Abstract
A kind of the general finite difference schemes with intrinsic parallelism for the boundary value problem of the quasilinear parabolic system is studied without assuming heuristically that the original boundary value problem has the unique smooth vector solution. By the method of a priori estimation of the discrete solutions of the nonlinear difference systems, and the interpolation formulas of the various norms of the discrete functions and the fixed-point technique in finite dimensional Euclidean space, the existence and uniqueness of the discrete vector solutions of the nonlinear difference system with intrinsic parallelism are proved. Moreover the unconditional stability of the general finite difference schemes with intrinsic parallelism is justified in the sense of the continuous dependence of the discrete vector solution of the difference schemes on the discrete data of the original problems in the discrete W 2 (2,1) norms. Finally the convergence of the discrete vector solutions of the certain difference schemes with intrinsic parallelism to the unique generalized solution of the original quasilinear parabolic problem is proved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Evans, D. J., Alternating group explicit method for the diffusion equations, Appl. Math. Modeling, 1985, 19: 201–206.
Zhang, B. L., Li, W. Z., On Alternating segment Crank-Nicolson scheme, Parallel Computing, 1994, 20: 897–902.
Han, Z., Fu, H. Y., Shen, L. J., Pure alternating segment explicit-implicit method for the diffusion equations, Intern. J. Computer Math., 1993, 51: 8–15.
Yuan, G. W., Shen, L. J., Zhou, Y. L., Unconditional stability of alternating difference schemes with intrinsic parallelism for two dimensional parabolic systems, Numerical Methods for Partial Differential Equations, 1999, 15: 625–636.
Yuan, G. W., Shen, L. J., Zhu, S. H., Unconditional stability of parallel difference schemes with variable time steplengthes for heat equations, International Journal of Computer Mathematics, 2000, 75: 315–322.
Yuan, G. W., Shen, L. J., Zhou, Y. L., Unconditional stability of parallel alternating difference schemes for semilinear parabolic systems, Applied Mathematics and Computation, 2001, 117(2–3): 267–283.
Zhou, Y. L., Finite difference method with intrinsic parallelism for quasilinear parabolic systems, Beijing Mathematics, 1996, 2(1): 1–19.
Zhou, Y. L., General finite difference schemes with intrinsic parallelism for nonlinear parabolic system, Beijing Mathematics, 1996, 2(1): 20–36.
Zhou, Y. L., On the difference schemes with intrinsic parallelism for nonlinear parabolic systems, Chinese J. Num. Math. & Appl., 1996, 18(1): 66–78.
Zhou, Y. L., Yuan, G. W., General difference schemes with intrinsic parallelism for nonlinear parabolic systems, Science in China, Ser. A, 1997, 40(4): 357–365.
Zhou, Y. L., Shen, L. J., Yuan, G. W., Some practical difference schemes with intrinsic parallelism for nonlinear parabolic systems, Chinese J. Numer. Math. and Appl., 1997, 19(3): 46–57.
Yuan, G. W., Shen, L. J., Zhou, Y. L., Some practical methods of implicit domain decomposition for parabolic equations (I)—(II), Annual Report of LCP, Beijing, 2001, 472–499.
Yuan, G. W., Shen, L. J., Zhou, Y. L., Parallel difference schemes for parabolic problem, Proceeding of 2002 5th International Conference on Algorithms and Architectures for Parallel Processing, IEEE Computer Society, 238–242.
Zhou, Y. L., Applications of Discrete Functional Analysis to Finite Difference Method, Beijing: International Academic Publishers, 1990.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhou, Y., Yuan, G. & Shen, L. The unconditional stable and convergent difference methods with intrinsic parallelism for quasilinear parabolic systems. Sci. China Ser. A-Math. 47, 453–472 (2004). https://doi.org/10.1360/02ys0260
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1360/02ys0260