A GPIU method for fractional diffusion equations

Hai-Long Shen ,Neng-Hao Yu ,Xin-Hui Shao

doi:10.1186/s13662-020-02731-9

Hai-Long Shen , Neng-Hao Yu + Show 1 more

Open Access

https://doi.org/10.1186/s13662-020-02731-9

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Abstract

The fractional diffusion equations can be discretized by applying the implicit finite difference scheme and the unconditionally stable shifted Grünwald formula. Hence, the generating linear system has a real Toeplitz structure when the two diffusion coefficients are non-negative constants. Through a similarity transformation, the Toeplitz linear system can be converted to a generalized saddle point problem. We use the generalization of a parameterized inexact Uzawa (GPIU) method to solve such a kind of saddle point problem and give a new algorithm based on the GPIU method. Numerical results show the effectiveness and accuracy for the new algorithm.

Highlights

The fractional differential operator is suitable for describing the memory, genetic, mechanical and electrical properties of various materials
Compared with the classical integer differential operator, it can more concisely and accurately describe the biological, mechanical and physical processes with historical memory and spatial global correlation characteristics, such as abnormal diffusion of particles, quantization problem of non-local field theory, the fractional capacitance theory, universal voltage shunt, chaotic circuit analysis, physical semiconductors field, dispersion in porous media, physical and engineering issues related to fractal dimensions, and non-Newtonian fluid mechanics
Most numerical methods for finite difference equations (FDEs) can generate a full coefficient matrix, which requires the computational cost of O(N3) and the storage of O(N2), where N is the number of grid points [14]

Summary

Introduction

The fractional differential operator is suitable for describing the memory, genetic, mechanical and electrical properties of various materials. Most numerical methods for finite difference equations (FDEs) can generate a full coefficient matrix, which requires the computational cost of O(N3) and the storage of O(N2), where N is the number of grid points [14].

Results

Conclusion