Fractional Convection

2017 ◽  
Vol 13 (1) ◽  
Author(s):  
Changpin Li ◽  
Qian Yi ◽  
Jürgen Kurths
Keyword(s):  
Diffusion Equation ◽  
Numerical Scheme ◽  
Order Convergence ◽  
Discrete Form ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Second Order Convergence ◽  
First Time ◽  
Convection Dominated ◽  
Insight Into

In this study, we describe the fractional convection operator for the first time and present its discrete form with second-order convergence. A numerical scheme for the fractional-convection–diffusion equation is also constructed in order to get insight into the fractional convection behavior visually. Then, we study the fractional-convection-dominated diffusion equation which has never been considered, where the diffusion is normal and is characterized by the Laplacian. The interesting fractional convection phenomena are observed through numerical simulation. Moreover, we investigate the fractional-convection-dominated-diffusion equation which is studied for the first time either, where the convection and the diffusion are both in the fractional sense. The corresponding fractional convection phenomena are displayed via computer graphics as well.

scholarly journals A New Way to Generate an Exponential Finite Difference Scheme for 2D Convection-Diffusion Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Caihua Wang
Keyword(s):  
Difference Scheme ◽  
Taylor Series Expansion ◽  
Convergent Series ◽  
Positive Type ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Extrapolation Algorithm ◽  
Convection Dominated ◽  
Insight Into ◽  
Point Stencil

The idea of direction changing and order reducing is proposed to generate an exponential difference scheme over a five-point stencil for solving two-dimensional (2D) convection-diffusion equation with source term. During the derivation process, the higher order derivatives alongy-direction are removed to the derivatives alongx-direction iteratively using information given by the original differential equation (similarly fromx-direction toy-direction) and then instead of keeping finite terms in the Taylor series expansion, infinite terms which constitute convergent series are kept on deriving the exponential coefficients of the scheme. From the construction process one may gain more insight into the relations among the stencil coefficients. The scheme is of positive type so it is unconditionally stable and the convergence rate is proved to be of second-order. Fourth-order accuracy can be obtained by applying Richardson extrapolation algorithm. Numerical results show that the scheme is accurate, stable, and especially suitable for convection-dominated problems with different kinds of boundary layers including elliptic and parabolic ones. The idea of the method can be applied to a wide variety of differential equations.

scholarly journals Sinc-Galerkin method for solving the time fractional convection–diffusion equation with variable coefficients

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Li Juan Chen ◽  
MingZhu Li ◽  
Qiang Xu
Keyword(s):  
Diffusion Equation ◽  
Galerkin Method ◽  
Numerical Scheme ◽  
Variable Coefficients ◽  
Exponential Rate ◽  
Order Accuracy ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Exponential Rate Of Convergence ◽  
Time Fractional Derivative

Abstract In this paper, a new numerical algorithm for solving the time fractional convection–diffusion equation with variable coefficients is proposed. The time fractional derivative is estimated using the $L_{1}$ L 1 formula, and the spatial derivative is discretized by the sinc-Galerkin method. The convergence analysis of this method is investigated in detail. The numerical solution is $2-\alpha$ 2 − α order accuracy in time and exponential rate of convergence in space. Finally, some numerical examples are given to show the effectiveness of the numerical scheme.

scholarly journals A survey of numerical schemes for transportation equation

2021 ◽  
Vol 308 ◽  
pp. 01020
Author(s):  
Simin Yu
Keyword(s):  
Diffusion Equation ◽  
Transport Equation ◽  
Numerical Scheme ◽  
Fundamental Equation ◽  
Natural Phenomenon ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Linear Transport ◽  
Linear Transport Equation ◽  
Material Feature

The convection-diffusion equation is a fundamental equation that exists widely. The convection-diffusion equation consists of two processes: diffusion and convection. The convection-diffusion equation can also be called drift-diffusion equaintion. The convection – diffusion equation mainly characterizes natural phenomenon in which physical particles, energy are transferred in a system. The well-known linear transport equation is also one kind of convection-diffusion equation. The transport equation can describe the transport of a scalar field such as material feature, chemical reaction or temperature in an incompressible flow. In this paper, we discuss the famous numerical scheme, Lax-Friedrichs method, for the linear transport equation. The important ingredient of the design of the Lax-Friedrichs Method, namely the choice of the numerical fluxes will be discussed in detail. We give a detailed proof of the L1 stability of the Lax-Friedrichs scheme for the linear transport equation. We also address issues related to the implementation of this numerical scheme.

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