ADI Scheme Research Articles

A fourth-order compact ADI scheme for solving a two-dimensional time-fractional reaction-subdiffusion equation

A fourth-order compact ADI scheme for solving a two-dimensional time-fractional reaction-subdiffusion equation

Splitting ADI Scheme for Fractional Laplacian Wave Equations

Splitting ADI Scheme for Fractional Laplacian Wave Equations

Finite Difference Approximation with ADI Scheme for Two-Dimensional Keller-Segel Equations

Finite Difference Approximation with ADI Scheme for Two-Dimensional Keller-Segel Equations

A family of linearly weighted-$$\theta $$ compact ADI schemes for sine-Gordon equations in high dimensions

A family of linearly weighted-$$\theta $$ compact ADI schemes for sine-Gordon equations in high dimensions

A weighted ADI scheme with variable time steps for diffusion-wave equations

A weighted ADI scheme with variable time steps for diffusion-wave equations

High-order orthogonal spline collocation ADI scheme for a new complex two-dimensional distributed-order fractional integro-differential equation with two weakly singular kernels

We present an alternating direction implicit scheme for the numerical solution of the distributed-order fractional integro-differential equation with two weakly singular kernels. Orthogonal spline collocation with piecewise Hermite bicubics is used for spatial discretization. The weighted and shifted Grünwald difference formula is employed to discretize the distributed-order time-fractional derivative combined with the second-order quadrature convolution rule introduced by Lubich for the Riemann–Liouville fractional integral. We prove the stability of the proposed scheme and derive error estimates. We also describe an efficient implementation of the scheme and show the numerical results demonstrating the accuracy and convergence rates in norm.

Optimal Design of Optical Waveguide Devices Utilizing Beam Propagation Method with ADI Scheme

Optimal Design of Optical Waveguide Devices Utilizing Beam Propagation Method with ADI Scheme

Pointwise-in-time error estimate of an ADI scheme for two-dimensional multi-term subdiffusion equation

Pointwise-in-time error estimate of an ADI scheme for two-dimensional multi-term subdiffusion equation

MODELISATION AND NUMERICAL SIMULATION OF SALT GRADIENT SOLAR POND: THE ALTERNATING DIRECTION IMPLICIT METHOD

Solar energy as free and abundant energy is considered among the renewable energy that can replace disappearing fossil fuels. Many solar power systems can be used to capture this energy, but the majority does not have the ability to store this energy in different weather and time conditions. Therefore, salinity gradient solar ponds used in the present work can overcome this issue by simultaneously capturing and storing solar energy. A solar pond is being built at Annaba. In this present work we consider two identical solar ponds subjected to the same solar radiations during a period of 28 days, and exposed to the same climatic conditions, initial and at the limits. in the first pond, nanoparticles of conductive metal, namely copper, are injected with a concentration of $0.09\%$, while the second pond does not contain any nanoparticles. a comparative study was carried out on the temperature profile in the two solar ponds mentioned above to see the influence of nanoparticles on the thermal performance of the solar pond. The mathematical model adopted in this work is based on the equation of heat conduction in two dimensions with an external source of energy to the system. The method of finite differences with ADI scheme was used to determine the temperature distribution in two different directions according to the horizontal axis ox and to the vertical axis OZ representing the depth. An average temperature and insolation values for the last ten years were obtained using the data provided by Annaba saline station. Finally, a comprehensive study was carried out in order to highlight the convergence, consistency and stability properties of the discrete model representing the solar pond.

ADI Methods for Three-dimensional Fractional Diffusions

ADI methods can be generalized to solve numerically multidimensional fractional diffusion equations, which describe fluid flows through porous media better than classical diffusion equations. A new, unconditionally stable, second-order and well balanced in space, third-order in time ADI scheme has been constructed and its convergence accelerated by an extrapolation technique coupled with the PageRank algorithm.

