Setochno-kharakteristicheskiy metod povyshennogo poryadka dlya sistem giperbolicheskikh uravneniy s kusochno-postoyannymi koeffitsientami

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

A new approach is considered for increasing the order of accuracy of the grid-characteristic method in the region of coefficient jumps. The approach is based on piecewise polynomial interpolation for schemes of the second and third orders of accuracy for the case where the interface between the media is consistent with a finite-difference grid. The method is intended for numerical simulation of the propagation of dynamic wave disturbances in heterogeneous media. Systems of hyperbolic equations with variable coefficients are used to describe the considered physical processes. The description of the numerical method and the results of its testing are given.

About the authors

N. I Khokhlov

Moscow Institute of Physics and Technology

Email: khokhlov.ni@mipt.ru
Dolgoprudnyi, Moscow oblast, 141700, Russia

I. B Petrov

Moscow Institute of Physics and Technology

Author for correspondence.
Email: petrov@mipt.ru
Dolgoprudnyi, Moscow oblast, 141700, Russia

References

  1. LeVeque R.J. Finite Volume Methods for Hyperbolic Problems. Cambridge, 2002.
  2. Brekhovskikh L.M., Godin O.A. Acoustics of Layered Media I. Berlin; Heidelberg, 1990.
  3. Moczo P., Kristek J., Galis M. The Finite-Difference Modelling of Earthquake Motions: Waves and Ruptures. Cambridge, 2014.
  4. Li Z., Ito K. The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains. Siam, 2006.
  5. Xu J. Estimate of the convergence rate of finite element solutions to elliptic equations of second order with discontinuous coefficients // arXiv Prepr. arXiv1311.4178. 2013.
  6. Adjerid S., Ben-Romdhane M., Lin T. Higher degree immersed finite element methods for second-order elliptic interface problems // Int. J. Numer. Anal. & Model. 2014. V. 11. № 3. P. 541-566.
  7. He X., Lin T., Lin Y., Zhang X. Immersed finite element methods for parabolic equations with moving interface // Numer. Methods Partial Differ. Equat. 2013. V. 29. № 2. P. 619-646.
  8. Tong F., Wang W., Feng X., Zhao J., Li Z. How to obtain an accurate gradient for interface problems? // J. Comput. Phys. 2020. V. 405. P. 109070.
  9. Lisitsa V., Podgornova O., Tcheverda V. On the interface error analysis for finite difference wave simulation // Comput. Geosci. 2010. V. 14. № 4. P. 769-778.
  10. Kaser M., Dumbser M. An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes-I. The two-dimensional isotropic case with external source terms // Geophys. J. Int. 2006. V. 166. № 2. P. 855-877.
  11. Wilcox L., Stadler G., Burstedde C., Ghattas O. A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media // J. Comput. Phys. 2010. V. 229. № 24. P. 9373-9396.
  12. Zhang C., LeVeque R.J. The immersed interface method for acoustic wave equations with discontinuous coefficients // Wave Motion. 1997. V. 25. № 3. P. 237-263.
  13. Тихонов А.Н., Самарский А.А. Однородные разностные схемы // Журн. вычислит. математики и мат. физики. 1961. Т. 1. № 1. С. 5-67.
  14. Piraux J., Lombard B. A new interface method for hyperbolic problems with discontinuous coefficients: one-dimensional acoustic example // J. Comput. Phys. 2001. V. 168. № 1. P. 227-248.
  15. Lombard B., Piraux J. Numerical treatment of two-dimensional interfaces for acoustic and elastic waves // J. Comput. Phys. 2004. V. 195. № 1. P. 90-116.
  16. Chiavassa G., Lombard B. Time domain numerical modeling of wave propagation in 2D heterogeneous porous media // J. Comput. Phys. 2011. V. 230. № 13. P. 5288-5309.
  17. Abraham D.S., Marques A.N., Nave J.C. A correction function method for the wave equation with interface jump conditions // J. Comput. Phys. 2018. V. 353. P. 281-299.
  18. Golubev V., Shevchenko A., Khokhlov N., Petrov I., Malovichko M. Compact grid-characteristic scheme for the acoustic system with the piece-wise constant coefficients // Int. J. Appl. Mech. 2022. V. 14. № 2. P. 2250002.
  19. Khokhlov N.I., Petrov I.B. On one class of high-order compact grid-characteristic schemes for linear advection // Russ. J. Numer. Anal. Math. Model. 2016. V. 31. № 6. P. 355-368.
  20. Favorskaya A.V., Zhdanov M.S., Khokhlov N.I., Petrov I.B. Modelling the wave phenomena in acoustic and elastic media with sharp variations of physical properties using the grid-characteristic method // Geophys. Prospect. 2018. V. 66. № 8. P. 1485-1502.
  21. Ito K., Takeuchi T. Immersed interface CIP for one dimensional hyperbolic equations // Commun. Comput. Phys. 2014. V. 16. № 1. P. 96-114.
  22. Stognii P.V., Khokhlov N.I., Petrov I.B. The numerical solution of the problem of the contact interaction in models with gas pockets // J. Phys. Conf. Ser. 2021. V. 1715. № 1. P. 012058.
  23. Golubev V.I., Khokhlov N.I., Nikitin I.S., Churyakov M.A. Application of compact grid-characteristic schemes for acoustic problems // J. Phys. Conf. Ser. 2020. V. 1479. № 1. P. 012058.
  24. Khokhlov N.I., Favorskaya A., Furgailo V. Grid-characteristic method on overlapping curvilinear meshes for modeling elastic waves scattering on geological fractures // Minerals. 2022. V. 12. № 12. P. 1597.
  25. Khokhlov N., Favorskaya A., Mitkovets I., Stetsyuk V. Grid-characteristic method using Chimera meshes for simulation of elastic waves scattering on geological fractured zones // J. Comput. Phys. 2021. V. 446. P. 110637.
  26. Kozhemyachenko A.A., Petrov I.B., Favorskaya A.V., Khokhlov N.I. Boundary conditions for modeling the impact of wheels on railway track // Comput. Math. Math. Phys. 2020. V. 60. № 9. P. 1539-1554.
  27. Favorskaya A.V., Khokhlov N.I., Petrov I.B. Grid-characteristic method on joint structured regular and curved grids for modeling coupled elastic and acoustic wave phenomena in objects of complex shape // Lobachevskii J. Math. 2020. V. 41. № 4. P. 512-525.
  28. Kholodov Y.A., Kholodov A.S., Tsybulin I. V. Construction of monotone difference schemes for systems of hyperbolic equations // Comput. Math. Math. Phys. 2018. V. 58. № 8. P. 1226-1246.
  29. Courant R., Isaacson E., Rees M. On the solution of nonlinear hyperbolic differential equations by finite differences // Commun. Pure Appl. Math. 1952. V. 5. № 3. P. 243-255.
  30. Lax P., Wendroff B. Systems of conservation laws // Commun. Pure Appl. Math. 1960. V. 13. № 2. P. 217-237.
  31. Русанов В.В. Разностные схемы третьего порядка точности для прямого вычисления разрывных решений // Докл. АН СССР. 1968. Т. 180. № 6. С. 1303-1305.
  32. Холодов А.С., Холодов Я.А. О критериях монотонности разностных схем для уравнений гиперболического типа // Журн. вычислит. математики и мат. физики. 2006. Т. 46. № 9. С. 1638-1667.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2023 Russian Academy of Sciences