Abstract
An algorithm is proposed to construct a sequence of evolutionary adaptive grids, each consisting of segments of uniform grids compatible with the Richardson extrapolation procedure. The necessary numerical accuracy is achieved by using Richardson's extrapolation method to increase the accuracy of the difference solution and by controlling the nonhomogeneity parameter on passing from one segment to the next. The numerical model used in the article is the diffusion-convection equation, whose solution contains large gradients. Numerical calculations show that the algorithm attains the prespecified accuracy on a nonuniform difference grid. Numerical examples support the universality of the proposed algorithm. High accuracy results can be obtained without changing the structure of the difference schemes approximating the original problem; only the grid spacing has to be changed.
Similar content being viewed by others
References
A. A. Samarskii and Yu. P. Popov, Difference Schemes in Gas Dynamics [in Russian], Nauka, Moscow (1977).
E. Dulan, J. Miller, and W. Shielders, Uniform Numerical Methods for Boundary-Layer Problems [Russian translation], Mir, Moscow (1983).
Yu. I. Shokin and N. N. Yanenko, Differential Approximation Method. Application in Gas Dynamics [in Russian], Nauka, Novosibirsk (1985).
A. A. Samarskii, Theory of Difference Schemes [in Russian], Nauka, Moscow (1977).
A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1978).
A. Kuprat, Creation and Annihilation of Nodes for the Moving Finite Element Method, dissertation, University of California, Berkeley (1992).
T. R. Harley, C. Z. Cheng, and S. C. Jardin, “The computation of resistive MHD instabilities in asixymmetric toroidal plasmas,” J. Comput. Phys.,103, No. 1, 43–62, (Nov. 1992).
G. I. Marchuk and V. V. Shaidurov, Accuracy Improvement of Solutions of Difference Schemes [in Russian], Nauka, Moscow (1979).
G. I. Marchuk, Methods of Computational Mathematics, [in Russian], Nauka, Moscow (1989).
Additional information
Translated from Chislennye Metody v Matematicheskoi Fizike, Moscow State University, pp. 22–36, 1998.
Rights and permissions
About this article
Cite this article
Andreev, V.F., Popov, A.M. Using Richardson's method to construct high-order accurate adaptive grids. Comput Math Model 10, 227–238 (1999). https://doi.org/10.1007/BF02358942
Issue Date:
DOI: https://doi.org/10.1007/BF02358942