scholarly journals Novel Second-Order Accurate Implicit Numerical Methods for the Riesz Space Distributed-Order Advection-Dispersion Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
X. Wang ◽  
F. Liu ◽  
X. Chen
Keyword(s):  
Numerical Methods ◽  
Quadrature Rule ◽  
Second Order ◽  
Riesz Space ◽  
Stability And Convergence ◽  
Dispersion Equations ◽  
Advection Dispersion ◽  
Alternating Direction ◽  
The Stability ◽  
Distributed Order

We derive and analyze second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations (RSDO-ADE) in one-dimensional (1D) and two-dimensional (2D) cases, respectively. Firstly, we discretize the Riesz space distributed-order advection-dispersion equations into multiterm Riesz space fractional advection-dispersion equations (MT-RSDO-ADE) by using the midpoint quadrature rule. Secondly, we propose a second-order accurate implicit numerical method for the MT-RSDO-ADE. Thirdly, stability and convergence are discussed. We investigate the numerical solution and analysis of the RSDO-ADE in 1D case. Then we discuss the RSDO-ADE in 2D case. For 2D case, we propose a new second-order accurate implicit alternating direction method, and the stability and convergence of this method are proved. Finally, numerical results are presented to support our theoretical analysis.

scholarly journals Some Second-Order σ Schemes Combined with an H1-Galerkin MFE Method for a Nonlinear Distributed-Order Sub-Diffusion Equation

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 187
Author(s):  
Yaxin Hou ◽  
Cao Wen ◽  
Hong Li ◽  
Yang Liu ◽  
Zhichao Fang ◽  
...  
Keyword(s):  
Finite Element ◽  
Diffusion Equation ◽  
Mixed Finite Element ◽  
Second Order ◽  
Stability And Convergence ◽  
Numerical Examples ◽  
The Stability ◽  
High Order Time ◽  
Distributed Order ◽  
Convergence Of The Scheme

In this article, some high-order time discrete schemes with an H 1 -Galerkin mixed finite element (MFE) method are studied to numerically solve a nonlinear distributed-order sub-diffusion model. Among the considered techniques, the interpolation approximation combined with second-order σ schemes in time is used to approximate the distributed order derivative. The stability and convergence of the scheme are discussed. Some numerical examples are provided to indicate the feasibility and efficiency of our schemes.

Stability and Convergence of Implicit Numerical Methods for a Class of Fractional Advection-Dispersion Models

2011 ◽  
Author(s):  
Fawang Liu ◽  
Pinghui Zhuang ◽  
Kevin Burrage
Keyword(s):  
Numerical Methods ◽  
Heavy Tails ◽  
Regional Scale ◽  
Wave Model ◽  
Anomalous Dispersion ◽  
Stability And Convergence ◽  
Dispersion Models ◽  
Diffusion Wave ◽  
Advection Dispersion ◽  
The Stability

In this paper, a class of fractional advection-dispersion models (FADM) is investigated. These models include five fractional advection-dispersion models: the immobile, mobile/immobile time FADM with a temporal fractional derivative 0 < γ < 1, the space FADM with skewness, both the time and space FADM and the time fractional advection-diffusion-wave model with damping with index 1 < γ < 2. They describe nonlocal dependence on either time or space, or both, to explain the development of anomalous dispersion. These equations can be used to simulate regional-scale anomalous dispersion with heavy tails, for example, the solute transport in watershed catchments and rivers. We propose computationally effective implicit numerical methods for these FADM. The stability and convergence of the implicit numerical methods are analyzed and compared systematically. Finally, some results are given to demonstrate the effectiveness of our theoretical analysis.

scholarly journals Stable and Convergent Finite Difference Schemes on NonuniformTime Meshes for Distributed-Order Diffusion Equations

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1975
Author(s):  
M. Luísa Morgado ◽  
Magda Rebelo ◽  
Luís L. Ferrás
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Numerical Results ◽  
Diffusion Equations ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
Uniform Mesh ◽  
Numerical Schemes ◽  
The Stability ◽  
Distributed Order

In this work, stable and convergent numerical schemes on nonuniform time meshes are proposed, for the solution of distributed-order diffusion equations. The stability and convergence of the numerical methods are proven, and a set of numerical results illustrate that the use of particular nonuniform time meshes provides more accurate results than the use of a uniform mesh, in the case of nonsmooth solutions.

Legendre Spectral Collocation Technique for Advection Dispersion Equations Included Riesz Fractional

2021 ◽  
Vol 6 (1) ◽  
pp. 9
Author(s):  
Mohamed M. Al-Shomrani ◽  
Mohamed A. Abdelkawy
Keyword(s):  
Numerical Methods ◽  
Numerical Algorithm ◽  
Spectral Collocation ◽  
Theoretical Formulation ◽  
Numerical Examples ◽  
Dispersion Equations ◽  
Advection Dispersion ◽  
Collocation Technique ◽  
Collocation Approach

The advection–dispersion equations have gotten a lot of theoretical attention. The difficulty in dealing with these problems stems from the fact that there is no perfect answer and that tackling them using local numerical methods is tough. The Riesz fractional advection–dispersion equations are quantitatively studied in this research. The numerical methodology is based on the collocation approach and a simple numerical algorithm. To show the technique’s performance and competency, a comprehensive theoretical formulation is provided, along with numerical examples.

Stability and Convergence of Two New Implicit Numerical Methods for the Fractional Cable Equation

2009 ◽  
Author(s):  
F. Liu ◽  
Q. Yang ◽  
I. Turner
Keyword(s):  
Numerical Methods ◽  
Partial Derivative ◽  
Stability And Convergence ◽  
Cable Equation ◽  
Neuronal Dynamics ◽  
Neuronal Dendrites ◽  
Space Step ◽  
The Stability ◽  
Fundamental Equations ◽  
Fractional Cable Equation

The cable equation is one the most fundamental equations for modeling neuronal dynamics. Cable equations with fractional order temporal operators have been introduced to model electrotonic properties of spiny neuronal dendrites. In this paper we consider the following fractional cable equation involving two fractional temporal derivatives: ∂u(x,t)∂t=0Dt1−γ1κ∂2u(x,t)∂x2−μ02Dt1−γ2u(x,t)+f(x,t), where 0 &lt; γ1,γ2 &lt; 1, κ &gt; 0, and μ02 are constants, and 0Dt1−γu(x,t) is the Rieman-Liouville fractional partial derivative of order 1 − γ. Two new implicit numerical methods with convergence order O(τ + h2) and O(τ2 + h2) for the fractional cable equation are proposed respectively, where τ and h are the time and space step sizes. The stability and convergence of these methods are investigated using the energy method. Finally, numerical results are given to demonstrate the effectiveness of both implicit numerical methods. These techniques can also be applied to solve other types of anomalous subdiffusion problems.

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