Volume 31, Issue 5
Two-Grid Characteristic Finite Volume Methods for Nonlinear Parabolic Problem

J. Comp. Math., 31 (2013), pp. 470-487.

Published online: 2013-10

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• 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.

• Keywords

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

35Q55, 65N30, 76D05

• BibTex
• RIS
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@Article{JCM-31-470, author = {}, title = {Two-Grid Characteristic Finite Volume Methods for Nonlinear Parabolic Problem}, journal = {Journal of Computational Mathematics}, year = {2013}, volume = {31}, number = {5}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1304-m4288}, url = {http://global-sci.org/intro/article_detail/jcm/9748.html} }
TY - JOUR T1 - Two-Grid Characteristic Finite Volume Methods for Nonlinear Parabolic Problem JO - Journal of Computational Mathematics VL - 5 SP - 470 EP - 487 PY - 2013 DA - 2013/10 SN - 31 DO - http://doi.org/10.4208/jcm.1304-m4288 UR - https://global-sci.org/intro/article_detail/jcm/9748.html KW - Two-grid, Characteristic finite volume method, Nonlinear parabolic problem, Error estimate, Numerical example. AB -

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.

Tong Zhang. (1970). Two-Grid Characteristic Finite Volume Methods for Nonlinear Parabolic Problem. Journal of Computational Mathematics. 31 (5). 470-487. doi:10.4208/jcm.1304-m4288
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