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Crank-Nicolson Quasi-Compact Scheme for the Nonlinear Two-Sided Spatial Fractional Advection-Diffusion Equations

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Journal of Applied Mathematics and Physics 2024年第4期12卷 1089-1100页

作者： Dechao Gao Zeshan Qiu Lizan Wang Jianxin Li Department of Basic Education Xinjiang University of Political Science and Law Tumushuke China

The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference operators and combining the compact technique, in the time direction is discretized by the Crank-Nicolson method. Through the energy method, the Stability and convergence of the numerical scheme in the sense of L2-norm are proved, and the convergence order is . Some examples are given to show that our numerical scheme is effective.

关键词： Crank-Nicolson Quasi-Compact Scheme Fractional Advection-Diffusion Equations Nonlinear Stability and convergence

带分数阶Robin边界条件的时间-空间分数阶扩散方程的有限差分方法

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Chinese Quarterly Journal of Mathematics 2024年第1期39卷 18-30页

作者：唐忠华房少梅 College of Mathematics and Informatics South China Agricultural UniversityGuangzhou 510642China

In this paper, an efficient numerical method is proposed to solve the Caputo-Riesz fractional diffusion equation with fractional Robin boundary conditions. We approximate the Riesz space fractional derivatives using the fractional central difference scheme with second-order accurate. A priori estimation of the solution of the numerical scheme is given, and the Stability and convergence of the numerical scheme are analyzed.Finally, a numerical example is used to verify the accuracy and efficiency of the numerical method.

关键词： Fractional boundary conditions Stability and convergence Caputo-Riesz fractional diffusion equation

Fast Second-Order Evaluation for Variable-Order Caputo Fractional Derivative with Applications to Fractional Sub-Diffusion Equations

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Numerical Mathematics(Theory,Methods and Applications) 2022年第1期15卷 200-226页

作者： Jia-Li Zhang Zhi-Wei Fang Hai-Wei Sun Department of Mathematics University of MacaoMacaoSARChina

In this paper,we propose a fast second-order approximation to the variable-order(VO)Caputo fractional derivative,which is developed based on L2-1σformula and the exponential-sum-approximation technique.The fast evaluation method can achieve the second-order accuracy and further reduce the computational cost and the acting memory for the VO Caputo fractional derivative.This fast algorithm is applied to construct a relevant fast temporal second-order and spatial fourth-order scheme(F L2-1σscheme)for the multi-dimensional VO time-fractional sub-diffusion equations.Theoretically,F L2-1σscheme is proved to fulfill the similar properties of the coefficients as those of the well-studied L2-1σscheme.Therefore,F L2-1σscheme is strictly proved to be unconditionally stable and convergent.A sharp decrease in the computational cost and the acting memory is shown in the numerical examples to demonstrate the efficiency of the proposed method.

关键词： Variable-order Caputo fractional derivative exponential-sum-approximation method fast algorithm time-fractional sub-diffusion equation Stability and convergence

A TWO-GRID FINITE ELEMENT APPROXIMATION FOR NONLINEAR TIME FRACTIONAL TWO-TERM MIXED SUB-DIFFUSION AND DIFFUSION WAVE EQUATIONS

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Journal of Computational Mathematics 2022年第6期40卷 936-954页

作者： Yanping Chen Qiling Gu Qingfeng Li Yunqing Huang School of Mathematical Sciences South China Normal UniversityGuangzhou 510631China School of Mathematics and Computational Science Xiangtan UniversityXiangtan 411199China

In this paper,we develop a two-grid method(TGM)based on the FEM for 2D nonlinear time fractional two-term mixed sub-diffusion and diffusion wave equations.A two-grid algorithm is proposed for solving the nonlinear system,which consists of two steps:a nonlinear FE system is solved on a coarse grid,then the linearized FE system is solved on the fine grid by Newton iteration based on the coarse solution.The fully discrete numerical approximation is analyzed,where the Galerkin finite element method for the space derivatives and the finite difference scheme for the time Caputo derivative with orderα∈(1,2)andα1∈(0,1).Numerical stability and optimal error estimate O(h^(r+1)+H^(2r+2)+τ^(min{3−α,2−α1}))in L^(2)-norm are presented for two-grid scheme,where t,H and h are the time step size,coarse grid mesh size and fine grid mesh size,respectively.Finally,numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm.

关键词： Two-grid method Finite element method Nonlinear time fractional mixed sub-diffusion and diffusion-wave equations L1-CN scheme Stability and convergence

A Compact Difference Scheme on Graded Meshes for the Nonlinear Fractional Integro-differential Equation with Non-smooth Solutions

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Acta Mathematicae Applicatae Sinica 2022年第3期38卷 601-613页

作者： Da-kang CEN Zhi-bo WANG Yan MO School of Mathematics and Statistics Guangdong University of TechnologyGuangzhou 510006China

In this paper,a compact finite difference scheme for the nonlinear fractional integro-differential equation with weak singularity at the initial time is developed,with O(N^(-(2-α))+M^(-4))accuracy order,where N;M denote the numbers of grids in temporal and spatial direction,α ∈(0,1)is the fractional order.To recover the full accuracy based on the regularity requirement of the solution,we adopt the L1 method and the trapezoidal product integration(PI)rule with graded meshes to discretize the Caputo derivative and the Riemann-Liouville integral,respectively,further handle the nonlinear term carefully by the Newton linearized method.Based on the discrete fractional Gr¨onwall inequality and preserved discrete coefficients of Riemann-Liouville fractional integral,the Stability and convergence of the proposed scheme are analyzed by the energy method.Theoretical results are also confirmed by a numerical example.

