discrete implicit difference scheme, we use the standard central finite difference method (FDM) in space. The stability and convergence of the BE ADI difference scheme are proved rigorously with the convergence order in which h and τ are corresponding on the step size of space and time respectively. The ADI algorithm greatly reduces the computational cost of the three-dimensional problem. At last, several numerical results are given to verify that the numerical results are in agreement with our theoretical analysis.

Fast BDF2 ADI methods for the multi-dimensional tempered fractional integrodifferential equation of parabolic type

In this paper, we consider the numerical solutions of the multi-dimensional tempered fractional integrodifferential equation. First, the second-order backward differentiation formula (BDF2) and the second-order convolution quadrature rule are utilized for the temporal discretization, and the finite difference method (FDM) and alternating direction implicit (ADI) technique are used for the spatial discretization, in which the ADI algorithm is mainly employed to reduce the computational cost of high-dimensional non-local problems. Then, the fully discrete BDF2 ADI difference scheme for multi-dimensional tempered non-local problems can be obtained. After that, the stability and convergence of the BDF2 ADI difference scheme in multi-dimensional cases are proved by means of the energy method. In the numerical experiments, some examples validate the theoretical analysis.

Propagation of local spatial solitons in power-law nonlinear PT-symmetric potentials based on finite difference

We consider the (2+1)-dimensional nonlinear Schrödinger equation with power-law nonlinearity under the parity-time-symmetry potential by using the Crank–Nicolson alternating direction implicit difference scheme, which can also be used to solve general boundary problems under the premise of ensuring accuracy. We use linear Fourier analysis to verify the unconditional stability of the scheme. To demonstrate the effectiveness of the scheme, we compare the numerical results with the exact soliton solutions. Moreover, by using the scheme, we test the stability of the solitons under the small environmental disturbances.

A second-order ADI difference scheme based on non-uniform meshes for the three-dimensional nonlocal evolution problem

This work constructs and analyzes a nonlocal evolution equation with a weakly singular kernel in three-dimensional space. In the temporal direction, the Crank-Nicolson (CN) method and product-integration (PI) rule are employed, from which the non-uniform meshes are used to eliminate the singular behaviour of the exact solution at t=0. Then, a fully discrete scheme is obtained by the spatial discretization based on the finite difference method. Simultaneously, an alternating direction implicit (ADI) algorithm is designed to reduce the computational cost. The stability in L2 norm and convergence are derived via the energy method, in which the convergence orders are O(k2+h2) with certain suitable assumptions, where k and h are corresponding space-time step sizes, respectively. Numerical results confirm the theoretical analysis.

Fourth-order alternating direction implicit difference scheme to simulate the space-time Riesz tempered fractional diffusion equation

The current paper proposes a new high-order finite difference scheme with low computational complexity to solve the space-time fractional tempered diffusion equation. At the first stage, the time derivative has been approximated by a difference scheme with second-order accuracy. Furthermore, in the next step, a compact operator has been employed to discretize the space derivative with fourth-order accuracy. After deriving the time-discrete scheme, its stability is analysed. So, a suitable term is added to the main difference scheme. By adding this term, we could construct the main ADI scheme. In the final stage, the convergence order of the full-discrete scheme based upon the ADI formulation is proved. The convergence order of the constructed technique is . The numerical results show the efficiency of the new technique.

Pointwise error estimate of an alternating direction implicit difference scheme for two-dimensional time-fractional diffusion equation

An alternating direction implicit (ADI) difference method is adopted to solve the two-dimensional time-fractional diffusion equation with Dirichlet boundary condition whose solution has some weak singularity at initial time. L1 scheme on uniform mesh is used to discretize the temporal Caputo fractional derivative. Pointwise-in-time error estimate is given for the fully discrete ADI scheme, where the error bound does not blowup when α (the order of fractional derivative) approaches 1−. It is shown both in theoretically and numerically that the temporal convergence order of the ADI scheme is O(τ2α+τtnα−1) at time t=tn; hence the scheme is globally O(τα) accurate in temporal direction, but it is O(τmin⁡{2α,1}) when t is away from 0.

