Abstract
On the basis of a mixed statement (velocity-strain), we complete the development of a general theory of completely conservative adjoint-coordinated difference schemes for dynamic problems of linear elasticity and viscoelasticity. In particular, our explicitly solvable discrete models permit controlling the total energy imbalance and have the same parallelization degree as the conventional explicit schemes.
Similar content being viewed by others
References
Popov, Yu.P. and Samarskii, A.A., Completely Conservative Difference Schemes for the Equations of Gas Dynamics in Euler’s Variables, Zh. Vychisl. Mat. Mat. Fiz., 1970, vol. 10, no. 4, pp. 990–998.
Samarskii, A.A. and Popov, Yu.P., Raznostnye skhemy gazovoi dinamiki (Difference Schemes for Gas Dynamics), Moscow: Nauka, 1975.
Konovalov, A.N., Discrete Models in Dynamical Problem of Linear Elasticity and Conservation Laws, Differ. Uravn., 2012, vol. 48, no. 7, pp. 990–996.
Samarskii, A.A., Teoriya raznostnykh skhem (Theory of Difference Schemes), Moscow: Nauka, 1983.
Konovalov, A.N., Adjoint-Consistent Approximations and Efficient Discrete Implementations for a Dynamic Problem of Linear Elasticity, Differ. Uravn., 2010, vol. 46, no. 7, pp. 1004–1010.
Samarskii, A.A. and Gulin, A.V., Ustoichivost’ raznostnykh skhem (Stability of Difference Schemes), Moscow: Editorial URSS, 2004.
Konovalov, A.N. and Sorokin, S.B, The Structure of the Equations of Elasticity Theory. A Statistical Problem, Preprint Akad. Nauk USSR, Novosibirsk: Comput. Center, 1986, no. 66.
Landau, L.D. and Lifshits, E.M., Mekhanika sploshnykh sred (Continuum Mechanics), Moscow, 1954.
Patsyuk, V.I., Asymptotic Behavior of a Certain Viscoelastic Medium, Preprint Akad. Nauk SSSR, Novosibirsk: Comput. Center, 1981, no. 288.
Marchuk, G.I., Metody rasshchepleniya (Decomposition Methods), Moscow: Nauka, 1988.
Konovalov, A.N., Iterative Methods in Problems of the Theory of Elasticity, Dokl. Akad. Nauk, 1995, vol. 340, no. 5, pp. 589–591.
Bukina, T.A., Operator Alternating-Triangular Method for the Three-Dimensional Static Problem in Elasticity Theory, Bull. Nov. Comput. Center Numer. Anal., 1994, vol. 6, pp. 29–36.
Author information
Authors and Affiliations
Additional information
Original Russian Text © A.N. Konovalov, 2013, published in Differentsial’nye Uravneniya, 2013, Vol. 49, No. 7, pp. 885–896.
Rights and permissions
About this article
Cite this article
Konovalov, A.N. Completely conservative difference schemes for dynamic problems of linear elasticity and viscoelasticity. Diff Equat 49, 857–868 (2013). https://doi.org/10.1134/S0012266113070082
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0012266113070082