Abstract
In this paper, we have considered a numerical difference approximation for solving two-dimensional Riesz space fractional convection-diffusion problem with source term over a finite domain. The convection and diffusion equation can depend on both spatial and temporal variables. Crank-Nicolson scheme for time combined with weighted and shifted Grünwald-Letnikov difference operator for space are implemented to get second order convergence both in space and time. Unconditional stability and convergence order analysis of the scheme are explained theoretically and experimentally. The numerical tests are indicated that the Crank-Nicolson scheme with weighted shifted Grünwald-Letnikov approximations are effective numerical methods for two dimensional two-sided space fractional convection-diffusion equation.
Highlights
Differential equation described fractional partial differential equations are appropriated to explain complex problems like viscoelasticity, electroanalytical chemistry, biology, fluid mechanics, engineering [1], physics [1,2], fractional operators [3] and flows in porous media [4,5,6,7,8]
This study has focused to have temporal and spatial second order convergence estimates for two dimensional two-sided space fractional convection–diffusion equations based on accurate finite difference method without extrapolation approach
The scheme has been judged using the Crank-Nicolson Peaceman Rachford alternating direction implicit (CNADI) method with the novel weighted Shifted Grünwald–Letnikov difference approximation (WSGD) and the algorithm has been supported with numerical simulation
Summary
Differential equation described fractional partial differential equations are appropriated to explain complex problems like viscoelasticity, electroanalytical chemistry, biology, fluid mechanics, engineering [1], physics [1,2], fractional operators [3] and flows in porous media [4,5,6,7,8]. In Reference [38], comparative study of the finite element and difference method for two dimension space fractional advection–dispersion equation has been considered by modeling non-Fickian solute transport in groundwater. Linear spline approximation for Riemann-Liouville fractional derivative and CNADI finite difference method for time discretization are applied for solving two-dimensional two-sided space factional convection-diffusion equation was explained (read the details in Reference [40]). We need to construct weighted and shifted Grünwald-Letnikov difference operator (WSGD) with the Crank-Nicolson-ADI (CNADI)method for two-sided two dimension space fractional convection–diffusion problem to have second order both in time and space. The scheme has been judged using the Crank-Nicolson Peaceman Rachford alternating direction implicit (CNADI) method with the novel weighted Shifted Grünwald–Letnikov difference approximation (WSGD) and the algorithm has been supported with numerical simulation.
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