Efficient second-order ADI difference schemes for three-dimensional Riesz space-fractional diffusion equations

In this paper, a three-dimensional time-dependent Riesz space-fractional diffusion equation is considered, and an alternating direction implicit (ADI) difference scheme is proposed, in which a weighted and shifted Grünwald difference scheme (see Tian et al. (2015) [33]) is utilized for the discretizations of space-fractional derivatives, and a fractional-Douglas-Gunn type ADI method is utilized for the discretization of time derivative. The method is proved to be unconditionally stable and convergent with second-order accuracy both in time and space with respect to a weighted discrete energy norm. Efficient implementation of the method is carefully discussed, and then based on fast matrix-vector multiplications, a fast conjugate gradient (FCG) solver for the resulting symmetric positive definite linear algebraic system is developed. Numerical experiments support the theoretical analysis and show strong effectiveness and efficiency of the method for large-scale modeling and simulations. Finally, a linearized ADI scheme based on second-order extrapolation method is developed and tested for the nonlinear Riesz space-fractional diffusion equation.

A fourth-order compact ADI scheme for two-dimensional Riesz space fractional nonlinear reaction–diffusion equation

In this paper, a second-order backward difference formula (abbr. BDF2) is used to approximate first-order time partial derivative, the Riesz fractional derivatives are approximated by fourth-order compact operators, a class of new alternating-direction implicit difference scheme (abbr. ADI) is constructed for two-dimensional Riesz space fractional nonlinear reaction–diffusion equation. Stability and convergence of the numerical method are analysed. Numerical experiments demonstrate that the proposed method is effective.

Alternating direction implicit difference scheme for the multi-term time-fractional integro-differential equation with a weakly singular kernel

We present an alternating direction implicit (ADI) difference scheme for the multi-term time-fractional integro-differential equation with a weakly singular kernel. The Caputo fractional derivative is discretized by L1 method and the integro-differential term is discretized by a convolution quadrature suggested by Lubich. The stability and optimal convergence order estimate of the difference scheme are derived. Numerical experiments verify our theory analysis.

An ADI difference scheme based on fractional trapezoidal rule for fractional integro-differential equation with a weakly singular kernel

In this paper, we propose a fast and efficient numerical method to solve the two-dimensional integro-differential equation with a weakly singular kernel. The numerical method are considered by finite difference approach for spatial discretization and alternating direction implicit (ADI) method in time, combined with the second-order fractional quadrature convolution rule introduced by Lubich and the classical L1 approximation for Caputo fractional derivative. The detailed analysis shows that the proposed scheme is unconditionally stable and convergent with the convergence order O(τmin{1+α,2−α}+h12+h22). Some numerical results are also given to confirm our theoretical prediction.

A backward Euler alternating direction implicit difference scheme for the three‐dimensional fractional evolution equation

A backward Euler alternating direction implicit (ADI) difference scheme is formulated and analyzed for the three‐dimensional fractional evolution equation. In our method, the Riemann‐Liouville fractional integral term is treated by means of first order convolution quadrature suggested by Lubich. Meanwhile, an ADI technique is adopted to reduce the multidimensional problem to a series of one‐dimensional problems. A fully discrete difference scheme is constructed with space discretization by finite difference method. Two new inner products and corresponding norms are defined to analyze the scheme. The verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature. Numerical experiments are reported to demonstrate the efficiency of our scheme.

High-order algorithm for the two-dimension Riesz space-fractional diffusion equation

ABSTRACTIn this paper, applying a novel second-order numerical approximation formula for the Riesz derivative and Crank–Nicolson technique for the temporal derivative, a numerical algorithm is constructed for the two-dimensional spatial fractional diffusion equation with convergence order , where τ, and are the temporal and spatial step sizes, respectively. It is proved that the proposed algorithm is unconditionally stable and convergent by using the energy method. Meanwhile, by adding the high-order perturbation items for the above numerical scheme, an alternating direction implicit difference scheme is also constructed. Finally, some numerical results are presented to demonstrate the validity of theoretical analysis and show the accuracy and effectiveness of the method described herein.

