Abstract
For a linearized finite-difference scheme approximating the Dirichlet problem for a multidimensional quasilinear parabolic equation with unbounded nonlinearity, we establish pointwise two-sided solution estimates consistent with similar estimates for the differential problem. These estimates are used to prove the convergence of finite-difference schemes in the grid L 2 norm.
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References
Vladimirov, V.S., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1981.
Samarskii, A.A., Teoriya raznostnykh skhem (Theory of Finite-Difference Schemes), Moscow: Nauka, 1989.
Samarskii, A.A. and Gulin, A.V., Chislennye metody (Numerical Methods), Moscow: Nauka, 1989.
Samarskii, A.A. and Vabishchevich, P.N., Chislennye metody resheniya zadach konvektsii–diffuzii (Numerical Methods for Convection–Diffusion Problems), Moscow: URSS, 2009.
Matus, P., Hieu, L.M., and Vulkov, L.G., Analysis of second order difference schemes on non-uniform grids for quasilinear parabolic equations, J. Comput. Appl. Math., 2017, vol. 310, pp. 186–199.
Matus, P., On convergence of difference schemes for IBVP for quasilinear parabolic equations with generalized solutions, Comput. Methods Appl. Math., 2014, vol. 14, no. 3, pp. 361–371.
Matus, P.P., Tuyen, Vo Thi Kim, and Gaspar, F., Monotone finite-difference schemes for linear parabolic equations with mixed boundary conditions, Dokl. Nats. Akad. Nauk Belarusi, 2016, vol. 58, no. 5, pp. 18–22.
Ladyzhenskaya, O.A., Solution of the first boundary problem in the large for quasi-linear parabolic equations, Tr. Mosk. Mat. Obs., 1958, vol. 7, pp. 149–177.
Ladyzhenskaya, O.A., Solonnikov, V.A., and Ural’tseva, N.N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa (Linear and Quasi-Linear Equations of Parabolic Type), Moscow: Nauka, 1967.
Farago, I., Horvath, R., and Korotov, S., Discrete maximum principles for F Esolutions of nonstationary diffusion–reaction problems with mixed boundary conditions, Numer. Methods Partial Differ. Equations, 2011, vol. 27, no. 3, pp. 702–720.
Farago, I. and Horvath, R., Discrete maximum principle and adequate discretizations of linear parabolic problems, SIAM J. Sci. Comput., 2006, vol. 28, no. 6, pp. 2313–2336.
Friedman, A., Partial Differential Equations of Parabolic Type, Englewood Cliffs: Prentice-Hall, 1964.
Koleva, M.N. and Vulkov, L.G., A second-order positivity preserving numerical method for Gamma equation, Appl. Math. Comput., 2013, vol. 220, pp. 722–734.
Jandacka, M. and Sevcovic, D., On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile, J. Appl. Math., 2005, vol. 3, pp. 235–258.
Repin, S.I., A Posteriori Estimates for Partial Differential Equations, Berlin: de Gruyter, 2008.
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Original Russian Text © P.P. Matus, D.B. Poliakov, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 7, pp. 991–1000.
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Matus, P.P., Poliakov, D.B. Consistent two-sided estimates for the solutions of quasilinear parabolic equations and their approximations. Diff Equat 53, 964–973 (2017). https://doi.org/10.1134/S0012266117070126
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DOI: https://doi.org/10.1134/S0012266117070126