The design of conservative finite element discretisations for the vectorial modified KdV equation

We design a consistent Galerkin scheme for the approximation of the vectorial modified Korteweg-de Vries equation. We demonstrate that the scheme conserves energy up to machine precision. In this sense the method is consistent with the energy balance of the continuous system. This energy balance ensures there is no numerical dissipation allowing for extremely accurate long time simulations free from numerical artifacts. Various numerical experiments are shown demonstrating the asymptotic convergence of the method with respect to the discretisation parameters. Some simulations are also presented that correctly capture the unusual interactions between solitons in the vectorial setting

Numerical superposition of Gaussian beams over propagating domain for high frequency waves and high-order invariant-preserving methods for dispersive waves

This thesis is devoted to efficient numerical methods and their implementations for two classes of wave equations. The first class is linear wave equations in very high frequency regime, for which one has to use some asymptotic approach to address the computational challenges. We focus on the use of the Gaussian beam superposition to compute the semi--classical limit of the Schr {o}dinger equation. The second class is dispersive wave equations arising in modeling water waves. For the Whitham equation, so-called the Burgers--Poisson equation, we design, analyze, and implement local discontinuous Galerkin methods to compute the energy conservative solutions with high-order of accuracy. Our Gaussian beam (GB) approach is based on the domain-propagation GB superposition algorithm introduced by Liu and Ralston [Multiscale Model. Simul., 8(2), 2010, 622--644]. We construct an efficient numerical realization of the domain propagation-based Gaussian beam superposition for solving the Schr odinger equation. The method consists of several significant steps: a semi-Lagrangian tracking of the Hamiltonian trajectory using the level set representation, a fast search algorithm for the effective indices associated with the non-trivial grid points that contribute to the approximation, an accurate approximation of the delta function evaluated on the Hamiltonian manifold, as well as efficient computation of Gaussian beam components over the effective grid points. Numerical examples in one and two dimensions demonstrate the efficiency and accuracy of the proposed algorithms. For the Burgers--Poisson equation, we design, analyze and test a class of local discontinuous Galerkin methods. This model, proposed by Whitham [Linear and Nonlinear Waves, John Wiley & Sons, New York, 1974] as a simplified model for shallow water waves, admits conservation of both momentum and energy as two invariants. The proposed numerical method is high order accurate and preserves two invariants, hence producing solutions with satisfying long time behavior. The $L^2$-stability of the scheme for general solutions is a consequence of the energy preserving property. The optimal order of accuracy for polynomial elements of even degree is proven. A series of numerical tests is provided to illustrate both accuracy and capability of the method

A numerical study of variational discretizations of the Camassa-Holm equation

We present two semidiscretizations of the Camassa-Holm equation in periodic domains based on variational formulations and energy conservation. The first is a periodic version of an existing conservative multipeakon method on the real line, for which we propose efficient computation algorithms inspired by works of Camassa and collaborators. The second method, and of primary interest, is the periodic counterpart of a novel discretization of a two-component Camassa-Holm system based on variational principles in Lagrangian variables. Applying explicit ODE solvers to integrate in time, we compare the variational discretizations to existing methods over several numerical examples.Comment: 45 pages, 14 figure

On error estimates for Galerkin finite element methods for the Camassa-Holm equation

We consider the Camassa-Holm (CH) equation, a nonlinear dispersive wave equation that models one-way propagation of long waves of moderately small amplitude. We discretize in space the periodic initial-value problem for CH (written in its original and in system form), using the standard Galerkin finite element method with smooth splines on a uniform mesh, and prove optimal-order $L^{2}$-error estimates for the semidiscrete approximation. We also consider an initial-boundary-value problem on a finite interval for the system form of CH and analyze the convergence of its standard Galerkin semidiscretization. Using the fourth-order accurate, explicit, "classical" Runge-Kutta scheme for time-stepping, we construct a highly accurate, stable, fully discrete scheme that we employ in numerical experiments to approximate solutions of CH, mainly smooth travelling waves and nonsmooth solitons of the `peakon' type