A second order BDF alternating direction implicit difference scheme for the two-dimensional fractional evolution equation

A second order backward differentiation formula (BDF) alternating direction implicit (ADI) difference scheme is formulated and analyzed for the two-dimensional fractional evolution equation. In this method, standard central difference approximation used for the spatial discretization and the time stepping – an alternating direction implicit scheme based on second order convolution quadrature suggested by Lubich and second order BDF are considered. The stability and convergence of the second order BDF ADI difference scheme in L2 norm are derived by the energy method. Numerical experiments in total agreement with our analysis are reported.

Two Alternating Direction Implicit Difference Schemes for Solving the Two-Dimensional Time Distributed-Order Wave Equations

Two alternating direction implicit difference schemes are established for solving a class of two-dimensional time distributed-order wave equations. The schemes are proved to be unconditionally stable and convergent in the maximum norm with the convergence orders $$O(\tau ^2+h_1^2+h_2^2+\Delta \gamma ^2)$$O(ź2+h12+h22+Δź2) and $$O(\tau ^2+h_1^4+h_2^4+\Delta \gamma ^4),$$O(ź2+h14+h24+Δź4), respectively, where $$\tau , h_i\; (i=1,2)$$ź,hi(i=1,2) and $$\Delta \gamma $$Δź are the step sizes in time, space and distributed order. Also, several numerical experiments are carried out to validate the theoretical results.

Split-step alternating direction implicit difference scheme for the fractional Schrödinger equation in two dimensions

In this paper, we propose a conservative and effective difference scheme for solving the two dimensional nonlinear space-fractional Schrödinger equation with the Riesz fractional derivative. The scheme is constructed by combining the split-step method for handling the nonlinearity with the alternating direction implicit (ADI) method for resolving the multi-dimensions difficulty. The Riesz space-fractional derivative is approximated by the second order accurate fractional centered difference. Based on matrix analysis, we show that in the discrete sense the scheme conserves the mass and energy for linear problems and conserves the mass for nonlinear problems. The unconditional convergence is proved rigorously in the linear case. Numerical tests are performed to support our theoretical results and show the efficiency of the proposed scheme.

The high-order compact numerical algorithms for the two-dimensional fractional sub-diffusion equation

In this paper, performing the average operators on the space variables, a numerical scheme with third-order temporal convergence for the two-dimensional fractional sub-diffusion equation is considered, for which the unconditional stability and convergence in L1(L∞)-norm are strictly analyzed for α ∈ (0, 0.9569347] by using the discrete energy method. Therewith, adding small perturbation terms, we construct a compact alternating direction implicit difference scheme for the two-dimensional case. Finally, some numerical results have been given to show the computational efficiency and numerical accuracy of both schemes for all α ∈ (0, 1).

Two Alternating Direction Implicit Difference Schemes for Two-Dimensional Distributed-Order Fractional Diffusion Equations

Two alternating direction implicit difference schemes are derived for two-dimensional distributed-order fractional diffusion equations. It is proved that the schemes are unconditionally stable and convergent in a discrete $$L^1(L^\infty )$$L1(L?) norm with the convergence orders $$O(\tau ^2|\ln \tau |+h_1^2+h_2^2+\Delta \alpha ^2)$$O(?2|ln?|+h12+h22+Δ?2) and $$O(\tau ^2|\ln \tau |+h_1^4+h_2^4+\Delta \alpha ^4),$$O(?2|ln?|+h14+h24+Δ?4), respectively, where $$ \tau , h_i \;(i=1,2)$$?,hi(i=1,2) and $$\Delta \alpha $$Δ? are the step sizes in time, space and distributed order. Several numerical examples are given to confirm the theoretical results.