Spectral Collocation Solutions to a Class of Pseudo-parabolic Equations
In this paper we solve by method of lines (MoL) a class of pseudo-parabolic PDEs defined on the real line. The method is based on the sinc collocation (SiC) in order to discretize the spatial derivatives as well as to incorporate the asymptotic behavior of solution at infinity. This MoL casts an initial value problem attached to these equations into a stiff semi-discrete system of ODEs with mass matrix independent of time. A TR-BDF2 finite difference scheme is then used in order to march in time.
The method does not truncate arbitrarily the unbounded domain to a finite one and does not assume the periodicity. These are two omnipresent, but non-natural, ingredients used to handle such problems.
The linear stability of MoL is proved using the pseudospectrum of the discrete linearized operator. Some numerical experiments are carried out along with an estimation of the accuracy in conserving two invariants. They underline the efficiency and robustness of the method. The convergence order of MoL is also established.
KeywordsPseudo-parabolic equation Infinite domain Camassa-Holm Peakon Sinc collocation TR-BDF2 Linear stability Pseudospectrum
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