关键词： nonlinear fractional integro-differential equation graded meshes discrete fractional Gr?nwall inequality compact difference scheme Stability and convergence

TWO-STEP SCHEME FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

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Journal of Computational Mathematics 2023年第2期41卷 287-304页

作者： Qiang Han Shaolin Ji Zhongtai Securities Institute for Financial Studies Shandong UniversityJinan 250100China

In this paper,a stochastic linear two-step scheme has been presented to approximate backward stochastic differential equations(BSDEs).A necessary and sufficient condition is given to judge the L 2-stability of our numerical schemes.This stochastic linear two-step method possesses a family of 3-order convergence schemes in the sense of strong stability.The coefficients in the numerical methods are inferred based on the constraints of strong stability and n-order accuracy(n∈N^(+)).Numerical experiments illustrate that the scheme is an efficient probabilistic numerical method.

关键词： Backward stochastic differential equation Stochastic linear two-step scheme Local truncation error Stability and convergence

Construction and analysis of the quadratic finite volume methods on tetrahedral meshes

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Science China Mathematics 2023年第4期66卷 855-886页

作者： Peng Yang Xiang Wang Yonghai Li School of Mathematics Jilin UniversityChangchun 130012China BNU-UIC Research Center for Mathematics Beijing Normal UniversityZhuhai 519087China

We construct and analyze a family of quadratic finite volume method(FVM)schemes over tetrahedral meshes.In order to prove the stability and the error estimate,we propose the minimum V-angle condition on tetrahedral meshes,and the surface and volume orthogonal conditions on dual meshes.Through the technique of element analysis,the local stability is equivalent to a positive definiteness of a 9 × 9 element matrix,which is difficult to analyze directly or even numerically.With the help of the surface orthogonal condition and the congruent transformation,this element matrix is reduced into a block diagonal matrix,and then we carry out the stability result under the minimum V-angle condition.It is worth mentioning that the minimum V-angle condition of the tetrahedral case is very different from a simple extension of the minimum angle condition for triangular meshes,while it is also convenient to use in practice.Based on the stability,we prove the optimal H^(1) and L^(2) error estimates,respectively,where the orthogonal conditions play an important role in ensuring the optimal L^(2) convergence rate.Numerical experiments are presented to illustrate our theoretical results.

关键词： finite volume method tetrahedral mesh orthogonal condition minimum V-angle condition Stability and convergence

DISCRETE ENERGY ANALYSIS OF THE THIRD-ORDER VARIABLE-STEP BDF TIME-STEPPING FOR DIFFUSION EQUATIONS

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Journal of Computational Mathematics 2023年第2期41卷 325-344页

作者： Hong-lin Liao Tao Tang Tao Zhou Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing 211106China Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA) MIITNanjing 211106China Division of Science and Technology BNU-HKBU United International CollegeZhuhai 519087China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing100190China

This is one of our series works on discrete energy analysis of the variable-step BDF schemes.In this part,we present Stability and convergence analysis of the third-order BDF(BDF3)schemes with variable steps for linear diffusion equations,see,e.g.,[SIAM J.Numer.Anal.,58:2294-2314]and[Math.Comp.,90:1207-1226]for our previous works on the BDF2 scheme.To this aim,we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877,by which we can establish a discrete energy dissipation law.Mesh-robust Stability and convergence analysis in the L^(2) norm are then obtained.Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios.We also present numerical tests to support our theoretical results.

关键词： Diffusion equations Variable-step third-order BDF scheme Discrete gradient structure Discrete orthogonal convolution kernels Stability and convergence

Numerical Methods for Semilinear Fractional Diffusion Equations with Time Delay

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Advances in Applied Mathematics and Mechanics 2022年第1期14卷 56-78页

作者： Shuiping Yang Yubin Liu Hongyu Liu Chao Wang School of Mathematics and Big Data Science Huizhou UniversityHuizhouGuangdong 516007China Department of Mathematics City University of Hong KongKowloonHong Kong Department of Mathematics Southern University of Science and TechnologyShenzhenGuangdong 518055China Guangdong Provincial Key Laboratory of Computational Science and Material Design Southern University of Science and TechnologyShenzhenGuangdong 518055China

In this paper,we consider the numerical solutions of the semilinear Riesz space-fractional diffusion equations(RSFDEs)with time delay,which constitute an important class of differential equations of practical significance.We develop a novel implicit alternating direction method that can effectively and efficiently tackle the RSFDEs in both two and three dimensions.The numerical method is proved to be uniquely solvable,stable and convergent with second order accuracy in both space and time.Numerical results are presented to verify the accuracy and efficiency of the proposed numerical scheme.

关键词： Semilinear Riesz space fractional diffusion equations with time delay implicit alternating direction method Stability and convergence