Abstract
An algorithmic approach to construction of finite difference schemes on regular grids developed by the first two authors is applied to quasilinear evolution equations in one spatial variable. The approach combines the finite volumes method, numerical integration, and difference elimination, which is done by means of computer algebra. As a concrete example, a difference scheme for the Korteweg-de Vries equation is constructed. This scheme is strongly consistent and absolutely stable. The numerical behavior of the scheme obtained is illustrated by solving a Cauchy problem.
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Compatible Spatial Discretizations, The IMA Volumes in Mathematics and its Applications, vol. 142, Arnold, D.N., Bochev, P.B., Lehoucq, R.B., Nicolaides, R.A., and Shashkov, M., Eds., Springer, 2006.
Gerdt, V.P., Blinkov, Yu.A., and Mozzhilkin, V.V., Gröbner bases and generation of difference schemes for partial differential equations, Symmetry, Integrability and Geometry: Methods and Applications, 2006, vol. 2, paper 051. arXiv:math.RA/0605334.
Blinkov, Yu.A. and Mozzhilkin, V.V., Generation of difference schemes for the Burgers equation by constructing Gröbner bases, Program. Comput. Software, 2006, vol. 32, no. 2, pp. 114–117.
Gerdt, V.P. and Robertz, D., Computation of difference Gröbner bases, Comput. Sci. J. Moldova, 2012, vol. 20, no. 2, pp. 203–226.
Gerdt, V.P. and Robertz, D., Consistency of finite difference approximations for linear PDE systems and its algorithmic verification, Proc. of ISSAC 2010, Watt, S.M., Ed., New York: Association for Computing Machinery, 2010, pp. 53—59.
Gerdt, V.P., Consistency analysis of finite difference approximations to PDE systems, Lect. Notes Comput. Sci., 2011, vol. 7125, pp. 28–42. arXiv:math.AP/1107. 4269
Amodio, P., Blinkov, Yu.A., Gerdt, V.P., and La Scala, R., On consistency of finite difference approximations to the Navier—Stokes equations, Lect. Notes Comput. Sci., 2013, vol. 8136, pp. 46–60. arXiv:math.NA/ 1307.0914
Kudryashov, N.A., Metody nelineinoi matematicheskoi fiziki (Methods of Nonlinear Mathematical Physics), Dolgoprudnyi: Intellekt, 2010, 2nd ed.
Integrability, Mikhailov, A.V., Ed., Lect. Notes Phys., 2009, vol. 767.
Lee, C.T. and Lee, C.C., Symbolic computation on a second-order KdV equation, J. Symbolic Computation, 2015, vol. 74, pp. 70–95.
Belashov, V.Yu. and Vladimirov, S.V., Solitary Waves in Dispersive Complex Media: Theory, Simulation, Applications, Berlin: Springer, 2005.
Samarskii, A.A., Teoriya raznostnykh skhem (Theory of Difference Schemes), Moscow: Nauka, 1989, 3d ed.
Rosinger, E.E., Nonlinear Equivalence, Reduction of PDEs to ODEs and Fast Convergent Numerical Methods, London: Pitman, 1983.
Thomas, J.W., Numerical Partial Differential Equations: Finite Difference Methods, New York: Springer, 1995.
Strikwerda, J.C., Finite Difference Schemes and Partial Differential Equations, Philadelphia: SIAM, 2004, 2nd ed.
Seiler, W.M., Involution: The formal theory of differential equations and its applications in computer algebra, in Algorithms and Computation in Mathematics, Heidelberg: Springer, 2010.
Shokin, Yu.I., Metod differentsial’nogo priblizheniya (Differential Approximation Method), Novosibirsk: Nauka, 1979.
Gerdt, V.P., Involutive algorithms for computing Gröbner bases, in Computational Commutative and Non-Commutative Algebraic Geometry, Cojocaru, S., Pfister, G., and Ufnarovski, V., Eds., NATO Science Series, IOS Press, 2005, pp. 199–225. arXiv:math.AC/050111.
Gerdt, V.P. and Blinkov, Yu.A., On computer algebraaided stability analysis of difference schemes generated by means of Gröbner bases. Computer algebra and differential equations, Acta Academiae Aboensis, 2007, ser. B, vol. 67, no. 2, pp. 168–177.
Greig, I.S. and Morris, J.Ll., A hopscotch method for the Korteweg-de Vries equation, J. Comput. Phys., 1976, vol. 20, pp. 64–80.
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Original Russian Text © Yu.A. Blinkov, V.P. Gerdt, K.B. Marinov, 2017, published in Programmirovanie, 2017, Vol. 43, No. 2.
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Blinkov, Y.A., Gerdt, V.P. & Marinov, K.B. Discretization of quasilinear evolution equations by computer algebra methods. Program Comput Soft 43, 84–89 (2017). https://doi.org/10.1134/S0361768817020049
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DOI: https://doi.org/10.1134/S0361768817020049