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Zh. Vychisl. Mat. Mat. Fiz., 2007, Volume 47, Number 11, Pages 1913–1936 (Mi zvmmf224)  

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 time-dependent 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 convection-diffusion problems described by the nonlinear Burgers and Buckley-Leverette equations. A detailed analysis of differential approximations and numerical results shows that the idea of using an arbitrary time-dependent system of coordinates for adapted grid generation in combination with the principle of quasi-stationarity makes the dynamic adaptation method universal, effective, and algorithmically simple. The universality is achieved due to the use of an arbitrary time-dependent system of coordinates that moves at a velocity determined by the unknown solution. This universal approach makes it possible to generate adapted grids for time-dependent 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 quasi-stationarity, grids adapted to the solution, parabolic equation, differential approximation, finite difference scheme, nonlinear heat transfer, nonlinear convection-diffusion equation.

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English version:
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
\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 1913--1936
\jour Comput. Math. Math. Phys.
\yr 2007
\vol 47
\issue 11
\pages 1833--1855

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    This publication is cited in the following articles:
    1. P. V. Breslavskiy, V. I. Mazhukin, “Dynamic adaptation method in gasdynamic simulations with nonlinear heat conduction”, Comput. Math. Math. Phys., 48:11 (2008), 2102–2115  mathnet  crossref  mathscinet  isi
    2. Mazhukin V.I. Shapranov A.V. Mazhukin A.V. Koroleva O.N., “Mathematical Formulation of a Kinetic Version of Stefan Problem For Heterogeneous Melting/Crystallization of Metals”, Math. Montisnigri, 36 (2016), 58–77  mathscinet  zmath  isi
    3. Mazhukin V.I. Mazhukin A.V. Demin M.M. Shapranov A.V., “Nanosecond Laser Ablation of Target Al in a Gaseous Medium: Explosive Boiling”, Appl. Phys. A-Mater. Sci. Process., 124:3 (2018), 237  crossref  isi  scopus
    4. Karamzin Yu. Kudryashova T. Polyakov S., “On One Class of Flow Schemes For the Convection-Diffusion Type Equation”, Math. Montisnigri, 41 (2018), 21–32  isi
    5. Orlov A.A., Tsimbalyuk A.F., Malyugin V R., Kotelnikova A.A., Leontyeva D.A., “Influence of Pressure in the Collector and the Refrigerant Temperature to Dynamics of Filling Tanks With Smooth Inner Walls”, AIP Conference Proceedings, 2101, eds. Martoyan G., Godymchuk A., Rieznichenko L., Amer Inst Physics, 2019, 020001-1  crossref  isi
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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