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.
Similar content being viewed by others
References
J. F. Thompson, Z. U. A. Warsi, and C. W. Mastin, “Boundary-Fitted Coordinate Systems for Numerical Solution to Partial Differential Equations. A Review,” J. Comput. Phys. 47(1), 1–108 (1982).
J. F. Thompson, Z. U. A. Warsi, and C. W. Mastin, Numerical Grid Generation: Foundations and Applications (North-Holland, New York, 1985).
Modeling, Mesh Generation, and Adaptive Numerical Methods for Partial Differential Equations, Ed. by I. Babuska, J. E. Flaherty, W. D. Henshaw, J. E. Hopcroft (Springer, New York, 1995).
Proceedings of the 4th–9th International Conferences on Numerical Grid Generation in Computational Field Simulations, 1994–2005.
Applied Geometry, Grid Generation, and Highly Accurate Computations, Trudy Vserossiiskoi Konferentsii, Ed. by Yu. G. Evtushenko, M. K. Kerimov, and V. A. Garanzha (Vychisl. Tsentr Ross. Akad. Nauk, Moscow, 2004), Vol. 1 [in Russian].
J. U. Brackbill and J. Saltzman, “Adaptive Zoning for Singular Problems in Two Dimensions,” J. Comput. Phys. 46, 342–368 (1982).
D. A. Anderson, “Equidistribution Schemes, Poisson Generators, and Adaptive Grids,” Appl. Math. Comput. 24, 211–227 (1987).
K. Matsuno and H. A. Dwyer, “Adaptive Methods for Elliptic Grid Generation,” J. Comput. Phys. 77, 40–52 (1988).
S. A. Ivanenko and G. P. Prokopov, “Methods of Adaptive Harmonic Grid Generation,” Zh. Vychisl. Mat. Mat. Fiz. 37, 643–662 (1997) [Comput. Math. Math. Phys. 37, 627–645 (1997)].
M. J. Berger and P. Colella, “Local Adaptive Mesh Refinement for Shock Hydrodynamics,” J. Comput. Phys. 82, 64–84 (1989).
M. J. Berger, “Data Structures for Adaptive Grid Generation,” SIAM J. Sci. Statist. Comput. 3, 904–916 (1986).
J. M. Hyman and S. Li, Iterative and Dynamic Control of Adaptive Mesh Refinement with Nested Hierarchical Grids (Los Alamos Lab., 1998) Report No. 5462.
A. Andersen, X. Zheng, and V. Cristini, “Adaptive Unstructured Volume Remeshing. I: The Method,” J. Comput. Phys. 208, 616–625 (2005).
R. R. Nourgaliev, T. N. Dinh, and T. G. Theofanous, “Adaptive Characteristics-Based Matching for Compressible Multifluid Dynamics,” J. Comput. Phys. 213(2), 500–529 (2006).
N. A. Dar’in and V. I. Mazhukin, “An Approach to Generation of Adaptive Grids,” Dokl. Akad. Nauk SSSR 298, 64–68 (1988).
N. A. Dar’in and V. I. Mazhukin, “An Approach to the Generation of Adaptive Grids for Nonstationary Problems,” Zh. Vychisl. Mat. Mat. Fiz. 28, 454–460 (1988).
N. A. Dar’in, V. I. Mazhukin, and A. A. Samarskii, “A Finite Difference Method for Solving One-Dimensional Gas Dynamics Problems on Adaptive Grids,” Dokl. Akad. Nauk SSSR 302, 1078–1081 (1988).
V. I. Mazhukin and L. Yu. Takoeva, “Principles of Generation of Dynamic Grids That Adapt to Solutions of One-Dimensional Boundary Value Problems,” Mat. Modelir. 2(3), 101–118 (1990).
V. I. Mazhukin, A. A. Samarskii, O. Kastel’yanos, and A. V. Shapranov, “The Dynamic Adaptation Method for Nonstationary Problems with High Gradients,” Mat. Modelir. 5(4), 32–56 (1993).
P. V. Breslavskii and V. I. Mazhukni, “The Dynamic Adaptation Method in Gas Dynamics,” Mat. Modelir. 7(12), 48–78 (1995).
W. H. Hui, P. Y. Li, and Z. W. Li, “A Unified Coordinate System for Solving the Two-Dimensional Euler Equations,” J. Comput. Phys. 153, 596–637 (1999).
W. H. Hui and S. Kudriakov, “A Unified Coordinate System for Solving the Three-Dimensional Euler Equations,” J. Comput. Phys. 172, 235–260 (2001).
