scholarly journals Lattice Boltzmann model for the Riesz space fractional reaction-diffusion

2018 ◽  
Vol 22 (4) ◽  
pp. 1831-1843 ◽  
Author(s):  
Hou-Ping Dai ◽  
Zhou-Shun Zheng ◽  
Wei Tan
Keyword(s):  
Diffusion Equation ◽  
Lattice Boltzmann Method ◽  
Spatial Correlation ◽  
Lattice Boltzmann ◽  
Reaction Diffusion ◽  
Riesz Space ◽  
Reaction Diffusion Equation ◽  
Difference Approximation ◽  
Boltzmann Method ◽  
Fractional Reaction

In this paper, a Riesz space fractional reaction-diffusion equation with non-linear source term is considered on a finite domain. This equation is commonly used to describe anomalous diffusion in thermal science. To solve the diffusion equation, a new fractional lattice Boltzmann method is proposed. Firstly, a difference approximation for the global spatial correlation of Riesz fractional derivative is derived by applying the numerical discretization technique, and a brief convergence analysis is presented. Then the global spatial correlation process is inserted into the evolution process of the standard lattice Boltzmann method. With combining Taylor expansion, Chapman-Enskog expansion and the multi-scales expansion, the governing evolution equation is recovered from the continuous Boltzmann equation. Three numerical examples are provided to confirm our theoretical analysis and illustrate the effectiveness of our method at last.

Evaluation of Lattice Boltzmann Method for Reaction-Diffusion Process in a Porous SOFC Anode Microstructure

2012 ◽  
Author(s):  
Hedvig Paradis ◽  
Bengt Sundén
Keyword(s):  
Porous Media ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Reaction Diffusion ◽  
Reaction Rates ◽  
Transport Processes ◽  
Microscopic Level ◽  
Reaction Diffusion Equation ◽  
Interaction Sites ◽  
Boltzmann Method

In the microscale structure of a porous electrode, the transport processes are among the least understood areas of SOFC. The purpose of this study is to evaluate the Lattice Boltzmann Method (LBM) for a porous microscopic media and investigate mass transfer processes with electrochemical reactions by LBM at a mesoscopic and microscopic level. Part of the anode structure of an SOFC for two components is evaluated qualitatively for two different geometry configurations of the porous media. The reaction-diffusion equation has been implemented in the particle distribution function used in LBM. The LBM code in this study is written in the programs MATLAB and Palabos. It has here been shown that LBM can be effectively used at a mesoscopic level ranging down to a microscopic level and proven to effectively take care of the interaction between the particles and the walls of the porous media. LBM can also handle the implementation of reaction rates where these can be locally specified or as a general source term. It is concluded that LBM can be valuable for evaluating the risk of local harming spots within the porous structure to reduce these interaction sites. In future studies, the information gained from the microscale modeling can be coupled to a macroscale CFD model and help in development of a smooth structure for interaction of the reforming reaction and the electrochemical reaction rates. This can in turn improve the cell performance.

scholarly journals Solving the Caputo Fractional Reaction-Diffusion Equation on GPU

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Jie Liu ◽  
Chunye Gong ◽  
Weimin Bao ◽  
Guojian Tang ◽  
Yuewen Jiang
Keyword(s):  
Diffusion Equation ◽  
Programming Model ◽  
Reaction Diffusion ◽  
Finite Difference Approximation ◽  
Reaction Diffusion Equation ◽  
Difference Approximation ◽  
Constant Vector ◽  
Parallel Solver ◽  
Vector Addition ◽  
Fractional Reaction

We present a parallel GPU solution of the Caputo fractional reaction-diffusion equation in one spatial dimension with explicit finite difference approximation. The parallel solution, which is implemented with CUDA programming model, consists of three procedures: preprocessing, parallel solver, and postprocessing. The parallel solver involves the parallel tridiagonal matrix vector multiplication, vector-vector addition, and constant vector multiplication. The most time consuming loop of vector-vector addition and constant vector multiplication is optimized and impressive performance improvement is got. The experimental results show that the GPU solution compares well with the exact solution. The optimized GPU solution on NVIDIA Quadro FX 5800 is 2.26 times faster than the optimized parallel CPU solution on multicore Intel Xeon E5540 CPU.

A parallel algorithm for the Riesz fractional reaction-diffusion equation with explicit finite difference method

2013 ◽  
Vol 16 (3) ◽  
Author(s):  
Chunye Gong ◽  
Weimin Bao ◽  
Guojian Tang
Keyword(s):  
Diffusion Equation ◽  
Parallel Algorithm ◽  
Reaction Diffusion ◽  
Fractional Diffusion ◽  
Riesz Space ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Parallel Solver ◽  
Serial Algorithm ◽  
Fractional Reaction

AbstractThe fractional reaction-diffusion equations play an important role in dynamical systems. Indeed, it is time consuming to numerically solve differential fractional diffusion equations. In this paper, we present a parallel algorithm for the Riesz space fractional diffusion equation. The parallel algorithm, which is implemented with MPI parallel programming model, consists of three procedures: preprocessing, parallel solver and postprocessing. The parallel solver involves the parallel matrix vector multiplication and vector vector addition. As to the authors’ knowledge, this is the first parallel algorithm for the Riesz space fractional reaction-diffusion equation. The experimental results show that the parallel algorithm is as accurate as the serial algorithm. The parallel algorithm on single Intel Xeon X5540 CPU runs 3.3-3.4 times faster than the serial algorithm on single CPU core. The parallel efficiency of 64 processes is up to 79.39% compared with 8 processes on a distributed memory cluster system.

scholarly journals Numerical Analysis WSGD Scheme for One- and Two-Dimensional Distributed Order Fractional Reaction–Diffusion Equation with Collocation Method via Fractional B-Spline

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Mohammad Ramezani
Keyword(s):  
Diffusion Equation ◽  
Collocation Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
B Spline ◽  
Computational Work ◽  
Distributed Order ◽  
Fractional Reaction

AbstractThe main propose of this paper is presenting an efficient numerical scheme to solve WSGD scheme for one- and two-dimensional distributed order fractional reaction–diffusion equation. The proposed method is based on fractional B-spline basics in collocation method which involve Caputo-type fractional derivatives for $$0 < \alpha < 1$$ 0 < α < 1 . The most significant privilege of proposed method is efficient and quite accurate and it requires relatively less computational work. The solution of consideration problem is transmute to the solution of the linear system of algebraic equations which can be solved by a suitable numerical method. The finally, several numerical WSGD Scheme for one- and two-dimensional distributed order fractional reaction–diffusion equation.

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