Two-Grid Characteristic Finite Volume Methods for Nonlinear Parabolic Problem

Two-Grid Characteristic Finite Volume Methods for Nonlinear Parabolic Problem

Year:    2013

Journal of Computational Mathematics, Vol. 31 (2013), Iss. 5 : pp. 470–487

Abstract

In this work, two-grid characteristic finite volume schemes for the nonlinear parabolic problem are considered. In our algorithms, the diffusion term is discretized by the finite volume method, while the temporal differentiation and advection terms are treated by the characteristic scheme. Under some conditions about the coefficients and exact solution, optimal error estimates for the numerical solution are obtained. Furthermore, the two-grid characteristic finite volume methods involve solving a nonlinear equation on coarse mesh with mesh size $H$, a large linear problem for the Oseen two-grid characteristic finite volume method on a fine mesh with mesh size $h = O(H^2)$ or a large linear problem for the Newton two-grid characteristic finite volume method on a fine mesh with mesh size $h = O(|log h|^{1/2}H^3)$. These methods we studied provide the same convergence rate as that of the characteristic finite volume method, which involves solving one large nonlinear problem on a fine mesh with mesh size $h$. Some numerical results are presented to demonstrate the efficiency of the proposed methods.

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jcm.1304-m4288

Journal of Computational Mathematics, Vol. 31 (2013), Iss. 5 : pp. 470–487

Published online:    2013-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    18

Keywords:    Two-grid Characteristic finite volume method Nonlinear parabolic problem Error estimate Numerical example.

  1. Two-Grid Finite Element Methods for the Steady Navier-Stokes/Darcy Model

    Zhao, Jing | Zhang, Tong

    East Asian Journal on Applied Mathematics, Vol. 6 (2016), Iss. 1 P.60

    https://doi.org/10.4208/eajam.080215.111215a [Citations: 7]
  2. A posteriorierror estimates of stabilized finite volume method for the Stokes equations

    Zhang, Tong | Mu, Lin | Yuan, JinYun

    Mathematical Methods in the Applied Sciences, Vol. 39 (2016), Iss. 1 P.32

    https://doi.org/10.1002/mma.3457 [Citations: 3]
  3. On the Convergence of a Crank–Nicolson Fitted Finite Volume Method for Pricing American Bond Options

    Gan, Xiaoting | Xu, Dengguo

    Mathematical Problems in Engineering, Vol. 2020 (2020), Iss. P.1

    https://doi.org/10.1155/2020/1052084 [Citations: 3]
  4. Two level penalty finite element methods for the stationary incompressible magnetohydrodynamics problem

    Zhang, Tong | Tao, Zhenzhen

    Computers & Mathematics with Applications, Vol. 70 (2015), Iss. 10 P.2355

    https://doi.org/10.1016/j.camwa.2015.09.003 [Citations: 2]
  5. Iterative penalty finite element methods for the steady incompressible magnetohydrodynamic problem

    Deng, Jien | Tao, Zhenzhen | Zhang, Tong

    Computational and Applied Mathematics, Vol. 36 (2017), Iss. 4 P.1637

    https://doi.org/10.1007/s40314-016-0323-y [Citations: 1]
  6. Decoupled two‐grid finite element method for the time‐dependent natural convection problem I: Spatial discretization

    Zhang, Tong | Yuan, JinYun | Si, ZhiYong

    Numerical Methods for Partial Differential Equations, Vol. 31 (2015), Iss. 6 P.2135

    https://doi.org/10.1002/num.21987 [Citations: 22]
  7. Stability and convergence of two‐grid Crank‐Nicolson extrapolation scheme for the time‐dependent natural convection equations

    Liang, Hongxia | Zhang, Tong

    Mathematical Methods in the Applied Sciences, Vol. 42 (2019), Iss. 18 P.6165

    https://doi.org/10.1002/mma.5713 [Citations: 3]
  8. Two-level finite element variational multiscale method based on bubble functions for the steady incompressible MHD flow

    Zhang, Tong | Qian, Yanxia | HuangFu, YuGao

    International Journal of Computer Mathematics, Vol. 94 (2017), Iss. 3 P.515

    https://doi.org/10.1080/00207160.2015.1115023 [Citations: 3]
  9. Characteristic block-centred finite difference methods for nonlinear convection-dominated diffusion equation

    Li, Xiaoli | Rui, Hongxing

    International Journal of Computer Mathematics, Vol. 94 (2017), Iss. 2 P.386

    https://doi.org/10.1080/00207160.2015.1109641 [Citations: 16]
  10. Two novel decoupling algorithms for the steady Stokes-Darcy model based on two-grid discretizations

    Zhang, Tong | Yuan, Jinyun

    Discrete & Continuous Dynamical Systems - B, Vol. 19 (2014), Iss. 3 P.849

    https://doi.org/10.3934/dcdsb.2014.19.849 [Citations: 15]
  11. Parallel two-grid finite element method for the time-dependent natural convection problem with non-smooth initial data

    Liang, Hongxia | Zhang, Tong

    Computers & Mathematics with Applications, Vol. 77 (2019), Iss. 8 P.2221

    https://doi.org/10.1016/j.camwa.2018.12.002 [Citations: 3]
  12. A two-level finite volume method for the unsteady Navier–Stokes equations based on two local Gauss integrations

    Zhang, Tong | Yang, Jinhua

    Journal of Computational and Applied Mathematics, Vol. 263 (2014), Iss. P.377

    https://doi.org/10.1016/j.cam.2013.12.041 [Citations: 7]