×
Bragin, Michael D.; Rogov, Boris V.

Iterative approximate factorization for difference operators of high-order bicompact schemes for multidimensional nonhomogeneous hyperbolic systems. (English. Russian original) Zbl 1371.65079

Dokl. Math. 95, No. 2, 140-143 (2017); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 473, No. 3, 7-11 (2017).
Summary: An iterative method for solving equations of multidimensional bicompact schemes based on an approximate factorization of their difference operators is proposed for the first time. Its algorithm is described as applied to a system of two-dimensional nonhomogeneous quasilinear hyperbolic equations. The convergence of the iterative method is proved in the case of the two-dimensional homogeneous linear advection equation. The performance of the method is demonstrated on two numerical examples. It is shown that the method preserves a high (greater than the second) order of accuracy in time and performs 3–4 times faster than Newton’s method. Moreover, the method can be efficiently parallelized.

Cited in 5 Documents

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L72 Second-order quasilinear hyperbolic equations

Keywords:

iterative method; bicompact schemes; difference operators; algorithm; convergence; quasilinear hyperbolic equations; numerical examples
Cite Review PDF
Full Text: DOI

References:

[1] J. J. Stoker, Water Waves: The Mathematical Theory with Applications (Wiley, New York, 1992). · Zbl 0812.76002 · doi:10.1002/9781118033159
[2] G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1999). · Zbl 0940.76002 · doi:10.1002/9781118032954
[3] R. Brun, Introduction to Reactive Gas Dynamics (Oxford Univ. Press, Oxford, New York, 2009). · Zbl 1167.82001 · doi:10.1093/acprof:oso/9780199552689.001.0001
[4] C. Cerecignani, The Boltzmann Equation and Its Applications (Springer-Verlag, New York, 1988). · Zbl 0646.76001 · doi:10.1007/978-1-4612-1039-9
[5] Glimm, J., No article title, Lect. Notes Phys., 344, 177-186 (1986)
[6] M. Renardy, W. J. Hrusa, and J. A. Nohel, Mathematical Problems in Viscoelasticity (Longman Sci. Tech., Essex, New York, 1987). · Zbl 0719.73013
[7] A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Chapman and Hall/CRC, London, 2001; Fizmatlit, Moscow, 2012). · Zbl 0926.35011
[8] Kholodov, A. S., Numerical methods for solving hyperbolic equations and systems, 141-174 (2008), Moscow
[9] Mikhailovskaya, M. N.; Rogov, B. V., No article title, Comput. Math. Math. Phys., 52, 578-600 (2012) · Zbl 1274.35235 · doi:10.1134/S0965542512040124
[10] Rogov, B. V., No article title, Dokl. Math., 86, 582-586 (2012) · Zbl 1255.65161 · doi:10.1134/S1064562412040448
[11] N. N. Yanenko, The Method of Fractional Steps: The Solution of Problems of Mathematical Physics in Several Variables (Nauka, Novosibirsk, 1967; Springer-Verlag, Berlin, 1971). · Zbl 0183.18201
[12] Houwen, P. J.; Sommeijer, B. P., No article title, J. Comput. Appl. Math., 128, 447-466 (2001) · Zbl 0974.65089 · doi:10.1016/S0377-0427(00)00523-9
[13] Skvortsov, L. M., No article title, Mat. Model., 14, 3-17 (2002) · Zbl 1029.65080
[14] E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics (Springer, Berlin, 2009). · Zbl 1227.76006 · doi:10.1007/b79761
[15] Bragin, M. D.; Rogov, B. V., No article title, Comput. Math. Math. Phys., 56, 947-961 (2016) · Zbl 1350.65085 · doi:10.1134/S0965542516060099
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.