Abstract
Nowadays, the grid-characteristic method is gaining popularity for linear hyperbolic systems of equations. This method is commonly used for the simulation of the wave propagation process in deformable media. This problem requires specific properties of the used numerical method: a high approximation order and a monotonicity. One of the robust monotonicity criteria is the grid-characteristic one, proposed by A. S. Kholodov. This work is devoted to the adaptation of this approach for broaden spatial stencils. On the seven-point stencil the fifth order linear scheme and quasi-monotonic schemes were constructed. Their behavior was evaluated on the linear transport equation with the constant coefficient. Based on the problem with the initial condition of a complex form, the obtained numerical solution was compared with the other ones. The achieved approximation order was calculated based on the numerical experiment results. The detailed analysis was used to choose the scheme with the best behavior.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS1995080223010146/MediaObjects/12202_2023_7062_Fig1_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS1995080223010146/MediaObjects/12202_2023_7062_Fig2_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS1995080223010146/MediaObjects/12202_2023_7062_Fig3_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS1995080223010146/MediaObjects/12202_2023_7062_Fig4_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS1995080223010146/MediaObjects/12202_2023_7062_Fig5_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS1995080223010146/MediaObjects/12202_2023_7062_Fig6_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS1995080223010146/MediaObjects/12202_2023_7062_Fig7_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS1995080223010146/MediaObjects/12202_2023_7062_Fig8_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS1995080223010146/MediaObjects/12202_2023_7062_Fig9_HTML.png)
REFERENCES
S. K. Godunov and E. I. Romenskii, Elements of Continuum Mechanics and Conservation Laws (Nauch. Kniga, Novosibirsk, 1998; Springer, New York, 2003). https://doi.org/10.1007/978-1-4757-5117-8
H. Nishikawa and K. Kitamura, ‘‘Very simple, carbuncle-free, boundary-layer-resolving, rotated-hybrid Riemann solvers,’’ J. Comput. Phys. 227, 2560–2581 (2008). https://doi.org/10.1016/j.jcp.2007.11.003
I. B. Petrov, A. V. Vasyukov, K. A. Beklemysheva, E. S. Onuchin, and N. A. Tovarnova, ‘‘On numerical modeling of fiber deformation and destruction under impact load,’’ Dokl. Math. 105, 207–211 (2022). https://doi.org/10.1134/S1064562422030061
H. Ali, H. Soleimani, N. Yahya, M. K. Baig, and A. Rostami, ‘‘Finite element method for modelling of two phase fluid flow in porous media,’’ J. Phys.: Conf. Ser. 1123, 012002 (2018). https://doi.org/10.1088/1742-6596/1123/1/012002
M. Tavelli, S. Chiocchetti, E. Romenski, A.-A. Gabriel, and M. Dumbser, ‘‘Space-time adaptive ADER discontinuous Galerkin schemes for nonlinear hyperelasticity with material failure,’’ J. Comput. Phys. 422, 109758 (2020).
V. Golubev, A. Shevchenko, N. Khokhlov, I. Petrov, and M. Malovichko, ‘‘Compact grid-characteristic scheme for the acoustic system with the piece-wise constant coefficients,’’ Int. J. Appl. Mech. 14, 2250002 (2022). https://doi.org/10.1142/S1758825122500028
I. Nikitin and V. Golubev, ‘‘Higher order schemes for problems of dynamics of layered media with nonlinear contact conditions,’’ Smart Innov. Syst. Technol. 274, 273–287 (2022). https://doi.org/10.1007/978-981-16-8926-0_19
V. I. Golubev, M. V. Muratov, E. K. Guseva, D. S. Konov, and I. B. Petrov, ‘‘Thermodynamic and mechanical problems of ice formations: Numerical simulation results,’’ Lobachevskii J. Math. 43, 970–979 (2022). https://doi.org/10.1134/S1995080222070113
S. K. Godunov, ‘‘A difference scheme for numerical solution of discontinuous solution of hydrodynamic equations,’’ Mat. Sb. 47, 271–306 (1959).
C. Berthon, ‘‘Stability of the MUSCL schemes for the Euler equations,’’ Commun. Math. Sci. 3, 133–157 (2005). https://doi.org/10.4310/CMS.2005.v3.n2.a3
C. W. Shu, B. Cockburn, C. Johnson, and E. Tadmor, ‘‘Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws,’’ in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Ed. by A. Quarteroni, Lect. Notes Math. 1697, 325–432 (1998). https://doi.org/10.1007/BFb0096355
A. Harten, ‘‘High resolution schemes for hyperbolic conservation laws,’’ J. Comput. Phys. 135, 260–278 (1997).
V. A. Titarev and E. F. Toro, ‘‘ADER: Arbitrary high order Godunov approach,’’ J. Sci. Comput. 17, 609–618 (2002). https://doi.org/10.1023/A:1015126814947
A. S. Kholodov and Ya. A. Kholodov, ‘‘Monotonicity criteria for difference schemes designed for hyperbolic equations,’’ Comput. Math. Math. Phys. 46, 1560–1588 (2006). https://doi.org/10.1134/S0965542506090089
A. S. Kholodov, ‘‘The construction of difference schemes of increased order of accuracy for equations of hyperbolic type,’’ USSR Comput. Math. Math. Phys. 20, 234–253 (1980). https://doi.org/10.1016/0041-5553(80)90017-8
V. I. Golubev, E. K. Guseva, and I. B. Petrov, ‘‘Application of quasi-monotonic schemes in seismic arctic problems,’’ Smart Innov. Syst. Technol. 274, 289–307 (2022). https://doi.org/10.1007/978-981-16-8926-0_20
I. B. Petrov, V. I. Golubev, and E. K. Guseva, ‘‘Hybrid grid-characteristic schemes for arctic seismic problems,’’ Dokl. Math. 104, 374–379 (2021). https://doi.org/10.31857/S2686954321060138
Funding
This work was carried out with the financial support of the Russian Science Foundation, project no. 21-71-10015.
Author information
Authors and Affiliations
Corresponding authors
Additional information
(Submitted by A. V. Lapin)
Rights and permissions
About this article
Cite this article
Guseva, E.K., Golubev, V.I. & Petrov, I.B. Linear, Quasi-Monotonic and Hybrid Grid-Characteristic Schemes for Hyperbolic Equations. Lobachevskii J Math 44, 296–312 (2023). https://doi.org/10.1134/S1995080223010146
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080223010146