Two structure-preserving schemes with fourth-order accuracy for the modified Kawahara equation

Abstract

In this article, we present two high-order structure-preserving difference schemes for the modified Kawahara equation, which are named as Scheme I and Scheme II, respectively. Scheme I is a compact fourth-order difference scheme with a seven-point stencil and preserves discrete mass, while Scheme II is a standard fourth-order difference scheme with a nine-point stencil and preserves discrete energy. The proposed two schemes are three-level implicit and the numerical convergence order is $O(\tau ^{2}+h^{4})$. The unconditional stability of Scheme I and Scheme II is proven by von Neumann’s analysis. According to the Lax equivalence theorem, the convergence of the two schemes is also presented. The errors and rates of convergence, the discrete conservative mass $Q^{n}$ and energy $E^{n}$ are compared with those from other schemes. At last, some numerical experiments are given to demonstrate that the two proposed schemes are accurate and efficient for handling the single and multi-solitary waves propagating over a long period.