Geometric Numerical Integration

The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods

Laplace Transform Inversion and Time-Discretization Methods for Evolution Equations

In this dissertation, we introduce Post-Widder-type inversion methods for the Laplace transform based on A-stable rational approximations of the exponential function. Since the results hold for Banach-space-valued functions, they yield efficient time-discretization methods for evolution equations of convolution type; e.g., linear first and higher order abstract Cauchy problems, inhomogeneous Cauchy problems, delay equations, Volterra and integro-differential equations, and problems that can be re-written as an abstract Cauchy problem on an appropriate state space

Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review

A review of diagonally implicit Runge-Kutta (DIRK) methods applied to rst-order ordinary di erential equations (ODEs) is undertaken. The goal of this review is to summarize the characteristics, assess the potential, and then design several nearly optimal, general purpose, DIRK-type methods. Over 20 important aspects of DIRKtype methods are reviewed. A design study is then conducted on DIRK-type methods having from two to seven implicit stages. From this, 15 schemes are selected for general purpose application. Testing of the 15 chosen methods is done on three singular perturbation problems. Based on the review of method characteristics, these methods focus on having a stage order of two, sti accuracy, L-stability, high quality embedded and dense-output methods, small magnitudes of the algebraic stability matrix eigenvalues, small values of aii, and small or vanishing values of the internal stability function for large eigenvalues of the Jacobian. Among the 15 new methods, ESDIRK4(3)6L[2]SA is recommended as a good default method for solving sti problems at moderate error tolerances

Pseudospectral methods and numerical continuation for the analysis of structured population models

In this thesis new numerical methods are presented for the analysis of models in population dynamics. The methods approximate equilibria and bifurcations in a certain class of so called structured population models. Chapter 1 consists of an introduction to structured population dynamics, where the state of the art is presented through a classical consumer-resource model [44]. The necessity of new numerical methods for analyzing structured population models is discussed and motivated by their applications to life sciences. In Chapter 2 [44] is extended to a more general class in which a structured population with a unique state at birth interacts with an environment of unstruc- tured populations and interaction variables. Equilibrium types are defined, the model is linearized and a characteristic equation is obtained. Finally, a discussion about equilibria and bifurcations under parameter variation is included. In Chapter 3 a new pseudospectral method for the computation of eigenvalues of linear VFE/DDE systems is presented. The technique consists of constructing a finite approximation of the infinitesimal generator of the solution semigroup. The spectral convergence of the method is proved, and a piecewise variation which speeds up the computations presented and validated with toy models. An exten- sion to deal with structured population models is proposed and validated with the model in [44]. Chapter 4 is devoted to the numerical continuation of equilibrium branches and bifurcation curves under parameter variation for models of the class presented in Chapter 2. A new technique for the curve continuation is presented, where a reduction of the dimension and a simplification of the equilibrium conditions result in new test functions for the detection of transcritical bifurcations, reducing the computational cost. The methods were implemented in the development of routines that were tested and validated with models from the literature

Analytical solutions of orientation aggregation models, multiple solutions and path following with the Adomian decomposition method

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.In this work we apply the Adomian decomposition method to an orientation aggregation problem modelling the time distribution of filaments. We find analytical solutions under certain specific criteria and programmatically implement the Adomian method to two variants of the orientation aggregation model. We extend the utility of the Adomian decomposition method beyond its original capability to enable it to converge to more than one solution of a nonlinear problem and further to be used as a corrector in path following bifurcation problems