A. N. Gil’manov, “Application of Dynamically Adaptive Grids to the Analysis of Flows with a Multiscale Structure,” Zh. Vychisl. Mat. Mat. Fiz. 41, 311–326 (2001) [Comput. Math. Math. Phys. 41, 289–303 (2001)].
D. V. Rudenko and S. V. Utyuzhnikov, “Use of Dynamically Adaptive Grids for Modeling Three-Dimensional Unsteady Gas Flows with High Gradients,” Zh. Vychisl. Mat. Mat. Fiz. 42, 395–409 (2002) [Comput. Math. Math. Phys. 42, 377–390 (2002)].
H. Tang and T. Tang, “Adaptive Mesh Methods for One-and Two-Dimensional Hyperbolic Conservation Laws,” SIAM J. Numer. Anal. 41, 487–515 (2003).
P. V. Breslavskii and V. I. Mazhukin, “Dynamically Adapted Grids for Interacting Discontinuous Solutions,” Zh. Vychisl. Mat. Mat. Fiz. 47, 717–737 (2007) [Comput. Math. Math. Phys. 47, 687–706 (2007)].
M. M. Demin, V. I. Mazhukin, and A. A. Shapranov, “Dynamic Adaptation Method for a Laminar Combustion Problem,” Zh. Vychisl. Mat. Mat. Fiz. 41, 648–661 (2001) [Comput. Math. Math. Phys. 41, 609–621 (2001)].
M. M. Demin, A. V. Shapranov, and I. Smurov, “The Method of Construction Dynamically Adapting Grids for Problems of Unstable Laminar Combustion,” Numer. Heat Transfer. Part B: Fundamentals 44, 387–415 (2003).
A. V. Lykov, Theory of Heat Conduction (Vysshaya Shkola, Moscow, 1967) [in Russian].
Ya. B. Zel’dovich and A. S. Kompaneets, “On the Theory of Heat Propagation in the Case when the Thermal Conductivity Depends on Temperature,” in On the 70th Anniversary of the Birth of A. F. Ioffe (Akad. Nauk SSSR, Moscow, 1950), pp. 61–71 [in Russian].
A. A. Samarskii and I. M. Sobol’, “Examples of Numerical Calculation of Temperature Waves,” Zh. Vychisl. Mat. Mat. Fiz. 3, 702–719 (1963).
P. P. Volosevich and E. I. Levanov, Self-Similar Solutions in Gas Dynamics and Heat Transfer (Moscow Inst. For Physics and Technology, Moscow, 1997) [in Russian].
G. I. Barenblatt and M. I. Vishik, “On Finite Rate of Propagation in Problems of Nonstationary Filtration of Fluids,” Prikl. Mat. Mekh. 20, 411–417 (1956).
R. E. Warming and B. J. Hyett, “The Modified Equation Approach to the Stability and Accuracy Analysis of Finite-Difference Methods,” J. Comput. Phys. 14, 159–179 (1974).
Yu. I. Shokin, First Differential Approximation (Nauka, Novosibirsk, 1979) [in Russian].
A. A. Samarskii, Theory of Finite Difference Schemes (Nauka, Moscow, 1989; Marcel Dekker, New York, 2001).
V. F. Vasilevskii and V. I. Mazhukin, “Numerical Calculation of Temperature Waves with Weak Discontinuities Using Dynamically Adapted Grids,” Differ. Uravn. 25, 1188–1193 (1989).
D. A. Anderson, J. C. Tannehill, and R. H. Pletcher, Computational Fluid Mechanics and Heat Transfer (Hemisphere, New York, 1984).
J. B. Bell and G. R. Shubin, “An Adaptive Grid Finite Difference Method for Conservation Law,” J. Comput. Phys. 52, 569–591 (1983).
E. R. Benton and G. W. Platzman, “A Table of the One-Dimensional Burgers Equation,” Quarterly Appl. Math. 30, 195–212 (1972).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.V. Mazhukin, V.I. Mazhukin, 2007, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2007, Vol. 47, No. 11, pp. 1913–1936.
Rights and permissions
About this article
Cite this article
Mazhukin, A.V., Mazhukin, V.I. Dynamic adaptation for parabolic equations. Comput. Math. and Math. Phys. 47, 1833–1855 (2007). https://doi.org/10.1134/S0965542507110097
Received:
Issue Date:
DOI: https://doi.org/10.1134/S0965542507110097