Compact difference scheme with high accuracy for one dimensional unsteady quasi-linear biharmonic problem of second kind: Application to physical problems

R. K. Mohanty ; D. Kaur

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 21 (2018) no. 1, pp. 65-82.

Résumé

The present paper uses a new two level implicit difference formula for the numerical study of one dimensional unsteady biharmonic equation with appropriate initial and boundary conditions. The proposed difference scheme is second order accurate in time and third order accurate in space on a non-uniform grid and in case of a uniform mesh, it is of order two in time and four in space. The approximate solutions are computed without using any transformation and linearization. The simplicity of the proposed scheme lies in its three point spatial discretization which yields a block tri-diagonal matrix structure without the use of any fictitious nodes for handling the boundary conditions. The proposed scheme is directly applicable to singular problems, which is the main utility of our work. The method is shown to be unconditionally stable for a model linear problem for a uniform mesh. The efficacy of the proposed approach has been tested on several physical problems, including complex fourth-order nonlinear equations like the Kuramoto–Sivashinsky equation and the extended Fisher–Kolmogorov equation, where comparison is made with some earlier work. It is clear from the numerical experiments that the obtained results are not only in good agreement with the exact solutions but also competitive with the solutions derived in earlier research studies.

Bibliographie

[1] Akrivis G., Smyrlis Y.-S., “Implicit-explicit BDF methods for the Kuramoto–Sivashinsky equation”, Appl. Numer. Math., 51 (2004), 151–169 | DOI | MR

[2] Aronson D. G., Weinberger H. F., “Multidimensional nonlinear diffusion arising in population genetics”, Adv. Math., 30 (1978), 33–76 | DOI | MR

[3] Danumjaya P., Pani A. K., Finite element methods for the extended Fisher–Kolmogorov (EFK) equation, Research Report: IMG-RR-2002-3, Industrial Mathematics Group, Department of Mathematics, IIT, Bombay, 2002 | MR

[4] Danumjaya P., Pani A. K., “Orthogonal cubic spline collocation method for the extended Fisher–Kolmogorov equation”, J. Comput. Appl. Math., 174 (2005), 101–117 | DOI | MR

[5] Dee G. T., van Saarloos W., “Bistable systems with propagating fronts leading to pattern formation”, Phys. Rev. Lett., 60 (1988), 2641–2644 | DOI

[6] Doss L. J. T., Nandini A. P., “An $H^1$-Galerkin mixed finite element method for the extended Fisher–Kolmogorov equation”, Int. J. Numer. Anal. Model. Ser. B, 3 (2012), 460–485 | MR

[7] Fan E., “Extended tanh-function method and its applications to nonlinear equations”, Phys. Lett. A, 277 (2000), 212–218 | DOI | MR

[8] Ganaie I. A., Arora S., Kukreja V. K., “Cubic Hermite collocation solution of Kuramoto–Sivashinsky equation”, Int. J. Comput. Math., 93 (2016), 223–235 | DOI | MR

[9] Hageman L. A., Young D. M., Applied Iterative Methods, Dover Publication, New York, 2004 | MR

[10] Haq Sirajul, Bibi Nagina, Tirmizi S. I. A., Usman M., “Meshless method of lines for the numerical solution of generalized Kuramoto–Sivashinsky equation”, Appl. Math. Comput., 217 (2010), 2404–2413 | MR

[11] Hooper A. P., Grimshaw R., “Nonlinear instability at the interface between two viscous fluids”, Phys. Fluids, 28 (1985), 37–45 | DOI

[12] Kelley C. T., Iterative Methods for Linear and Non-linear Equations, SIAM Publications, Philadelphia, 1995 | MR

[13] Khater A. H., Temsah R. S., “Numerical solutions of the generalized Kuramoto–Sivashinsky equation by Chebyshev spectral collocation methods”, Comput. Math. Appl., 56 (2008), 1465–1472 | DOI | MR

[14] Kuramoto Y., Tsuzuki T., “Persistent propagation of concentration waves in dissipative media far from thermal equilibrium”, Prog. Theor. Phys., 55 (1976), 356–569 | DOI

[15] Lai Huilin, Ma Changfeng, “Lattice Boltzmann method for the generalized Kuramoto–Sivashinsky equation”, Phys. A, 388 (2009), 1405–1412 | DOI | MR

[16] Mitchell A. R., Computational Methods in Partial Differential Equations, John Wiley Sons, New York, 1969 | MR

[17] Mittal R. C., Arora G., “Quintic B-spline collocation method for numerical solution of the Kuramoto–Sivashinsky equation”, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 2798–2808 | DOI | MR

[18] Mohanty R. K., “An accurate three spatial grid-point discretization of $O(k^2+h^4)$ for the numerical solution of one-space dimensional unsteady quasi-linear biharmonic problem of second kind”, Appl. Math. Comput., 140 (2003), 1–14 | MR

[19] Mohanty R. K., Kaur Deepti, “High accuracy implicit variable mesh methods for numerical study of special types of fourth order non-linear parabolic equations”, Appl. Math. Comput., 273 (2016), 678–696 | MR

[20] Mohanty R. K., Kaur Deepti, “A class of quasi-variable mesh methods based on off-step discretization for the numerical solution of fourth-order quasi-linear parabolic partial differential equations”, Adv. Difference Equ., 2016, Paper No. 326, 29 pp. | DOI | MR

[21] Mohanty R. K., Kaur Deepti, “Numerov type variable mesh approximations for 1D unsteady quasi-linear biharmonic problem: application to Kuramoto–Sivashinsky equation”, Numer. Algor., 74 (2017), 427–459 | DOI | MR

[22] Mohanty R. K., McKee Sean, Kaur Deepti, “A class of two-level implicit unconditionally stable methods for a fourth order parabolic equation”, Appl. Math. Comput., 309 (2017), 272–280 | MR

[23] Saprykin S., Demekhin E. A., Kalliadasis S., “Two-dimensional wave dynamics in thin films. I. Stationary solitary pulses”, Phys. Fluids, 17:11 (2005), 117105, 16 pp. | DOI | MR

[24] Sivashinsky G., “Nonlinear analysis of hydrodynamic instability in laminar flames. Part I. Derivation of basic equations”, Acta Astronaut., 4 (1977), 1117–1206 | DOI | MR

[25] Stephenson J. W., “Single cell discretizations of order two and four for biharmonic problems”, J. Comput. Phys., 55 (1984), 65–80 | DOI | MR

[26] Tatsumi T., “Irregularity, regularity and singularity of turbulence”, Turbulence and chaotic phenomena in fluids, Proc. Inter. Symposium (Kyoto, Japan, 1983), North-Holland, 1984, 1–10 | MR

[27] Uddin Marjan, Haq Sirajul, Siraj-ul-Islam, “A mesh-free numerical method for solution of the family of Kuramoto–Sivashinsky equations”, Appl. Math. Comput., 212 (2009), 458–469 | MR

[28] Yan Xu, Shu Chi-Wang, “Local discontinuous Galerkin methods for the Kuramoto–Sivashinsky equations and the Ito-type coupled KdV equations”, Comput. Methods Appl. Mech. Eng., 195 (2006), 3430–3447 | DOI | MR

[29] Ye Lina, Yan Guangwu, Li Tingting, “Numerical method based on the lattice Boltzmann model for the Kuramoto–Sivashisky equation”, J. Sci. Comput., 49 (2011), 195–210 | DOI | MR

[30] Zimmermann W., “Propagating fronts near a Lifshitz point”, Phys. Rev. Lett., 66 (1991), 1546 | DOI