Abstract
A large number of problems in physics and technology lead to boundary value or initial boundary value problems for linear and nonlinear partial differential equations. Moreover, the number of problems that have an analytical solution is limited. These are problems in canonical domains such as, for example, a rectangle, circle, or ball, and usually for equations with constant coefficients. In practice, it is often necessary to solve problems in very complex areas and for equations with variable coefficients, often nonlinear. This leads to the need to look for approximate solutions using various numerical methods. A fairly effective method for the numerical solution of problems in mathematical physics is the finite difference method or the grid method, which makes it possible to reduce the approximate solution of partial differential equations to the solution of systems of algebraic equations. The article studied the most popular numerical methods of the first, second, third and fourth order of accuracy. All of these circuits have been compared with exact solutions.
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References
Кoвeня, B.M.: Paзнocтныe мeтoды peшeния мнoгoмepныx зaдaч (2004)
Кoвeня, B.M., Чиpкoв, Д.B.: Meтoды кoнeчныx paзнocтeй и кoнeчныx oбъeмoв для peшeния зaдaч мaтeмaтичecкoй физики. Hoвocибиpcк: Hoвocибиpcкий гocyдapcтвeнный yнивepcитeт, pp. 7–8 (2013)
Aндepcoн, Д., Taннexилл, Д., Плeтчep, P.: Bычиcлитeльнaя гидpoмexaникa и тeплooбмeн: B 2-x т.: Пep. c aнгл. Mиp (1990)
Warming, R.F., Hyett, B.J.: The modified equation approach to the stability and accuracy analysis of finite-difference methods. J. Comput. Phys. 14(2), 159–179 (1974)
Lax, P.D.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math. 7(1), 159–193 (1954)
Thomas, L.H.: Elliptic problems in linear difference equations over a network. Watson Sci. Comput. Lab. Rept. Columbia Univ. New York 1, 71 (1949)
Madaliev, E., Madaliev, M., Adilov, K., Pulatov, T.: Comparison of turbulence models for two-phase flow in a centrifugal separator. In: E3S Web of Conferences 2021, vol. 264, p. 1009 (2021)
Mirzoev, A.A., Madaliev, M., Sultanbayevich, D.Y.: Numerical modeling of non-stationary turbulent flow with double barrier based on two liquid turbulence model. In: 2020 International Conference on Information Science and Communications Technologies (ICISCT), pp. 1–7 (2020)
Abdulkhaev, Z.E., Abdurazaqov, A.M., Sattorov, A.M.: Calculation of the transition processes in the pressurized water pipes at the start of the pump unit. JournalNX 7(05), 285–291 (2021). https://doi.org/10.17605/OSF.IO/9USPT
Lax, P.: Systems of conservation laws. Los Alamos National Lab NM (1959)
MacCormack, R.W.: The effect of viscosity in hypervelocity impact cratering. J. Spacecr. Rockets 40(5), 757–763 (2003)
Warming, R.F., Kutler, P., Lomax, H.: Second-and third-order noncentered difference schemes for nonlinear hyperbolic equations. AIAA J. 11(2), 189–196 (1973)
Abarbanel, S., Gottlieb, D., Turkelm, E.: Difference schemes with fourth order accuracy for hyperbolic equations. SIAM J. Appl. Math. 29, 329–351 (1975)
Malikov, Z.M., Madaliev, M.E.: Numerical simulation of two-phase flow in a centrifugal separator. Fluid. Dyn. 55, 1012–1028 (2020). https://doi.org/10.1134/S0015462820080066
Nazarov, F.K., Malikov, Z.M., Rakhmanov, N.M.: Simulation and numerical study of two-phase flow in a centrifugal dust catcher. J. Phys. Conf. Ser. 1441(1), 12155 (2020)
Malikov, Z.M., Madaliev, M.E.: New two-fluid turbulence model-based numerical simulation of flow in a flat suddenly expanding channel. Her. Bauman Mosc. State Tech. Univ. Ser. Nat. Sci. 4(97), 24–39 (2021). (In Russian). https://doi.org/10.18698/1812-3368-2021-4-24-39
Abdukarimov, B., O’tbosarov, S., Abdurazakov, A.: Investigation of the use of new solar air heaters for drying agricultural products. In: E3S Web of Conferences, vol. 264, p. 1031 (2021)
Erkinjonson, M.M.: Numerical calculation of an air centrifugal separator based on the SARC turbulence model. J. Appl. Comput. Mech. 7(2) (2021). https://doi.org/10.22055/JACM.2020.31423.1871
Hamdamov, M.M., Mirzoyev, A.A., Buriev, E.S., Tashpulatov, N.: Simulation of non-isothermal free turbulent gas jets in the process of energy exchange. In: E3S Web of Conferences, vol. 264, p. 01017 (2021)
Fayziev, R.A., Hamdamov, M.M.: Model and program of the effect of incomplete combustion gas on the economy. In: ACM International Conference Proceeding Series, pp. 401–406 (2021)
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Madaliev, M.E.U., Fayziev, R.A., Buriev, E.S., Mirzoev, A.A. (2023). Comparison of Finite Difference Schemes of Different Orders of Accuracy for the Burgers Wave Equation Problem. In: Koucheryavy, Y., Aziz, A. (eds) Internet of Things, Smart Spaces, and Next Generation Networks and Systems. NEW2AN 2022. Lecture Notes in Computer Science, vol 13772. Springer, Cham. https://doi.org/10.1007/978-3-031-30258-9_2
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