A Comparison of Parallel Algorithms for Numerical Solution of Parabolic Problems with Fractional Power Elliptic Operators

Raimondas Čiegis,Ignas Dapšys,Remigijus Čiegis

doi:10.3390/axioms11030098

Raimondas Čiegis, Ignas Dapšys + Show 1 more

Open Access

https://doi.org/10.3390/axioms11030098

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Abstract

In this article we construct parallel solvers analyze the efficiency and accuracy of general parallel solvers for three dimensional parabolic problems with the fractional power of elliptic operators. The proposed discrete method are targeted for general non-constant elliptic operators, the second motivation for the usage of such schemes arises when non-uniform space meshes are essential. Parallel solvers are required to solve the obtained large size systems of linear equations. The detailed scalability analysis is done in order to compare the efficiency of prposed parallel algorithms. Results of computational experiments are presented and analyzed.

Highlights

Fractional differential equations are used to simulate non-standard diffusive processes when the diffusion processes can’t be desribed by classical mathematical models
It is interesting to stress, that the classical spectral algorithm and some modifications can be used as computational tools to solve parabolic type problems with nonlocal operator Aαh, making the complexity of the nonlocal algorithms the same as for spectral methods targeted to solve parabolic problems with standard elliptic operators
The efficiency of the parallel algorithm changes only very slightly. It follows from the presented results of computational experiments that the efficiency of the parallel AAA algorithm is robust with respect to fractional power parameter, the order of AAA algorithm parameter m and discrete time step τ variations

Summary

Introduction

Fractional differential equations are used to simulate non-standard diffusive processes when the diffusion processes can’t be desribed by classical mathematical models. More examples are given in [5,6] It is well-known that the fractional power of an elliptic operator Aαh can be defined in a non-unique way. It is interesting to stress, that the classical spectral algorithm and some modifications can be used as computational tools to solve parabolic type problems with nonlocal operator Aαh, making the complexity of the nonlocal algorithms the same as for spectral methods targeted to solve parabolic problems with standard elliptic operators Clearly, this strategy is computationally efficient only if the differential problem is solved in a rectangular domain, if the complete set of eigenfunctions of operator Ah are known in advance, and if the FFT technique can be applied.

Problem Formulation

The AAA Algorithm

Additive Splitting Scheme

Parallel Algorithms

Numerical Experiments

Conclusions