Stable spectral collocation solutions to a class of Benjamin Bona Mahony initial value problem


We are concerned with stable spectral collocation solutions to non-periodic Benjamin Bona Mahony (BBM), modified BBM and Benjamin Bona Mahony-Burgers (BBM-B) initial value problems on the real axis. The spectral collocation is based alternatively on the scaled Hermite and sinc functions. In order to march in time we use several one step and linear multistep finite difference schemes such that the method of lines (MoL) involved is stable in sense of Lax. The method based on Hermite functions ensures the correct behavior of the solutions at large spatial distances and in long time periods. In order to prove the stability we use the pseudospectra of the linearized spatial discretization operators. The extent at which the energy integral of BBM model is conserved over time is analyzed for Hermite collocation along with various finite difference schemes. This analysis has been fairly useful in optimizing the scaling parameter. The effectiveness of our approach has been confirmed by some numerical experiments.


Călin-Ioan Gheorghiu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)


Regularized long wave equation; scaled Hermite collocation; method of lines; Lax stability; energy conservation.


See the expanding block below.

Paper coordinates

C.I. Gheorghiu, Stable spectral collocation solutions to a class of Benjamin Bona Mahony initial value problems. Appl. Math. Comp., 273 (2016) 1090-1099
doi: 10.1016/j.amc.2015.10.078


not available yet.

About this paper

Print ISSN


Online ISSN
Google Scholar Profile

google scholar link

[1] T.B. Benjamin, J.L. Bona, J.J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser.A. 272 (1972) 47–78.

[2] J.L. Bona, W.G. Pritchard, L.R. Scott, Comparison of two model equations for long waves, AMS Lect. Appl. Math. 20 (1983) 235–267.

[3] J.P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd, DOVER, 2000.

[4] D. Dutykh, T. Katsaounis, D. Mitsotakis, Finite volume schemes for dispersive wave propagation and runup, J. Comput. Phys. 230 (2011) 3035–3061.

[5] D. Funaro, Polynomial Approximation of Differential Equations, Springer Verlag, Berlin, 1992.

[6] C.I. Gheorghiu, Laguerre collocation solutions to boundary layer type problems, Numer. Algor. 64 (2013) 385–401.

[7] C.I. Gheorghiu, Pseudospectral solutions to some singular nonlinear BVPs, Appl. Nonlinear Mech. Numer. Algorithm. (2015), doi:10.1007/s11075-014-9834-z

[8] C.I. Gheorghiu, Spectral Methods for Non-Standard Eigenvalue Problems. Fluid and Structural Mechanics and Beyond, Springer Briefs in Mathematics, Springer, Cham Heidelberg New York Dordrecht London, 2014.

[9] D.J. Higham, L.N. Trefethen, Stifness of ODEs, BIT 33 (1993) 285–303.

[10] A.-K. Kassam, L.N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAM J. Sci. Comp. 26 (2005) 1214–1233.

[11] H.G. Marshall, J.P. Boyd, Solitons in a continuously stratified equatorial ocean, J. Phys. Oceanogr. 17 (1987) 1016-1031.

[12] F.W.J. Olver, D.M. Lozier, R.F. Boisvert, C.W. Clark (Eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010.

[13] K. Omrani, The convergence of fully discrete Galerkin approximations for the Benjamin-Bona-Mahony (BBM) equation, Appl. Math. Comput. 180 (2006) 614–621.

[14] K. Omrani, M. Ayadi, Finite difference discretization of the Benjamin-Bona-Mahony-Burgers Equation, Numer. Methods Partial Differential Eq. 24 (2008) 239–248.

[15] A. Quarteroni, F. Saleri, Scientific Computing with MATLAB and Octave, 2nd Ed., Springer, Springer-Verlag Berlin Heidelberg, 2006.

[16] S.C. Reddy, L.N. Trefethen, Stability of the method of lines, Numer. Math. 62 (1992) 235–267.

[17] F. Stenger, Summary of Sinc numerical methods, J. Comput. Appl. Math. 121 (2000) 379–420.

[18] L.N. Trefethen, Spectral methods in MATLAB, SIAM (2000).

[19] L. Wahlbin, Error estimates for a Galerkin method for a class of model equations for long waves, Numer. Math. 23 (1975) 289–303.

[20] J.A.C. Weideman, S.C. Reddy, A MATLAB differentiation matrix suite, ACM Trans. Math. Software 26 (2000) 465–519.


Related Posts