Finite volume ADI scheme for hybrid dimension heat conduction problems set in a cross-shaped domain

Finite volume ADI scheme for hybrid dimension heat conduction problems set in a cross-shaped domain

Parallel 3D ADI Scheme for Partially Dimension Reduced Heat Conduction Problem

This model describes the heat equation in 3D domains, and this problem is reduced to a hybrid dimension problem, keeping the initial dimension only in some parts and reducing it to one-dimensional equation within the domains in some distance from the base regions. Such mathematical models are typical in industrial installations such as pipelines. Our aim is to add two additional improvements into this methodology. First, the economical ADI type finite volume scheme is constructed to solve the non-classical heat conduction problem. Special interface conditions are defined between 3D and 1D parts. It is proved that the ADI scheme is unconditionally stable. Second, the parallel factorization algorithm is proposed to solve the obtained systems of discrete equations. Due to both modifications the run-time of computations is reduced essentially. Results of computational experiments confirm the theoretical error analysis and scalability estimates of the parallel algorithm.

High dimensional Riesz space distributed-order advection-dispersion equations with ADI scheme in compression format

<abstract><p>A second order alternating direction implicit scheme for time-dependent Riesz space distributed-order advection-dispersion equations is applied to higher dimensions with the Tensor-Train decomposition technique. The solutions are solved in compressed format, the Tensor-Train format, and the errors accumulated due to compressions are analyzed to ensure convergence. Problems with low-rank data are tested, the results illustrated a steeper growth in the ranks of the numerical solutions than that in related works.</p></abstract>

Convergence analysis of the ADI scheme for parabolic problems using discrete harmonic functions

Convergence analysis of the ADI scheme for parabolic problems using discrete harmonic functions

A High-Order Compact Adi Scheme for Solving a Two-Dimensional Fractional Neutron Diffusion Model Governing Dynamical Behaviour of a Lead-Cooled Fast Reactor

A High-Order Compact Adi Scheme for Solving a Two-Dimensional Fractional Neutron Diffusion Model Governing Dynamical Behaviour of a Lead-Cooled Fast Reactor

A High-Order Compact Adi Scheme for Solving a Two-Dimensional Fractional Neutron Diffusion Model Governing Dynamical Behaviour of a Lead-Cooled Fast Reactor

A High-Order Compact Adi Scheme for Solving a Two-Dimensional Fractional Neutron Diffusion Model Governing Dynamical Behaviour of a Lead-Cooled Fast Reactor

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.

α-Robust H1-norm convergence analysis of ADI scheme for two-dimensional time-fractional diffusion equation

A fully discrete ADI scheme is proposed for solving the two-dimensional time-fractional diffusion equation with weakly singular solutions, where L1 scheme on graded mesh is adopted to tackle the initial singularity. An improved discrete fractional Grönwall inequality is employed to give an α-robust H1-norm convergence analysis of the fully discrete ADI scheme, where the error bound does not blow up when the order of fractional derivative α→1−. Numerical results show that the theoretical analysis is sharp.

HOC–ADI schemes for two-dimensional Ginzburg–Landau equation in superconductivity

In this paper, we apply the high-order compact scheme coupled with alternating direction implicit (HOC–ADI) method to solve the two-dimensional Ginzburg–Landau (2D GL) equation. Five HOC–ADI schemes with second order accuracy in time and fourth order in space are proposed for 2D GL equation. Scheme I and Scheme II are both nonlinear, which need nonlinear iteration.To overcome this shortcoming, Scheme III and Scheme IV are proposed in order to avoid nonlinear iteration. With the three-level ADI scheme and the method of extrapolation, we obtain a linearized scheme V. Some numerical experiments are shown to testify and compare the superiority of the new numerical schemes.

Finite volume approximation with ADI scheme and low-rank solver for high dimensional spatial distributed-order fractional diffusion equations

High dimensional conservative spatial distributed-order fractional diffusion equation is discretized by midpoint quadrature rule, Crank–Nicolson method, and a finite volume approximation, with alternating direction implicit scheme. The resulting scheme is shown to be consistent and unconditionally stable, hence convergent with order 3−α, where α is the maximum of the involving fractional orders. Moreover, if the initial condition and source term possess Tensor-Train format (TT-format) with low TT-ranks, the scheme can be solved in TT-format, such that higher dimensional cases can be considered. Perturbation analysis ensures that the accumulated errors due to data recompression do not affect the overall convergence order. Numerical examples with low TT-ranks initial conditions and source terms, and with dimensions up to 20 are tested.