A parallel CGS block-centered finite difference method for a nonlinear time-fractional parabolic equation

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In this article, a block-centered finite difference method for fractional Cattaneo equation is introduced and analyzed.The unconditional stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete L 2 norm with optimal order of convergenceboth for pressure and velocity are established on nonuniform rectangular grids. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

“…Next, we estimate lower bound of the second term on the left-hand side of Equation (33). Noting that Equations (16) and (17) and using Cauchy-Schwarz's inequality, we can obtain that…”

Section: Error Analysis For Discrete Schemementioning

confidence: 99%

mentioning

confidence: 99%

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A block‐centered finite difference method for fractional Cattaneo equation

Rui

Liu

2017

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“…In block‐centered finite difference methods were developed. Recently, a parallel CGS block‐centered finite difference method for a nonlinear time‐fractional parabolic equation has been studied . We consider block‐centered finite differences in this paper since these schemes are of practical use in implementation and can achieve superconvergence.…”

Section: Introductionmentioning

confidence: 99%

A fully conservative block‐centered finite difference method for Darcy–Forchheimer incompressible miscible displacement problem

Rui

2019

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In this paper, a block‐centered finite difference method is derived for the Darcy–Forchheimer incompressible miscible displacement problem in porous media by introducing an auxiliary flux variable to guarantee full mass conservation. Error estimates for the pressure, velocity, concentration, and auxiliary flux in different discrete norms are established rigorously and carefully on nonuniform grids. Finally, some numerical experiments are presented to show that the convergence rates are in agreement with the theoretical analysis.

“…Roughly speaking, the fractional models can be classified into two principal kinds: space-fractional differential equation and time-fractional one. Numerical methods and theory of solutions for fractional differential equations have been studied extensively by many researchers which mainly cover finite element methods [1][2][3][4], mixed finite element methods [5][6][7], finite difference methods [8][9][10][11][12][13][14], finite volume (element) methods [15,16], (local) discontinuous Galerkin ((L)DG) methods [17], spectral methods [18][19][20][21][22][23] and so forth.…”

Section: Introductionsmentioning

confidence: 99%

A fast‐high order compact difference method for the fractional cable equation

Liu

Cheng

2018

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The Cable equation is one of the most fundamental equations for modeling neuronal dynamics. In this article, we consider a high order compact finite difference numerical solution for the fractional Cable equation, which is a generalization of the classical Cable equation by taking into account the anomalous diffusion in the movement of the ions in neuronal system. The resulting finite difference scheme is unconditionally stable and converges with the convergence order of O ( normalτ min ( 1 + normalγ 1 , 1 + normalγ 2 ) + h 4 ) in maximum norm, 1‐norm and 2‐norm. Furthermore, we present a fast solution technique to accelerate Toeplitz matrix‐vector multiplications arising from finite difference discretization. This fast solution technique is based on a fast Fourier transform and depends on the special structure of coefficient matrices, and it helps to reduce the computational work from O ( M N 2 ) required by traditional methods to O ( M N log 2 N ) without using any lossy compression, where N = normalτ − 1 and τ is the size of time step, M = h − 1 and h is the size of space step. Moreover, we give a compact finite difference scheme and consider its stability analysis for two‐dimensional fractional Cable equation. The applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.