
This article is cited in 5 scientific papers (total in 5 papers)
Dynamic adaptation for parabolic equations
A. V. Mazhukin^{}, V. I. Mazhukin^{} ^{} Institute of Mathematical Modeling, Miusskaya pl. 4, Moscow, 125047, Russia
Abstract:
A dynamic adaptation method is presented that is based on the idea of using an arbitrary timedependent system of coordinates that moves at a velocity determined by the unknown solution. Using some model problems as examples, the generation of grids that adapt to the solution is considered for parabolic equations. Among these problems are the nonlinear heat transfer problem concerning the formation of stationary and moving temperature fronts and the convectiondiffusion problems described by the nonlinear Burgers and BuckleyLeverette equations. A detailed analysis of differential approximations and numerical results shows that the idea of using an arbitrary timedependent system of coordinates for adapted grid generation in combination with the principle of quasistationarity makes the dynamic adaptation method universal, effective, and algorithmically simple. The universality is achieved due to the use of an arbitrary timedependent system of coordinates that moves at a velocity determined by the unknown solution. This universal approach makes it possible to generate adapted grids for timedependent problems of mathematical physics with various mathematical features. Among these features are large gradients, propagation of weak and strong discontinuities in nonlinear transport and heat transfer problems, and moving contact and free boundaries in fluid dynamics. The efficiency is determined by automatically fitting the velocity of the moving nodes to the dynamics of the solution. The close relationship between the adaptation mechanism and the structure of the parabolic equations allows one to automatically control the nodes’ motion so that their trajectories do not intersect. This mechanism can be applied to all parabolic equations in contrast to the hyperbolic equations, which do not include repulsive components. The simplicity of the algorithm is achieved due to the general approach to the adaptive grid generation, which is independent of the form and type of the differential equations.
Key words:
dynamic adaptation, principle of quasistationarity, grids adapted to the solution, parabolic equation, differential approximation, finite difference scheme, nonlinear heat transfer, nonlinear convectiondiffusion equation.
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Computational Mathematics and Mathematical Physics, 2007, 47:11, 1833–1855
Bibliographic databases:
UDC:
519.635 Received: 22.05.2007
Citation:
A. V. Mazhukin, V. I. Mazhukin, “Dynamic adaptation for parabolic equations”, Zh. Vychisl. Mat. Mat. Fiz., 47:11 (2007), 1913–1936; Comput. Math. Math. Phys., 47:11 (2007), 1833–1855
Citation in format AMSBIB
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\by A.~V.~Mazhukin, V.~I.~Mazhukin
\paper Dynamic adaptation for parabolic equations
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2007
\vol 47
\issue 11
\pages 19131936
\mathnet{http://mi.mathnet.ru/zvmmf224}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=2405034}
\transl
\jour Comput. Math. Math. Phys.
\yr 2007
\vol 47
\issue 11
\pages 18331855
\crossref{https://doi.org/10.1134/S0965542507110097}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2s2.036448976231}
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http://mi.mathnet.ru/eng/zvmmf224 http://mi.mathnet.ru/eng/zvmmf/v47/i11/p1913
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