Uniformly Accurate Methods for Klein-Gordon type Equations

The main contribution of this thesis is the development of a novel class of uniformly accurate methods for Klein-Gordon type equations. Klein-Gordon type equations in the non-relativistic limit regime, i.e., $c\gg 1$, are numerically very challenging to treat, since the solutions are highly oscillatory in time. Standard Gautschi-type methods suffer from severe time step restrictions as they require a CFL-condition $c^2\tau<1$ with time step size $\tau$ to resolve the oscillations. Within this thesis we overcome this difficulty by introducing limit integrators, which allows us to reduce the highly oscillatory problem to the integration of a non-oscillatory limit system. This procedure allows error bounds of order $\mathcal{O}(c^{-2}+\tau^2)$ without any step size restrictions. Thus, these integrators are very efficient in the regime $c\gg 1$. However, limit integrators fail for small values of $c$. In order to derive numerical schemes that work well for small as well as for large $c$, we use the ansatz of "twisted variables", which allows us to develop uniformly accurate methods with respect to $c$. In particular, we introduce efficient and robust uniformly accurate exponential-type integrators which resolve the solution in the relativistic regime as well as in the highly oscillatory non-relativistic regime without any step size restriction. In contrast to previous works, we do not employ any asymptotic nor multiscale expansion of the solution. Compared to classical methods our new schemes allow us to reduce the regularity assumptions as they converge under the same regularity assumptions required for the integration of the corresponding limit system. In addition, the newly derived first- and second-order exponential-type integrators converge to the classical Lie and Strang splitting schemes for the limit system. Moreover, we present uniformly accurate schemes for the Klein-Gordon-Schrödinger and the Klein-Gordon-Zakharov system. For all uniformly accurate integrators we establish rigorous error estimates and underline their uniform convergence property numerically

Modeling water waves beyond perturbations

In this chapter, we illustrate the advantage of variational principles for modeling water waves from an elementary practical viewpoint. The method is based on a `relaxed' variational principle, i.e., on a Lagrangian involving as many variables as possible, and imposing some suitable subordinate constraints. This approach allows the construction of approximations without necessarily relying on a small parameter. This is illustrated via simple examples, namely the Serre equations in shallow water, a generalization of the Klein-Gordon equation in deep water and how to unify these equations in arbitrary depth. The chapter ends with a discussion and caution on how this approach should be used in practice.Comment: 15 pages, 1 figure, 39 references. This document is a contributed chapter to an upcoming volume to be published by Springer in Lecture Notes in Physics Series. Other author's papers can be downloaded at http://www.denys-dutykh.com

Practical use of variational principles for modeling water waves

This paper describes a method for deriving approximate equations for irrotational water waves. The method is based on a 'relaxed' variational principle, i.e., on a Lagrangian involving as many variables as possible. This formulation is particularly suitable for the construction of approximate water wave models, since it allows more freedom while preserving the variational structure. The advantages of this relaxed formulation are illustrated with various examples in shallow and deep waters, as well as arbitrary depths. Using subordinate constraints (e.g., irrotationality or free surface impermeability) in various combinations, several model equations are derived, some being well-known, other being new. The models obtained are studied analytically and exact travelling wave solutions are constructed when possible.Comment: 30 pages, 1 figure, 62 references. Other author's papers can be downloaded at http://www.denys-dutykh.com

Numerical schemes for general Klein–Gordon equations with Dirichlet and nonlocal boundary conditions

In this work, we address the problem of solving nonlinear general Klein–Gordon equations (nlKGEs). Different fourth- and sixth-order, stable explicit and implicit, finite difference schemes are derived. These new methods can be considered to approximate all type of Klein–Gordon equations (KGEs) including phi-four, forms I, II, and III, sine-Gordon, Liouville, damped Klein–Gordon equations, and many others. These KGEs have a great importance in engineering and theoretical physics.The higher-order methods proposed in this study allow a reduction in the number of nodes, which might also be very interesting when solving multi-dimensional KGEs. We have studied the stability and consistency of the proposed schemes when considering certain smoothness conditions of the solutions. Additionally, both the typical Dirichlet and some nonlocal integral boundary conditions have been studied. Finally, some numerical results are provided to support the theoretical aspects previously considered

Efficient high-order finite difference methods for nonlinear Klein–Gordon equations. I: Variants of the phi-four model and the form-I of the nonlinear Klein–Gordon equation

In this paper, the problem of solving some nonlinear Klein–Gordon equations (KGEs) is considered. Here, we derive different fourth- and sixth-order explicit and implicit algorithms to solve the phi-four equation and the form-I of the nonlinear Klein–Gordon equation. Stability and consistency of the proposed schemes are studied under certain conditions. Numerical results are presented and then compared with others obtained from some methods already existing in the scientific literature to explain the efficiency of the new algorithms. It is also shown that similar schemes can be proposed to solve many classes of nonlinear KGEs

Numerical study of the generalised Klein-Gordon equations

24 pages, 10 figures, 56 references. Other author's papers can be downloaded at http://www.denys-dutykh.com/International audienceIn this study, we discuss an approximate set of equations describing water wave propagating in deep water. These generalized Klein-Gordon (gKG) equations possess a variational formulation, as well as a canonical Hamiltonian and multi-symplectic structures. Periodic travelling wave solutions are constructed numerically to high accuracy and compared to a seventh-order Stokes expansion of the full Euler equations. Then, we propose an efficient pseudo-spectral discretisation, which allows to assess the stability of travelling waves and localised wave packets

Accelerating the Fourier split operator method via graphics processing units

Current generations of graphics processing units have turned into highly parallel devices with general computing capabilities. Thus, graphics processing units may be utilized, for example, to solve time dependent partial differential equations by the Fourier split operator method. In this contribution, we demonstrate that graphics processing units are capable to calculate fast Fourier transforms much more efficiently than traditional central processing units. Thus, graphics processing units render efficient implementations of the Fourier split operator method possible. Performance gains of more than an order of magnitude as compared to implementations for traditional central processing units are reached in the solution of the time dependent Schr\"odinger equation and the time dependent Dirac equation