Physical Sciences and Mathematics Commons^™

A Variable Nonlinear Splitting Algorithm For Reaction Diffusion Systems With Self- And Cross-Diffusion, Matthew Beauregard, Joshua L. Padgett

Faculty Publications

Self- and cross-diffusion are important nonlinear spatial derivative terms that are included into biological models of predator-prey interactions. Self-diffusion models overcrowding effects, while cross-diffusion incorporates the response of one species in light of the concentration of another. In this paper, a novel nonlinear operator splitting method is presented that directly incorporates both self- and cross-diffusion into a computational efficient design. The numerical analysis guarantees the accuracy and demonstrates appropriate criteria for stability. Numerical experiments display its efficiency and accuracy

A Nonlinear Splitting Algorithm For Systems Of Partial Differential Equations With Self-Diffusion, Matthew Beauregard, Joshua L. Padgett, Rana D. Parshad

Faculty Publications

Systems of reaction-diffusion equations are commonly used in biological models of food chains. The populations and their complicated interactions present numerous challenges in theory and in numerical approximation. In particular, self-diffusion is a nonlinear term that models overcrowding of a particular species. The nonlinearity complicates attempts to construct efficient and accurate numerical approximations of the underlying systems of equations. In this paper, a new nonlinear splitting algorithm is designed for a partial differential equation that incorporates self diffusion. We present a general model that incorporates self-diffusion and develop a numerical approximation. The numerical analysis of the approximation provides criteria for …

A Class Of Discontinuous Petrov–Galerkin Methods. Ii. Optimal Test Functions, Leszek Demkowicz, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

We lay out a program for constructing discontinuous Petrov–Galerkin (DPG) schemes having test function spaces that are automatically computable to guarantee stability. Given a trial space, a DPG discretization using its optimal test space counterpart inherits stability from the well posedness of the undiscretized problem. Although the question of stable test space choice had attracted the attention of many previous authors, the novelty in our approach lies in the fact we identify a discontinuous Galerkin (DG) framework wherein test functions, arbitrarily close to the optimal ones, can be locally computed. The idea is presented abstractly and its feasibility illustrated through …

Lanchester's Equations In Three Dimensions, Christina Spradlin, Greg Spradlin

Gregory S. Spradlin

Mean Field Effects For Counterpropagating Traveling Wave Solutions Of Reaction-Diffusion Systems, Andrew J. Bernoff, R. Kuske, B. J. Matkowsky, V. Volpert

All HMC Faculty Publications and Research

In many problems, e.g., in combustion or solidification, one observes traveling waves that propagate with constant velocity and shape in the x direction, say, are independent of y and z and describe transitions between two equilibrium states, e.g., the burned and the unburned reactants. As parameters of the system are varied, these traveling waves can become unstable and give rise to waves having additional structure, such as traveling waves in the y and z directions, which can themselves be subject to instabilities as parameters are further varied. To investigate this scenario we consider a system of reaction-diffusion equations with a …