High precision implicit method for 3D quasilinear hyperbolic equations on a dissimilar domain: application to 3D telegraphic equation. (English) Zbl 1524.65379
Summary: In this paper, we recommend a novel high accuracy compact three-level implicit numerical method of order two in time and four in space using unequal mesh for the solution of 3D quasi-linear hyperbolic equations on an irrational domain. The stability analysis of the model Telegraphic equation for unequal mesh has been discussed and it has been shown that the proposed method for Telegraphic equation is unconditionally stable. Operator splitting technique is used to solve 3D linear model equations. In this technique, we use very well-known tri-diagonal solver technique to solve a set of tri-diagonal matrices on an irrational domain. The projected scheme is examined on numerous physical problems on irrational domain to demonstrate the accurateness and efficiency of the suggested scheme.
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 |
| 35L20 | Initial-boundary value problems for second-order hyperbolic equations |
| 35L15 | Initial value problems for second-order hyperbolic equations |
| 35L72 | Second-order quasilinear hyperbolic equations |
| 65F05 | Direct numerical methods for linear systems and matrix inversion |
Keywords:
quasi-linear 3D hyperbolic equation; irrational domain; telegraphic equation; operator splitting technique; dissipative and van der Pol equations; Ohm’s lawReferences:
| [1] | Ciment, M.; Leventhal, S. H., Higher order compact implicit schemes for the wave equation, Math. Comput., 29, 985-994 (1975) · Zbl 0309.35043 |
| [2] | Ciment, M.; Leventhal, S. H., A note on the operator compact implicit method for the wave equation, Math. Comput., 32, 143-147 (1978) · Zbl 0373.35039 |
| [3] | Titarev, V. A.; Toro, E. F., ADER schemes for three-dimensional non-linear hyperbolic systems, J. Comput. Phys., 204, 715-736 (2005) · Zbl 1060.65641 |
| [4] | Toro, E. F.; Titarev, V. A., ADER schemes for scalar linear hyperbolic conservation laws with source terms in three-space dimensions, J. Comput. Phys., 202, 196-215 (2005) · Zbl 1061.65103 |
| [5] | Dai, W.; Song, H.; Su, S.; Nassar, R., A stable finite difference scheme for solving a hyperbolic two-step model in a 3D micro sphere exposed to ultrashort-pulsed lasers, Int. J. Numer. Methods Heat Fluid Flow, 16, 693-717 (2006) · Zbl 1182.80015 |
| [6] | Wang, C. G.; Fang, D., Blow-up of solution of Cauchy problem for three-dimensional damped nonlinear hyperbolic equation, Nonlinear Anal., 71, 358-372 (2009) · Zbl 1170.35472 |
| [7] | Srivastava, V. K.; Awasthi, M. K.; Chaurasia, R. K.; Tamsir, M., The telegraph equation and its solution by reduced differential transform method, Model. Simul. Eng., Article 746351 pp. (2013) |
| [8] | Srivastava, V. K.; Awasthi, M. K.; Kumar, Sunil, Analytical approximations of two and three dimensional time-fractional telegraphic equation by reduced differential transform method, Egypt. J. Basic Appl. Sci., 1, 60-66 (2014) |
| [9] | Srivastava, V. K.; Awasthi, M. K.; Chaurasia, R. K., Reduced differential transform method to solve two and three dimensional second-order hyperbolic telegraphic equations, J. King Saud Univ., Eng. Sci., 29, 166-171 (2017) |
| [10] | Qiu, H.; Zhang, Y., Decay of the 3D quasilinear hyperbolic equation with nonlinear damping, Adv. Math. Phys., 2017, Article 2708483 pp. (2017) · Zbl 1418.35039 |
| [11] | Zhang, Y., Initial boundary value problem for 3D quasilinear hyperbolic equations with nonlinear damping, Appl. Anal., 98, 2048-2063 (2019) · Zbl 1437.35096 |
| [12] | Messaoudi, S. A.; Talahmeh, A. A.; Al-Smail, J. H., Nonlinear damped wave equation: existence and blow-up, Comput. Math. Appl., 74, 3024-3041 (2017) · Zbl 1415.35061 |
| [13] | Prieto, F. U.; Gavete, L.; Benito, J.; Garcia, A.; Vargas, A. M., Solving the telegraphic equation in 2-D and 3-D using generalized finite difference method (GFDM), Eng. Anal. Bound. Elem., 112, 13-24 (2020) · Zbl 1464.65093 |
