The Tailored Finite Point Method For Convection-dominated Diffusion Equation

Traditional numerical methods for solving convection dominated problems always cause the phenomenon of numerical oscillations or numerical diffusions.In view of the numerical instability caused by convection-dominated problems,this paper presents the tailored finite point method(TFPM)for numerically solving singularly perturbed convection-diffusion equations with variable coefficients.The algorithmic format of the tailored finite point method is based on the local nature of the solution of the problem solved in each discrete point.It is an effective high-precision algorithm for solving the singular perturbation problem.For time fractional convection-dominated diffusion equations,the classical finite-difference method tends to produce non-physical oscillations under convection-dominated conditions,whereas the use of tailored finite point methods for spatial terms can effectively avoid numerical oscillations caused by too small diffusion coefficients.The main research work in this paper has the following aspects:Firstly,the basic principle and research history of tailored finite point method are introduced,and the definition and basic properties of fractional differential operator and numerical approximation of fractional derivative are introduced.The research background and research status of the convective diffusion equation and the time fractional convection-diffusion equation are introduced.Secondly,a tailoreded finite point numerical algorithm format was constructed aiming at the one-dimensional and two-dimensional convection-dominated diffusion equations and time fractional convection-dominated diffusion equations.For one-dimensional unsteady convection-diffusion equations,the explicit TFPM scheme,the implicit TFPM scheme,and the scheme based on exponential transformation equations are introduced.For the time fractional convection-diffusion equation,the TFPM discrete format based on G-L approximation and L1 approximation is introduced respectively.The relationship between the numerical simulation results and the space steps and time steps is discussed in detail.The convergence order of the algorithm is analyzed.Then the numerical simulation error results are compared with the finite difference method and the characteristic finite difference method,respectively.It fully proves the efficiency of this method.Finally,through theoretical analysis,the stability of the algorithm in this paper is discussed.For the one-dimensional and two-dimensional time fractional convection-diffusion equations,the stability of the TFPM discretization scheme based on G-L approximation and L1 approximation is discussed.Theoretically analyzes the feasibility and effectiveness of the algorithm.