MFE revisited : part 1: adaptive grid-generation using the heat equation

In this paper the moving-nite-element method (MFE) is used to solve the heat equation, with an articial time component, to give a non-uniform (steady-state) grid that is adapted to a given prole. It is known from theory and experiments that MFE, applied to parabolic PDEs, gives adaptive grids which satisfy an equidistribution type law. This property is used to create non-uniform nite-element grids that are dictated by second-order derivatives of the solution. The proposed procedure could be used to create an initial grid for MFE itself, to dene a regridding strategy for MFE in case of a distorted grid, or to prescribe a new adaptive grid method where the heat quation is used as a "monitor function"

Tensor-product adaptive grids based on coordinate transformations

AbstractIn this paper we discuss a two-dimensional adaptive grid method that is based on a tensor-product approach. Adaptive grids are a commonly used tool for increasing the accuracy and reducing computational costs when solving both partial differential equations (PDEs) and ordinary differential equations. A traditional and widely used form of adaptivity is the concept of equidistribution, which is well-defined and well-understood in one space dimension. The extension of the equidistribution principle to two or three space dimensions, however, is far from trivial and has been the subject of investigation of many researchers during the last decade. Besides the nonsingularity of the transformation that defines the nonuniform adaptive grid, the smoothness of the grid (or transformation) plays an important role as well. We will analyse these properties and illustrate their importance with numerical experiments for a set of time-dependent PDE models with steep moving pulses, fronts, and boundary layers

Adaptive Method of Lines for Magneto-Hydrodynamic PDE Models

An adaptive grid technique for use in the solution of multi-dimensional time-dependent PDEs is applied to several magnetohydrodynamic model problems. The technique employs the method-of-lines and can be viewed both in a continuous and semi-discrete setting. By using an equidistribution principle, it has the ability to track individual features of the physical solutions in the developing plasma ows. Moreover, it can be shown that the underlying grid varies smoothly in time and space. The results of several numerical experiments are presented which cover many aspects typifying nonlinear magneto- uid dynamics

Pattern formation in the 1-D Gray-Scott model

In this work, we analyze a pair of one-dimensional coupled reaction-diusion equations known as the Gray{Scott model, in which self-replicating patterns have been observed. We focus on stationary and traveling patterns, and begin by deriving the asymptotic scaling of the parameters and variables necessary for the analysis of these patterns. Single{pulse and multiple{pulse stationary waves are shown to exist in the appropriately{scaled equations on the innite line. A (single) pulse is a narrow interval in which the concentration U of one chemical is small, while that of the second, V , is large, and outside of which the concentration U tends (slowly) to the homogeneous steady state U 1, while V is everywhere close to V 0. In addition, we establish the existence of a plethora of periodic steady states consisting of periodic arrays of pulses interspersed by intervals in which the concentration V is exponentially small and U varies slowly. These periodic states are spatially inhomogeneous steady patterns whose length scales are determined exclusively by the reactions of the chemicals and their diusions, and not by other mechanisms such as boundary conditions. A complete bifurcation study of these solutions is presented. We also establish the non-existence of traveling solitary pulses in this system. This non-existence result re ects the system's degeneracy and indicates that some event, for example pulse-splitting, `must' occur when a pair of pulses moving apart from each other (as has been observed in simulations): these pulses evolve towards the non-existent traveling solitary pulses. The main mathematical techniques employed in this analysis of the stationary and traveling patterns are geometric singular perturbation theory and adiabatic Melnikov theory. Finally, the theoretical results are compared to those obtained from direct numerical simulation of the coupled partial dierential equations on a `very large' domain, using a moving grid code. It has been checked that the boundaries do not in uence the dynamics. A subset of the family of stationary single pulses appears to be stable. This subset determines the boundary of a region in parameter space in which the self-replicating process takes place. In that region, we observe that the core of a time-dependent self-replicating pattern turns out to be precisely a stationary periodic pulse-pattern of the type that we construct. Moreover, the simulations reveal some other essential components of the pulse-splitting process and provide an important guide to further analysis

Transition of liesegang precipitation systems: simulations with an adaptive grid pde method

The dynamics of the Liesegang type pattern formation is investigated in a centrally symmetric two-dimensional setup. According to the observations in real experiments, the qualitative change of the dynamics is exhibited for slightly different initial conditions. Two kinds of chemical mechanisms are studied; in both cases the pattern formation is described using a phase separation model including the Cahn-Hilliard equations. For the numerical simulations we make use of a recent adaptive grid PDE method, which successfully deals with the computationally critical cases such as steep gradients in the concentration distribution and investigation of long time behavior. The numerical simulations show a good agreement with the real experiments