Velocity-induced numerical solutions of reaction-diffusion systems on continuously growing domains

Reaction-diffusion systems have been widely studied in developmental biology, chemistry and more recently in financial mathematics. Most of these systems comprise nonlinear reaction terms which makes it difficult to find closed form solutions. It therefore becomes convenient to look for numerical solutions: finite difference, finite element, finite volume and spectral methods are typical examples of the numerical methods used. Most of these methods are locally based schemes. We examine the implications of mesh structure on numerically computed solutions of a well-studied reaction-diffusion model system on two-dimensional fixed and growing domains. The incorporation of domain growth creates an additional parameter – the grid-point velocity – and this greatly influences the selection of certain symmetric solutions for the ADI finite difference scheme when a uniform square mesh structure is used. Domain growth coupled with grid-point velocity on a uniform square mesh stabilises certain patterns which are however very sensitive to any kind of perturbation in mesh structure. We compare our results to those obtained by use of finite elements on unstructured triangular elements

Oscillatory instability in super-diffusive reaction -- diffusion systems: fractional amplitude and phase diffusion equations

Nonlinear evolution of a reaction--super-diffusion system near a Hopf bifurcation is studied. Fractional analogues of complex Ginzburg-Landau equation and Kuramoto-Sivashinsky equation are derived, and some of their analytical and numerical solutions are studied

Local Discontinuous Galerkin Methods Coupled with Implicit Integration Factor Methods for Solving Reaction-Cross-Diffusion Systems

We present a new numerical method for solving nonlinear reaction-diffusion systems with cross-diffusion which are often taken as mathematical models for many applications in the biological, physical, and chemical sciences. The two-dimensional system is discretized by the local discontinuous Galerkin (LDG) method on unstructured triangular meshes associated with the piecewise linear finite element spaces, which can derive not only numerical solutions but also approximations for fluxes at the same time comparing with most of study work up to now which has derived numerical solutions only. Considering the stability requirement for the explicit scheme with strict time step restriction (Δt=O(hmin2)), the implicit integration factor (IIF) method is employed for the temporal discretization so that the time step can be relaxed as Δt=O(hmin). Moreover, the method allows us to compute element by element and avoids solving a global system of nonlinear algebraic equations as the standard implicit schemes do, which can reduce the computational cost greatly. Numerical simulations about the system with exact solution and the Brusselator model, which is a theoretical model for a type of autocatalytic chemical reaction, are conducted to confirm the expected accuracy, efficiency, and advantages of the proposed schemes

Analysis of a reaction-diffusion system of ƛ -w type

The author studies two coupled reaction-diffusion equations of 'ƛ - w' type, on an open, bounded, convex domain Ω C R(^d) (d ≤ 3), with a boundary of class C(^2), and homogeneous Neumann boundary conditions. The equations are close to a supercritical Hopf bifurcation in the reaction kinetics, and are model equations for oscillatory reaction-diffusion equations. Global existence, uniqueness and continuous dependence on initial data of strong and weak solutions are proved using the classical Faedo-Galerkin method of Lions and compactness arguments. The work provides a complete case study for the application of this method to systems of nonlinear reaction-diffusion equations. The author also undertook the numerical analysis of the reaction-diffusion system. Results are presented for a fully-practical piecewise linear finite element method by mimicking results in the continuous case. Semi-discrete and fully-discrete error estimates are proved after establishing a priori bounds for various norms of the approximate solutions. Finally, the theoretical results are illustrated and verified via the numerical simulation of periodic plane waves in one space dimension, and preliminary results representing target patterns and spiral solutions presented in two space dimensions

Existence analysis for a reaction-diffusion Cahn-Hilliard-type system with degenerate mobility and singular potential modeling biofilm growth

The global existence of bounded weak solutions to a diffusion system modeling biofilm growth is proven. The equations consist of a reaction-diffusion equation for the substrate concentration and a fourth-order Cahn-Hilliard-type equation for the volume fraction of the biomass, considered in a bounded domain with no-flux boundary conditions. The main difficulties are coming from the degenerate diffusivity and mobility, the singular potential arising from a logarithmic free energy, and the nonlinear reaction rates. These issues are overcome by a truncation technique and a Browder-Minty trick to identify the weak limits of the reaction terms. The qualitative behavior of the solutions is illustrated by numerical experiments in one space dimension, using a BDF2 (second-order backward Differentiation Formula) finite-volume scheme

Analysis and simulation on dynamics of a partial differential system with nonlinear functional responses

We introduce a reaction–diffusion system with modified nonlinear functional responses. We first discuss the large-time behavior of positive solutions for the system. And then, for the corresponding steady-state system, we are concerned with the priori estimate, the existence of the nonconstant positive solutions as well as the bifurcations emitting from the positive constant equilibrium solution. Finally, we present some numerical examples to test the theoretical and computational analysis results. Meanwhile, we depict the trajectory graphs and spatiotemporal patterns to simulate the dynamics for the system. The numerical computations and simulated graphs imply that the available food resource for consumer is very likely not single