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Numerical solutions to the Rosenau-Kawahara equation for shallow water waves via pseudo-compact methods. (English) Zbl 1468.65101

Summary: This paper presents two linear finite difference schemes for the so-called Rosenau-Kawahara equation, modified from a linear scheme by J. Hu et al. [Adv. Math. Phys. 2014, Article ID 217393, 11 p. (2014; Zbl 1302.65191)], under a pseudo-compact method. Existence and uniqueness of solutions generated by both schemes are proved. It is shown that the first scheme possesses some conservation properties for mass and energy, whereas the other proposed scheme provides only mass conservation. Some discussions on stability are given, which reveal that numerical solutions are stable with respect to \(\|\cdot\|_{\infty}\). It is also shown that pseudo-compactness allows some terms in the schemes to reach fourth-order convergence, even though the numerical solutions is of second-order convergence overall. Furthermore, numerical simulations are illustrated confirming that our schemes induce some improvements over the existing scheme by Hu et al. [loc. cit.] on precision and cost.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
86A05 Hydrology, hydrography, oceanography

Citations:

Zbl 1302.65191
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References:

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