Abstract
In this paper, two classes of methods are developed for the solution of two space dimensional wave equations with a nonlinear source term. We have used non-polynomial cubic spline function approximations in both space directions. The methods involve some parameters, by suitable choices of the parameters, a new high accuracy three time level scheme of order O(h 4 + k 4 + τ 2 + τ 2 h 2 + τ 2 k 2) has been obtained. Stability analysis of the methods have been carried out. The results of some test problems are included to demonstrate the practical usefulness of the proposed methods. The numerical results for the solution of two dimensional sine-Gordon equation are compared with those already available in literature.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Strauss, W.: Nonlinear wave equations. CBMS 73, AMS (1989)
Mohanty, R.K.: An unconditionally stable difference scheme for the one space dimensional linear hyperbolic equation. Appl. Math. Lett. 17, 101–105 (2004)
Gao, F., Chi, C.: Unconditionally stable difference schemes for a one-space dimensional linear hyperbolic equation. Appl. Math. Comput. 187, 1272–1276 (2007)
Mohebbi, A., Dehghan, M.: High order compact solution of the one-space dimensional linear hyperbolic equation. Numer. Methods. Partial. Differ. Equ. 24, 1222–1235 (2008)
Raggett, G.F., Wilson, P.D.: A fully implicit finite difference approximation to the one-dimensional wave equation using a cubic spline technique. J. Inst. Math. Appl. 14, 75–77 (1974)
Rashidinia, J., Jalilian, R., Kazemi, V.: Spline methods for the solution of hyperbolic equations. Appl. Math. Comput. 190, 882–886 (2007)
Ding, H., Zhang, Y.: A new unconditionally stable compact difference scheme of for the 1D linear hyperbolic equation. Appl. Math. Comput. 207, 236–241 (2009)
Liu, H.W., Liu, L.B.: An unconditionally stable spline difference scheme of O(k 2 + h 4) for solving the second order 1D linear hyperbolic equation. Math. Comput. Model. 49, 1985–1993 (2009)
Mohanty, R.K., Gopal, V.: High accuracy cubic spline finite difference approximation for the solution of one-space dimensional non-linear wave equations. Appl. Math. Comput. 218(8), 4234–4244 (2011)
Mohanty, R.K., Gopal, V.: An off-step discretization for the solution of 1D mildly nonlinear wave equations with variable coefficients. J. Adv. Res. Sci. Comput. 04(02), 1–13 (2012)
Rashidinia, J., Mohammadi, R.: Tension spline approach for the numerical solution of nonlinear Klein-Gordon equation. Comput. Phys. Comm. 181(1), 78–91 (2010)
Rashidinia, J., Mohammadi, R.: Tension spline solution of nonlinear sine-Gordon equation. Numer. Algor. 56(1), 129–142 (2011)
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)
Mittal, R.C., Bhatia, R.: Numerical solution of second order one dimensional hyperbolic telegraph equation by cubic B-spline collocation method. Appl. Math. Comput. 220, 496–506 (2013)
Dosti, M., Nazemi, A.: Quartic B-spline collocation method for solving one-dimensional hyperbolic telegraph equation. J. Inf. Comput. Sci. 7(2), 083–090 (2012)
Mohanty, R.K., Jain, M.K.: An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation. Numer. Methods. Partial Differ. Equ. 17(6), 684–688 (2001)
Mohanty, R.K.: An operator splitting method for an unconditionally stable difference scheme for a linear hyperbolic equation with variable coefficients in two space dimensions. Appl. Math. Comput. 152, 799–806 (2004)
Ding, H.F., Zhang, Y.X.: A new fourth order compact finite difference scheme for the two-dimensional second order hyperbolic equations. J. Comput. Appl. Math 230, 626–632 (2009)
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)
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)
Dehghan, M., Salehi, R.: A method based on meshless approach for the numerical solution of the two space dimensional hyperbolic telegraph equation. Math. Method. Appl. Sci. 35(10), 1220–1233 (2012)
Piperno, S.: Symplectic local time-stepping in non-dissipative dgtd methods applied to wave propagation problems. ESAIM: Math. Modelling. Numer. Analysis 5, 815–841 (2006)
Shi, D.Y., Li, Z.Y.: Superconvergence analysis of the finite element method for nonlinear hyperbolic equations with nonlinear boundary condition. Applied Mathematics - A Journal of the Chinese Universities 4, 455–462 (2008)
Chabassier, J., Joly, P.: Energy preserving schemes for nonlinear hamiltonian systems of wave equations: application to the vibrating piano string. Computer Methods. Appl. Mechanic. Engineering 45, 2779–2795 (2010)
Chawla, M.M., Al-Zanaidi, M.A.: A linearly implicit one-step time integration scheme for nonlinear hyperbolic equations. Intern. J. Computer Math. 76, 349–361 (2001)
Chawla, M.M., Al-Zanaidi, M.A.: A linearly implicit one-step time integration scheme for nonlinear hyperbolic equations in two space dimensions. Intern. J. Computer Math. 80(3), 357–365 (2003)
Djidjeli, K., Price, W.G., Twizell, E.H.: Numerical solutions of a damped sine-Gordon equation in two space variables. J. Eng. Math. 29, 347–369 (1995)
Dehghan, M., Shokri, A.: A numerical method for solution of the two dimensional sine-Gordon equation using the radial basis functions. Math. Comput. Simulation 79, 700–715 (2008)
Jiwari, R., Pandit, S., Mittal, R.C.: Numerical simulation of two dimensional sine-Gordon solitons by differential quadrature method. Comput. Phys. Commun. 183, 600–616 (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zadvan, H., Rashidinia, J. Non-polynomial spline method for the solution of two-dimensional linear wave equations with a nonlinear source term. Numer Algor 74, 289–306 (2017). https://doi.org/10.1007/s11075-016-0149-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-016-0149-0
Keywords
- Non-polynomial spline approximation
- Two-dimensional wave equation
- Stability analysis
- Sine-Gordon equation