Michael Chapwanya

An instability theory for the formation of ribbed moraine, drumlins and mega-scale glacial lineations

We present a theory for the coupled flow of ice, subglacial water and subglacial sediment, which is designed to represent the processes which occur at the bed of an ice sheet. The ice is assumed to flow as a Newtonian viscous fluid, the water can flow between the till and the ice as a thin film, which may thicken to form streams or cavities, and the till is assumed to be transported, either through shearing by the ice, squeezing by pressure gradients in the till, or by fluvial sediment transport processes in streams or cavities. In previous studies, it was shown that the dependence of ice sliding velocity on effective pressure provided a mechanism for the generation of bedforms resembling ribbed moraine, while the dependence of fluvial sediment transport on water film depth provides a mechanism for the generation of bedforms resembling mega-scale glacial lineations. Here, we combine these two processes in a single model, and show that, depending largely on the granulometry of the till, instability can occur in a range of types which range from ribbed moraine through three-dimensional drumlins to mega-scale glacial lineations.

From enzyme kinetics to epidemiological models with Michaelis-Menten contact rate: Design of nonstandard finite difference schemes

We consider the basic SIR epidemiological model with the Michaelis-Menten formulation of the contact rate. From the study of the Michaelis-Menten basic enzymatic reaction, we design two types of Nonstandard Finite Difference (NSFD) schemes for the SIR model: Exact-related schemes based on the Lambert W function and schemes obtained by using Mickenss rules of more complex denominator functions for discrete derivatives and nonlocal approximations of nonlinear terms. We compare and investigate the performance of the two types of schemes by showing that they are dynamically consistent with the continuous model. Numerical simulations that support the theory and demonstrate computationally the power of NSFD schemes are presented.

A model for reactive porous transport during re-wetting of hardened concrete †

A mathematical model is developed that captures the transport of liquid water in hardened concrete, as well as the chemical reactions that occur between the imbibed water and the residual calcium-silicate compounds residing in the porous concrete matrix. The main hypothesis in this model is that the reaction product—calcium-silicate hydrate gel—clogs the pores within the concrete, thereby hindering water transport. Numerical simulations are employed to determine the sensitivity of the model solution to changes in various physical parameters, and compare to experimental results available in the literature.

Coupling finite volume and nonstandard finite difference schemes for a singularly perturbed Schrödinger equation

The Schrödinger equation is a model for many physical processes in quantum physics. It is a singularly perturbed differential equation where the presence of the small reduced Plancks constant makes the classical numerical methods very costly and inefficient. We design two new schemes. The first scheme is the nonstandard finite volume method, whereby the perturbation term is approximated by nonstandard technique, the potential is approximated by its mean value on the cell and the complex dependent boundary conditions are handled by exact schemes. In the second scheme, the deficiency of classical schemes is corrected by the inner expansion in the boundary layer region. Numerical simulations supporting the performance of the schemes are presented.

An explicit nonstandard finite difference scheme for the Allen–Cahn equation

We design explicit nonstandard finite difference schemes for the nonlinear Allen–Cahn reaction diffusion equation in the limit of very small interaction length . In the proposed scheme, the perturbation parameter is part of the argument of the functional step size, thereby minimizing the restrictions normally associated with standard explicit finite difference schemes. The derivation involves splitting the equation into the space-independent and the time-independent different models. An exact nonstandard scheme is proposed for the space-independent model and energy conservative schemes are proposed for the time-independent model. We show the power of the derived scheme over the existing schemes through several numerical examples.

On a fractional step-splitting scheme for the Cahn-Hilliard equation

Purpose – For a partial differential equation with a fourth-order derivative such as the Cahn-Hilliard equation, it is always a challenge to design numerical schemes that can handle the restrictive time step introduced by this higher order term. The purpose of this paper is to employ a fractional splitting method to isolate the convective, the nonlinear second-order and the fourth-order differential terms. Design/methodology/approach – The full equation is then solved by consistent schemes for each differential term independently. In addition to validating the second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution. Findings – The scheme is second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution. Originality/value – The authors be...

A MATHEMATICAL MODEL FOR EBOLA EPIDEMIC WITH SELF-PROTECTION MEASURES

The South African Research Chairs Initiative of the Department of Science and Technology and National Research Foundation: SARChI Chair in Mathematical Models and Methods in Bioengineering and Biosciences. TB and YAT acknowledge the support, in part, of DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS).

A 2D channel-clogging biofilm model

We present a model of biofilm growth in a long channel where the biomass is assumed to have the rheology of a viscous polymer solution. We examine the competition between growth and erosion-like surface detachment due to the flow. A particular focus of our investigation is the effect of the biofilm growth on the fluid flow in the pores, and the issue of whether biomass can grow sufficiently to shut off fluid flow through the pores, thus clogging the pore space. Net biofilm growth is coupled along the pore length via flow rate and nutrient transport in the pore flow. Our 2D model extends existing results on stability of 1D steady state biofilm thicknesses to show that, in the case of flows driven by a fixed pressure drop, full clogging of the pore can indeed happen in certain cases dependent on the functional form of the detachment term.

Finite difference discretisation of a model for biological nerve conduction

A nonstandard finite difference method is proposed for the discretisation of the semilinear FitzHugh-Nagumo reaction diffusion equation. The equation has been useful in describing, for example, population models, biological models, heat and mass transfer models, and many other applications. The proposed approach involves splitting the equation into the space independent and the time independent sub equation. Numerical simulations for the full equation are presented.

The bio-mathematics Chair at University of Pretoria in 2015

In this report we summarize the activities and achievements of the Bio-mathematics Chair at the University of Pretoria in 2015.