| [14] | Prieto, F. U.; Gavete, L.; Vargas, A. M.; Garcia, A.; Benito, J., Solving second-order non-linear hyperbolic partial differential equations using generalized finite difference method (GFDM), J. Comput. Appl. Math., 363, 1-21 (2020) · Zbl 1422.65188 |
| [15] | Biala, T. A.; Jator, S. N., A boundary value approach for solving three-dimensional elliptic and hyperbolic partial differential equations, SpringerPlus, 4, Article 588 pp. (2015) · Zbl 1382.65460 |
| [16] | Rim, D., Dimensional splitting of hyperbolic partial differential equations using the random transform, SIAM J. Sci. Comput., 40, A4184-A4207 (2018) · Zbl 1406.65071 |
| [17] | Mohanty, R. K.; Gopal, Venu, A new off-step high order approximation for the solution of three-space dimensional nonlinear wave equations, Appl. Math. Model., 37, 2802-2815 (2013) · Zbl 1352.65254 |
| [18] | Mohanty, R. K.; Singh, S.; Singh, S., A new high order space derivative discretization for 3D quasi-linear hyperbolic partial differential equations, Appl. Math. Comput., 232, 529-541 (2014) · Zbl 1410.65321 |
| [19] | Mohanty, R. K.; Khurana, Gunjan, A new fast algorithm based on half-step discretization for 3D quasilinear hyperbolic partial differential equations, Int. J. Comput. Methods, 16, Article 1850090 pp. (2019) · Zbl 1404.65099 |
| [20] | Mohanty, R. K., Stability interval for explicit difference schemes for multi-dimensional second-order hyperbolic equations with significant first order space derivative terms, Appl. Math. Comput., 190, 1683-1690 (2007) · Zbl 1122.65381 |
| [21] | Mohanty, R. K.; Jain, M. K.; Arora, U., An unconditionally stable ADI method for the linear hyperbolic equation in three space dimensions, Int. J. Comput. Math., 79, 133-142 (2002) · Zbl 0995.65093 |
| [22] | Mohanty, R. K., An operator splitting technique for an unconditionally stable difference method for a linear three space dimensional hyperbolic equation with variable coefficients, Appl. Math. Comput., 162, 549-557 (2005) · Zbl 1063.65084 |
| [23] | Dehghan, M., On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, Numer. Methods Partial Differ. Equ., 21, 24-40 (2005) · Zbl 1059.65072 |
| [24] | Dehghan, M.; Shokri, A., A numerical method for solving the hyperbolic telegraph equation, Numer. Methods Partial Differ. Equ., 24, 1080-1093 (2008) · Zbl 1145.65078 |
| [25] | Mohanty, R. K., New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations, Int. J. Comput. Math., 86, 2061-2071 (2009) · Zbl 1181.65112 |
| [26] | Dehghan, M.; Shokri, A., A meshless method for numerical solution of a linear hyperbolic equation with variable coefficients in two space dimensions, Numer. Methods Partial Differ. Equ., 25, 494-506 (2009) · Zbl 1159.65084 |
| [27] | Dehghan, M.; Mohebbi, A., High order implicit collocation method for the solution of two-dimensional linear hyperbolic equation, Numer. Methods Partial Differ. Equ., 25, 232-243 (2009) · Zbl 1156.65087 |
| [28] | Saadatmandi, A.; Dehghan, M., Numerical solution of hyperbolic telegraph equation using the Chebyshev Tau method, Numer. Methods Partial Differ. Equ., 26, 239-252 (2010) · Zbl 1186.65136 |
| [29] | Karaa, S., Unconditionally stable ADI scheme of higher-order for linear hyperbolic equations, Int. J. Comput. Math., 87, 3030-3038 (2010) · Zbl 1210.65161 |
| [30] | Lakestani, M.; Saray, B. N., Numerical solution of telegraph equation using interpolating scaling function, Comput. Math. Appl., 60, 1964-1972 (2010) · Zbl 1205.65288 |
| [31] | Dehghan, M.; Ghesmati, A., Combination of meshless local weak and strong (MLWS) forms to solve the two dimensional hyperbolic telegraph equation, Eng. Anal. Bound. Elem., 34, 324-336 (2010) · Zbl 1244.65147 |
| [32] | Dehghan, M.; Ghesmati, A., Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method, Eng. Anal. Bound. Elem., 34, 51-59 (2010) · Zbl 1244.65137 |
| [33] | Dehghan, M.; Salehi, R., A method based on meshless approach for the numerical solution of the two-space dimensional hyperbolic telegraphic equation, Math. Methods Appl. Sci., 35, 1220-1233 (2012) · Zbl 1250.35015 |
