On the rate of error growth in time for numerical solutions of nonlinear dispersive wave equations

We study the numerical error in solitary wave solutions of nonlinear dispersive wave equations. A number of existing results for discretizations of solitary wave solutions of particular equations indicate that the error grows quadratically in time for numerical methods that do not conserve energy, but grows only linearly for conservative methods. We provide numerical experiments suggesting that this result extends to a very broad class of equations and numerical methods.

Kato smoothing properties of a class of nonlinear dispersive wave equations on a periodic domain

The solutions of the Cauchy problem of the KdV equation on a periodic domain $\T$, \[ u_t +uu_x +u_{xxx} =0, \quad u(x,0)= \phi (x), \quad x\in \T, \ t\in \R,\] possess neither the sharp Kato smoothing property, \[ \phi \in H^s (\T) \implies \partial ^{s+1}_xu \in L^{\infty}_x (\T, L^2 (0,T)),\] nor the Kato smoothing property, \[ \phi \in H^s (\T) \implies u\in L^2 (0,T; H^{s+1} (\T)).\] Considered in this article is the Cauchy problem of the following dispersive equations posed on the periodic domain $\T$, \[ u_t +uu_x +u_{xxx} - g(x) (g(x) u)_{xx} =0, \qquad u(x,0)= \phi (x), \quad x\in \T, \ t>0 \, ,\ \qquad (1) \] where $g\in C^{\infty} (\T)$ is a real value function with the support \[ \mbox{$\omega = \{ x\in \T, \ g(x) \ne 0\}$.}\] It is shown that \begin{itemize} \item[(1)] if $\omega\ne \emptyset$, then the solutions of the Cauchy problem (1) possess the Kato smoothing property; \item[(2)] if $g$ is a nonzero constant function, then the solutions of the Cauchy problem (1) possess the sharp Kato smoothing property. \end{itemize}

Finite volume schemes for Boussinesq type equations

Finite volume schemes are commonly used to construct approximate solutions to conservation laws. In this study we extend the framework of the finite volume methods to dispersive water wave models, in particular to Boussinesq type systems. We focus mainly on the application of the method to bidirectional nonlinear, dispersive wave propagation in one space dimension. Special emphasis is given to important nonlinear phenomena such as solitary waves interactions.

Finite volume schemes for Boussinesq type equations

Finite volume schemes are commonly used to construct approximate solutions to conservation laws. In this study we extend the framework of the finite volume methods to dispersive water wave models, in particular to Boussinesq type systems. We focus mainly on the application of the method to bidirectional nonlinear, dispersive wave propagation in one space dimension. Special emphasis is given to important nonlinear phenomena such as solitary waves interactions.