An ADI extrapolated Crank-Nicolson orthogonal spline collocation method for nonlinear reaction-diffusion systems: a computational study

An alternating direction implicit (ADI) orthogonal spline collocation (OSC) method is described for the approximate solution of a class of nonlinear reaction-diffusion systems. Its efficacy is demonstrated on the solution of well-known examples of such systems, specifically the Brusselator, Gray-Scott, Gierer-Meinhardt and Schnakenberg models, and comparisons are made with other numerical techniques considered in the literature. The new ADI method is based on an extrapolated Crank-Nicolson OSC method and is algebraically linear. It is efficient, requiring at each time level only $O({\cal N})$ operations where ${\cal N}$ is the number of unknowns. Moreover,it is shown to produce approximations which are of optimal global accuracy in various norms, and to possess superconvergence properties

Computational Methods and Results for Structured Multiscale Models of Tumor Invasion

We present multiscale models of cancer tumor invasion with components at the molecular, cellular, and tissue levels. We provide biological justifications for the model components, present computational results from the model, and discuss the scientific-computing methodology used to solve the model equations. The models and methodology presented in this paper form the basis for developing and treating increasingly complex, mechanistic models of tumor invasion that will be more predictive and less phenomenological. Because many of the features of the cancer models, such as taxis, aging and growth, are seen in other biological systems, the models and methods discussed here also provide a template for handling a broader range of biological problems

Integration and conjugacy in knot theory

This thesis consists of three self-contained chapters. The first two concern quantum invariants of links and three manifolds and the third contains results on the word problem for link groups. In chapter 1 we relate the tree part of the Aarhus integral to the mu-invariants of string-links in homology balls thus generalizing results of Habegger and Masbaum. There is a folklore result in physics saying that the Feynman integration of an exponential is itself an exponential. In chapter 2 we state and prove an exact formulation of this statement in the language which is used in the theory of finite type invariants. The final chapter is concerned with properties of link groups. In particular we study the relationship between known solutions from small cancellation theory and normal surface theory for the word and conjugacy problems of the groups of (prime) alternating links. We show that two of the algorithms in the literature for solving the word problem, each using one of the two approaches, are the same. Then, by considering small cancellation methods, we give a normal surface solution to the conjugacy problem of these link groups and characterize the conjugacy classes. Finally as an application of the small cancellation properties of link groups we give a new proof that alternating links are non-trivial.Comment: University of Warwick Ph.D. thesi

Finite element simulation of three-dimensional free-surface flow problems

An adaptive finite element algorithm is described for the stable solution of three-dimensional free-surface-flow problems based primarily on the use of node movement. The algorithm also includes a discrete remeshing procedure which enhances its accuracy and robustness. The spatial discretisation allows an isoparametric piecewise-quadratic approximation of the domain geometry for accurate resolution of the curved free surface. The technique is illustrated through an implementation for surface-tension-dominated viscous flows modelled in terms of the Stokes equations with suitable boundary conditions on the deforming free surface. Two three-dimensional test problems are used to demonstrate the performance of the method: a liquid bridge problem and the formation of a fluid droplet

Finite element solution techniques for large-scale problems in computational fluid dynamics

Element-by-element approximate factorization, implicit-explicit and adaptive implicit-explicit approximation procedures are presented for the finite-element formulations of large-scale fluid dynamics problems. The element-by-element approximation scheme totally eliminates the need for formation, storage and inversion of large global matrices. Implicit-explicit schemes, which are approximations to implicit schemes, substantially reduce the computational burden associated with large global matrices. In the adaptive implicit-explicit scheme, the implicit elements are selected dynamically based on element level stability and accuracy considerations. This scheme provides implicit refinement where it is needed. The methods are applied to various problems governed by the convection-diffusion and incompressible Navier-Stokes equations. In all cases studied, the results obtained are indistinguishable from those obtained by the implicit formulations