Geodesic
Linear quasi-monotonous and hybrid grid-characteristic schemes for the numerical solution of linear acoustic problems
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 26 (2023) no. 2, pp. 135-147 Cet article a éte moissonné depuis la source Math-Net.Ru

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The system of linear acoustic equations is hyperbolic. It describes the process of the acoustic wave propagation in deformable media. An important property of the schemes used for the numerical solution is their high approximation order. This property allows one to simulate the perturbation propagation process over sufficiently large distances. Another important property is monotonicity of the schemes used, which prevents the appearance of non-physical solution oscillations. In this paper, we present linear quasi-monotone and hybrid grid-characteristic schemes for a linear transport equation and a one-dimensional acoustic system. They are constructed by a method of analysis in the space of unknown coefficients proposed by A.S. Kholodov and a grid-characteristic monotonicity criterion. Wide spatial stencils with five to seven nodes of the computational grid are considered. Reflection of a longitudinal wave with a sharp front from the interface between media with different parameters is used to compare the numerical solutions. 
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     title = {Linear quasi-monotonous and hybrid grid-characteristic schemes for the numerical solution of linear acoustic problems},
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E. K. Guseva; V. I. Golubev; I. B. Petrov. Linear quasi-monotonous and hybrid grid-characteristic schemes for the numerical solution of linear acoustic problems. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 26 (2023) no. 2, pp. 135-147. http://geodesic.mathdoc.fr/item/SJVM_2023_26_2_a1/

[1] Magomedov K.M., Kholodov A.S., Setochno-kharakteristicheskie chislennye metody, Izd-vo «Yurait», M., 2018 | MR

[2] Golubev V.I., Muratov M.V., Guseva E.K., Konov D.S., Petrov I.B., “Thermodynamic and mechanical problems of ice formations: numerical simulation results”, Lobachevskii J. Math., 43:4 (2022), 970–979 | DOI | MR | Zbl

[3] Beklemysheva K.A., Vasyukov A.V., Kazakov A.O., Petrov I.B., “Grid-characteristic numerical method for low-velocity impact testing of fiber-metal laminates”, Lobachevskii J. Math., 39:7 (2018), 874–883 | DOI | MR | Zbl

[4] Nikitin I.S., Golubev V.I., “Higher order schemes for problems of dynamics of layered media with nonlinear contact conditions”, Smart Innovation, Systems and Technologies, 274 (2022), 273–287 | DOI

[5] Golubev V.I., Shevchenko A.V., Petrov I.B., “Raising convergence order of grid-characteristic schemes for 2D linear elasticity problems using operator splitting”, Computer Research and Modeling, 14:4 (2022), 899–910 | DOI | MR

[6] Kholodov A.S., Kholodov Ya.A., “O kriteriyakh monotonnosti raznostnykh skhem dlya uravnenii giperbolicheskogo tipa”, Zhurn. vychisl. matem. i mat. fiziki, 46:9 (2006), 1638–1667 | MR

[7] Kolgan V., “Primenenie operatorov sglazhivaniya v raznostnykh skhemakh vysokogo poryadka tochnosti”, Zhurn. vychisl. matem. i mat. fiziki, 18:5 (1978), 1340–1345 | MR | Zbl

[8] Godunov S., “Raznostnyi metod chislennogo rascheta razryvnykh reshenii uravnenii gidrodinamiki”, Mat. sbornik, 47:89 (1959), 271–306 | Zbl

[9] Lax P., “Weak solutions of nonlinear hyperbolic equations and their numerical computation”, Selected Papers, 1 (2005), 198–232

[10] Fedorenko R.P., “Primenenie raznostnykh skhem vysokoi tochnosti dlya chislennogo resheniya giperbolicheskikh uravnenii”, Zhurn. vychisl. matem. i mat. fiziki, 2:6 (1962), 1122–1128 | Zbl

[11] Kholodov A.S., “O postroenii raznostnykh skhem povyshennogo poryadka tochnosti dlya uravnenii giperbolicheskogo tipa”, Zhurn. vychisl. matem. i mat. fiziki, 20:6 (1980), 1601–1620 | MR | Zbl

[12] Godunov S.K., Uravneniya matematicheskoi fiziki, Nauka, M., 1971 | MR

[13] Petrov I.B., Golubev V.I., Guseva E.K., “Gibridnye setochno-kharakteristicheskie skhemy dlya zadach arkticheskoi seismorazvedki”, Dokl. RAN. Matem., inform., prots. upravleniya, 501:3 (2021), 67–73 | DOI

[14] Golubev V.I., Guseva E.K., Petrov I.B., “Application of quasi-monotonic schemes in seismic arctic problems”, Smart Innovations, Systems and Technologies, 274 (2022), 289–307 | DOI

[15] Friedrichs K., “Symmetric positive linear differential equations”, Communications on Pure and Applied Mathematics, 11:3 (1958), 333–418 | DOI | MR | Zbl

[16] Rusanov V.V., “Difference schemes of the third order of accuracy for the forward calculation of discontinuous solutions”, Doklady Akademii Nauk, 180 (1968), 1303–1305 | MR | Zbl

[17] Lax P., Wendroff B., “Systems of conservation laws”, Communications on Pure and Applied Mathematics, 13:2 (1960), 217–237 | DOI | MR | Zbl

[18] Warming R.F., Beam R.M., “Upwind Second-Order Difference Schemes and Applications in Unsteady Aerodynamic Flow”, Proc. AIAA 2nd Comput. Fluid Dyn. Conference (Hartford, Connecticut, 1975) | MR

[19] Courant R., Isaacson E., Rees M., “On the solution of nonlinear hyperbolic differential equations by finite differences”, Communications on Pure and Applied Mathematics, 5:3 (1952), 243–255 | DOI | MR | Zbl