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Volume 11, Issue 2
Convergence of Iterative Difference Method with Nonuniform Meshes for Quasilinear Parabolic Systems

Yulin Zhou , Longjun Shen & Guangwei Yuan

J. Part. Diff. Eq., 11 (1998), pp. 163-172.

Published online: 1998-11

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  • Abstract
In this paper, we study the general difference schemes with nonuniform meshes for the following problem: u_t = A(x,t,u,u_x)u_{xx}, + f(x,t,u,u_x), 0 < x < l, 0 < t ≤ T \qquad (1) u(0,t) = u(l ,t) = 0, 0 < t ≤ T \qquad\qquad (2) u(x,0) = φ(x), 0 ≤ x ≤ l \qquad\qquad (3) where u, φ, and f are m-dimensional vector valued functions, u_t = \frac{∂u}{∂t}, u_x = \frac{∂u}{∂x}, u_{xx} = \frac{∂²u}{∂_x²}. In the practical computation, we usually use the method of iteration to calculate the approximate solutions for the nonlinear difference schemes. Here the estimates of the iterative sequence constructed from the iterative difference schemes for the problem (1)-(3) is proved. Moreover, when the coefficient matrix A = A(x, t, u) is independent of u_x, t he convergence of the approximate difference solution for the iterative difference schemes to the unique solution of the problem (1)-(3) is proved without imposing the assumption of heuristic character concerning the existence of the unique smooth solution for the original problem (1)-(3).
  • Keywords

Difference scheme nonlinear parabolic systems iteration convergence

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@Article{JPDE-11-163, author = {}, title = {Convergence of Iterative Difference Method with Nonuniform Meshes for Quasilinear Parabolic Systems}, journal = {Journal of Partial Differential Equations}, year = {1998}, volume = {11}, number = {2}, pages = {163--172}, abstract = { In this paper, we study the general difference schemes with nonuniform meshes for the following problem: u_t = A(x,t,u,u_x)u_{xx}, + f(x,t,u,u_x), 0 < x < l, 0 < t ≤ T \qquad (1) u(0,t) = u(l ,t) = 0, 0 < t ≤ T \qquad\qquad (2) u(x,0) = φ(x), 0 ≤ x ≤ l \qquad\qquad (3) where u, φ, and f are m-dimensional vector valued functions, u_t = \frac{∂u}{∂t}, u_x = \frac{∂u}{∂x}, u_{xx} = \frac{∂²u}{∂_x²}. In the practical computation, we usually use the method of iteration to calculate the approximate solutions for the nonlinear difference schemes. Here the estimates of the iterative sequence constructed from the iterative difference schemes for the problem (1)-(3) is proved. Moreover, when the coefficient matrix A = A(x, t, u) is independent of u_x, t he convergence of the approximate difference solution for the iterative difference schemes to the unique solution of the problem (1)-(3) is proved without imposing the assumption of heuristic character concerning the existence of the unique smooth solution for the original problem (1)-(3).}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5563.html} }
TY - JOUR T1 - Convergence of Iterative Difference Method with Nonuniform Meshes for Quasilinear Parabolic Systems JO - Journal of Partial Differential Equations VL - 2 SP - 163 EP - 172 PY - 1998 DA - 1998/11 SN - 11 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5563.html KW - Difference scheme KW - nonlinear parabolic systems KW - iteration KW - convergence AB - In this paper, we study the general difference schemes with nonuniform meshes for the following problem: u_t = A(x,t,u,u_x)u_{xx}, + f(x,t,u,u_x), 0 < x < l, 0 < t ≤ T \qquad (1) u(0,t) = u(l ,t) = 0, 0 < t ≤ T \qquad\qquad (2) u(x,0) = φ(x), 0 ≤ x ≤ l \qquad\qquad (3) where u, φ, and f are m-dimensional vector valued functions, u_t = \frac{∂u}{∂t}, u_x = \frac{∂u}{∂x}, u_{xx} = \frac{∂²u}{∂_x²}. In the practical computation, we usually use the method of iteration to calculate the approximate solutions for the nonlinear difference schemes. Here the estimates of the iterative sequence constructed from the iterative difference schemes for the problem (1)-(3) is proved. Moreover, when the coefficient matrix A = A(x, t, u) is independent of u_x, t he convergence of the approximate difference solution for the iterative difference schemes to the unique solution of the problem (1)-(3) is proved without imposing the assumption of heuristic character concerning the existence of the unique smooth solution for the original problem (1)-(3).
Yulin Zhou , Longjun Shen & Guangwei Yuan . (2019). Convergence of Iterative Difference Method with Nonuniform Meshes for Quasilinear Parabolic Systems. Journal of Partial Differential Equations. 11 (2). 163-172. doi:
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