John A. Mackenzie

Chemotaxis: A Feedback-Based Computational Model Robustly Predicts Multiple Aspects of Real Cell Behaviour

A simple feedback model of chemotaxis explains how new pseudopods are made and how eukaryotic cells steer toward chemical gradients.

Convergence analysis of finite difference approximations on equidistributed grids to a singularly perturbed boundary value problem

Abstract We derive e -uniform error estimates for two first-order upwind discretizations of a model inhomogeneous, second-order, singularly perturbed boundary value problem on a non-uniform grid. Here, e is the small parameter multiplying the highest derivative term. The grid is suggested by the equidistribution of a positive monitor function which is a linear combination of a constant floor and a power of the second derivative of the solution. Our analysis shows how the floor should be chosen to ensure e -uniform convergence and indicates the convergence behaviour for such grids. Numerical results are presented which confirm the e -uniform convergence rates.

A Moving Mesh Method for One-dimensional Hyperbolic Conservation Laws

We develop an adaptive method for solving one-dimensional systems of hyperbolic conservation laws that employs a high resolution Godunov-type scheme for the physical equations, in conjunction with a moving mesh PDE governing the motion of the spatial grid points. Many other moving mesh methods developed to solve hyperbolic problems use a fully implicit discretization for the coupled solution-mesh equations, and so suffer from a significant degree of numerical stiffness. We employ a semi-implicit approach that couples the moving mesh equation to an efficient, explicit solver for the physical PDE, with the resulting scheme behaving in practice as a two-step predictor-corrector method. In comparison with computations on a fixed, uniform mesh, our method exhibits more accurate resolution of discontinuities for a similar level of computational work.

A posteriori error analysis for numerical approximations of Friedrichs systems

Abstract. The global error of numerical approximations for symmetric positive systems in the sense of Friedrichs is decomposed into a locally created part and a propagating component. Residual-based two-sided local a posteriori error bounds are derived for the locally created part of the global error. These suggest taking the

Modeling Cell Movement and Chemotaxis Using Pseudopod-Based Feedback

L^2

On a uniformly accurate finite difference approximation of a singulary peturbed reaction-diffusion problem using grid equidistribution

-norm as well as weaker, dual norms of the computable residual as local error indicators. The dual graph norm of the residual

Improvement in screening performance and diagnosis of congenital hypothyroidism in Scotland 1979–2003

{\vec r}_h

The Efficient Generation of Simple Two-Dimensional Adaptive Grids

is further bounded from above and below in terms of the

A computational method for the coupled solution of reaction-diffusion equations on evolving domains and manifolds

L^2

Uniformly convergent high order finite element solutions of a singularly perturbed reaction-diffusion equation using mesh equidistribution

norm of