| [34] | Mohanty, R. K., New high accuracy super stable alternating direction implicit methods for two and three dimensional hyperbolic damped wave equations, Results Phys., 4, 156-163 (2014) |
| [35] | Sharifi, S.; Rashidinia, J., Numerical solution of hyperbolic telegraph equation by cubic B-spline collocation method, Appl. Math. Comput., 281, 28-38 (2016) · Zbl 1410.65403 |
| [36] | Yuzbasi, S., Numerical solutions of hyperbolic telegraph equation by using the Bessel functions of first kind and residual correction, Appl. Math. Comput., 287, 83-93 (2016) · Zbl 1410.65405 |
| [37] | Yuzbasi, S.; Karacayir, M., A Galerkin-like scheme to solve two-dimensional telegraph equation using collocation points in initial and boundary conditions, Comput. Math. Appl., 74, 3242-3249 (2017) · Zbl 1397.65211 |
| [38] | Yuzbasi, S., A collocation approach for solving two-dimensional second-order linear hyperbolic equations, Appl. Math. Comput., 338, 101-114 (2018) · Zbl 1427.65312 |
| [39] | Yuzbasi, S.; Karacayir, M., A Galerkin-type method to solve one-dimensional telegraph equation using collocation points in initial and boundary conditions, Int. J. Comput. Methods, 15, Article 1850031 pp. (2018) · Zbl 1404.65186 |
| [40] | Lin, J.; Chen, F.; Zhang, Y.; Lu, J., An accurate meshless collocation technique for solving two-dimensional hyperbolic telegraph equations in arbitrary domains, Eng. Anal. Bound. Elem., 108, 372-384 (2019) · Zbl 1464.65147 |
| [41] | Priyadarshini, I.; Mohanty, R. K., High resolution half-step compact numerical approximation for 2D quasilinear elliptic equations in vector form and the estimates of normal derivatives on an irrational domain, Soft Comput., 25, 9967-9991 (2021) · Zbl 1498.65188 |
| [42] | Priyadarshini, I.; Mohanty, R. K., High resolution compact numerical method for the system of 2D quasilinear elliptic boundary value problems and the solution of normal derivatives on an irrational domain with engineering applications, Eng. Comput., 38, S1, 539-560 (2022) |
| [43] | Lees, M., Alternating direction methods for hyperbolic differential equations, J. Soc. Ind. Appl. Math., 10, 610-616 (1962) · Zbl 0111.29204 |
| [44] | Gourlay, A. R.; Mitchell, A. R., A classification of split difference methods for hyperbolic equations in several space dimensions, SIAM J. Numer. Anal., 6, 62-71 (1969) · Zbl 0175.16203 |
| [45] | Mohanty, R. K., High accuracy difference schemes for a class of three space dimensional singular parabolic equations with variable coefficients, J. Comput. Appl. Math., 89, 39-51 (1997) · Zbl 0904.65085 |
| [46] | Dehghan, M., A new ADI technique for two-dimensional parabolic equation with an integral condition, Comput. Math. Appl., 43, 1477-1488 (2002) · Zbl 1001.65094 |
| [47] | Mohanty, R. K.; Ghosh, B. P., Absolute stability of an implicit method based on third-order off-step discretization for the initial-value problem on a graded mesh, Eng. Comput., 37, 809-822 (2021) |
| [48] | Mohanty, R. K.; Ghosh, B. P., On the absolute stability of a two-step third order method on a graded mesh for an initial-value problem, Comput. Appl. Math., 40, 35 (2021) · Zbl 1476.65136 |
| [49] | Hageman, L. A.; Young, D. M., Applied Iterative Methods (2004), Dover Publication: Dover Publication New York · Zbl 1059.65028 |
| [50] | Jain, M. K., Numerical Solution of Differential Equations: Finite Difference and Finite Element Methods (2018), New AGE International Publication: New AGE International Publication New Delhi · Zbl 1398.65003 |
| [51] | Mohanty, R. K.; Ghosh, B. P., High resolution operator compact implicit half-step approximation for 3D quasi-linear hyperbolic equations and ADI method for 3D telegraphic equation on an irrational domain, Appl. Numer. Math., 172, 446-474 (2022) · Zbl 1484.65185 |
| [52] | Polyanin, A. D.; Zaitsev, V. F., Nonlinear Partial Differential Equations (2003), Chapman and Hall/CRC: Chapman and Hall/CRC New York · Zbl 1024.35001 |
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.
