A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel
Copyright policy )A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel
In this paper, a two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel is proposed to reduce the computation time and improve the accuracy of the scheme developed by Xu et al. (Applied Numerical Mathematics 152 (2020) 169-184). The proposed scheme consists of three steps: First, a small nonlinear system is solved on the coarse grid using fix-point iteration. Second, the Lagrange's linear interpolation formula is used to arrive at some auxiliary values for analysis of the fine grid. Finally, a linearized Crank-Nicolson finite difference system is solved on the fine grid. Moreover, the algorithm uses a central difference approximation for the spatial derivatives. In the time direction, the time derivative and integral term are approximated by Crank-Nicolson technique and product integral rule, respectively. With the help of the discrete energy method, the stability and space-time second-order convergence of the proposed approach are obtained in $L^2$-norm. Finally, the numerical results agree with the theoretical analysis and verify the effectiveness of the algorithm.
- Henan Normal University China (People's Republic of)
- Huainan Normal University China (People's Republic of)
- Huainan Normal University China (People's Republic of)
- Ural Federal University Russian Federation
- Henan Normal University China (People's Republic of)
TIME TWO-GRID ALGORITHM, time two-grid algorithm, TWO-DIMENSIONAL, NUMERICAL EXPERIMENTS, STABILITY AND CONVERGENCE, Fractional derivatives and integrals, accurate second order, Finite difference methods for initial value and initial-boundary value problems involving PDEs, FOS: Mathematics, INTEGRODIFFERENTIAL EQUATIONS, Mathematics - Numerical Analysis, Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs, numerical experiments, Finite difference methods for boundary value problems involving PDEs, ITERATIVE METHODS, Numerical Analysis (math.NA), SECOND ORDERS, Fractional partial differential equations, ACCURATE SECOND ORDER, SECOND-ORDER SCHEME, NONLINEAR FRACTIONAL EVOLUTION EQUATION, Integro-partial differential equations, Error bounds for initial value and initial-boundary value problems involving PDEs, APPROXIMATION ALGORITHMS, Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs, stability and convergence, FRACTIONAL EVOLUTION EQUATIONS, Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs, nonlinear fractional evolution equation, TWO-GRID ALGORITHM, NONLINEAR EQUATIONS
TIME TWO-GRID ALGORITHM, time two-grid algorithm, TWO-DIMENSIONAL, NUMERICAL EXPERIMENTS, STABILITY AND CONVERGENCE, Fractional derivatives and integrals, accurate second order, Finite difference methods for initial value and initial-boundary value problems involving PDEs, FOS: Mathematics, INTEGRODIFFERENTIAL EQUATIONS, Mathematics - Numerical Analysis, Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs, numerical experiments, Finite difference methods for boundary value problems involving PDEs, ITERATIVE METHODS, Numerical Analysis (math.NA), SECOND ORDERS, Fractional partial differential equations, ACCURATE SECOND ORDER, SECOND-ORDER SCHEME, NONLINEAR FRACTIONAL EVOLUTION EQUATION, Integro-partial differential equations, Error bounds for initial value and initial-boundary value problems involving PDEs, APPROXIMATION ALGORITHMS, Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs, stability and convergence, FRACTIONAL EVOLUTION EQUATIONS, Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs, nonlinear fractional evolution equation, TWO-GRID ALGORITHM, NONLINEAR EQUATIONS
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