Abstract
In this paper, a stabilized numerical method with high accuracy is proposed to solve time-fractional singularly perturbed convection-diffusion equation with variable coefficients. The tailored finite point method (TFPM) is adopted to discrete equation in the spatial direction, while the time direction is discreted by the G-L approximation and the L1 approximation. It can effectively eliminate non-physical oscillation or excessive numerical dispersion caused by convection dominant. The stability of the scheme is verified by theoretical analysis. Finally, one-dimensional and two-dimensional numerical examples are presented to verify the efficiency of the method.
Highlights
1 Introduction Fractional calculus is a generalization of traditional integer-order calculus to noninteger order
The fractional integrals and derivatives are of nonlocal property because they are quasi-differential operators. They provide valuable tools for describing the memory and genetic properties of different materials and processes, as well as the dynamics of complex systems controlled by anomalous diffusion [1, 2]
The fractional calculus has a long history of rapid development and widespread application [3,4,5], and it is involved in nonlinear oscillating earthquakes [6], hydrodynamic models [7], continuous statistical mechanics [8], physical phenomena modeling [9], colored noise [10], solid mechanics [11], economics [12], anomalous transport [13], bioengineering [14] and many other aspects
Summary
Fractional calculus is a generalization of traditional integer-order calculus to noninteger order (fractional order). We use the tailored finite point method (TFPM) to solve the time-fractional convection-dominant diffusion problem with variable coefficient, and we find that this algorithm is very effective. Motivated by the advantages of TFPM, we adopt TFPM discrete in the spatial direction and use the G-L and L1 approximations of Caputo derivative discrete in the time direction in this paper to solve one-dimensional and two-dimensional time-fractional convection-dominated diffusion equations numerically.
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