Pseudospectral Methods for the Fractional Laplacian on R

118 p.In this thesis, first, we propose a novel pseudospectral method to approximate accurately and efficiently the fractional Laplacian without using truncation. More precisely, given a bounded regular function defined over R, we map the unbounded domain into a finite one, and represent the resulting function as a trigonometric series. Therefore, a key ingredient is the computation of the fractional Laplacian of an elementary trigonometric function. As an application of the method, we do the simulation of Fisher¿s equation with the fractionalLaplacian in the monostable case.In addition, using complex variable techniques, we compute explicitly, in terms of the 2F1 Gaussian hypergeometric function, the one-dimensional fractional Laplacian of the Higgins functions, the Christov functions, and their sine-like and cosine-like versions. After discussing the numerical difficulties in the implementation of the proposed formulas, we develop another method that gives exact results, by using variable precision arithmetic.Finally, we discuss some other numerical approximations of the fractional Laplacian using a fast convolution technique. While the resulting techniques are less accurate, they are extremely fast; furthermore, the results can be improved by the use of Richardson's extrapolation

Pseudospectral Methods for the Fractional Laplacian on R

In this thesis, first, we propose a novel pseudospectral method to approximate accu- rately and efficiently the fractional Laplacian without using truncation. More pre- cisely, given a bounded regular function defined over R, we map the unbounded domain into a finite one, and represent the resulting function as a trigonometric se- ries. Therefore, a key ingredient is the computation of the fractional Laplacian of an elementary trigonometric function. As an application of the method, we do the simulation of Fisher’s equation with the fractional Laplacian in the monostable case. In addition, using complex variable techniques, we compute explicitly, in terms of the 2 F1 Gaussian hypergeometric function, the one-dimensional fractional Laplacian of the Higgins functions, the Christov functions, and their sine-like and cosine-like versions. After discussing the numerical difficulties in the implementation of the proposed formulas, we develop another method that gives exact results, by using variable precision arithmetic. Finally, we discuss some other numerical approximations of the fractional Laplacian using a fast convolution technique. While the resulting techniques are less accu- rate, they are extremely fast; furthermore, the results can be improved by the use of Richardson’s extrapolation

Non-local regularisations of scalar conservation laws

112 p.This thesis focuses on studying equations related to a model problem derived in a Shallow-Waterlimit. These equations are non-local higher-order regularisations of a scalar conservation law with,typically, either a quadratic or a cubic non-linear flux. It is known that hyperbolic conservation lawsexhibit discontinuous solutions and, in general, weak solutions are non-unique. The classical way toderive uniqueness for such systems is by regularising with viscous terms, typically of second order, andthen perform the vanishing viscosity limit. However, other type of regularisations may arise dependingon the physical or modelling set-up. An example of such regularised equations is the model justmentioned. This is a generalised Korteweg-de Vries-Burgers equation with a non-local linear diffusion,which is an operator of the Riesz-Feller type, and a local and linear dispersion term.It is the aim of this thesis to advance in the analysis of this particular model. First, the purelyviscous version of the equation is studied and the vanishing viscosity limit is proved applying the doublescale technique of Kru¿kov. Subsequently, a generalisation of this result to a more general Riesz-Felleroperator is given as well as the asymptotic behaviour of travelling wave solutions in the tail. The secondpart of the thesis is devoted to proving the existence of travelling waves for the full model with a cubicnon-linearity. The existence of waves that violate the classical Lax condition is shown. Formally, thesesolutions would ensue in the limit of vanishing diffusion and dispersion, at the right rate, and give rise tonon-classical shocks. The work is completed with a study of large time behaviour in the purely viscouscase. It is concluded that, for the sub-critical case of a paradigm locally Lipschitz flux, the large timeasymptotic behaviour is given by the unique entropy solution of the scalar conservation law