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A POD-based reduced-order Crank-Nicolson/fourth-order alternating direction implicit (ADI) finite difference scheme for solving the two-dimensional distributed-order Riesz space-fractional diffusion equation. (English) Zbl 07249433
Summary: This paper introduces a high-order numerical procedure to solve the two-dimensional distributed-order Riesz space-fractional diffusion equation. In the proposed technique, first, a second-order numerical integration rule is employed to estimate the integral of the distributed-order Riesz space-fractional derivative. Then, the time derivative is discretized by a second-order difference scheme. Finally, the spatial direction is approximated by a difference formulation with fourth-order accuracy. The stability of the semi-discrete scheme is analyzed. We conclude that the difference between two consecutive time steps i.e. $$U_{i, j}^n - U_{i, j}^{n - 1}$$ is nearly zero when $$n \to \infty$$. So, a suitable term is added to the main difference scheme as by adding this term we could derive the main ADI scheme. Furthermore, to reduce the used CPU time, we combine the fourth-order ADI formulation with the proper orthogonal decomposition method and then we gain a POD based reduced-order compact ADI finite difference plane. In the next, the convergence order of the fully discrete formulation has been investigated. The numerical results show the efficiency of new technique. It must be noted that the finite difference method is an effective and robust numerical technique for solving nonlinear equations that the ADI approach can be combined with it to improve the numerical simulations.
##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65N06 Finite difference methods for boundary value problems involving PDEs 65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65D30 Numerical integration 35R11 Fractional partial differential equations 26A33 Fractional derivatives and integrals 35K57 Reaction-diffusion equations
FODE
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