Open AccessA linearized implicit pseudo‐spectral method for some model equations: the regularized long wave equations
Author(s) -
Djidjeli K.,
Price W. G.,
Twizell E. H.,
Cao Q.
Publication year - 2003
Publication title -
communications in numerical methods in engineering
Language(s) - English
Resource type - Journals
eISSN - 1099-0887
pISSN - 1069-8299
DOI - 10.1002/cnm.635
Subject(s) - mathematics , wave equation , generalization , spectral method , mathematical analysis , stability (learning theory) , space (punctuation) , equations of motion , fourier transform , numerical analysis , soliton , nonlinear system , computer science , classical mechanics , physics , quantum mechanics , machine learning , operating system
An efficient numerical method is developed for the numerical solution of non‐linear wave equations typified by the regularized long wave equation (RLW) and its generalization (GRLW). The method developed uses a pseudo‐spectral (Fourier transform) treatment of the space dependence together with a linearized implicit scheme in time. =10pt An important advantage to be gained from the use of this method, is the ability to vary the mesh length, thereby reducing the computational time. Using a linearized stability analysis, it is shown that the proposed method is unconditionally stable. The method is second order in time and all‐order in space. The method presented here is for the RLW equation and its generalized form, but it can be implemented to a broad class of non‐linear long wave equations (Equation (2)), with obvious changes in the various formulae. Test problems, including the simulation of a single soliton and interaction of solitary waves, are used to validate the method, which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm. Copyright © 2003 John Wiley & Sons, Ltd.