Mapundi Banda | University of Pretoria (original) (raw)

Papers by Mapundi Banda

Mathematics of computation, Jan 1, 2008

The Lattice Boltzmann equations are usually constructed to satisfy physical requirements like Gal... more The Lattice Boltzmann equations are usually constructed to satisfy physical requirements like Galilean invariance and isotropy as well as to possess a velocity-independent pressure, no compressible effects, just to mention a few. In this paper, a stability criterion for such constructions is introduced and is used to derive a new relation of the parameters in a parametrized 2-dimensional 9-velocity model.

SIAM Journal on Scientific Computing, 2006

ABSTRACT Lattice Boltzmann equations are usually constructed to satisfy some physical requirement... more ABSTRACT Lattice Boltzmann equations are usually constructed to satisfy some physical requirements such as Galilean invariance and isotropy, to possess a velocity-independent pressure and no compressible effects, and so on. In this paper, a stability requirement is introduced as a new criterion for the constructions. With this requirement, we derive some relations of parameters for several lattice Boltzmann models. Interestingly, these relations are satisfied by many choices of parameters used in the literature.

Industrial Mathematics has established itself as an important branch of professional mathematics.... more Industrial Mathematics has established itself as an important branch of professional mathematics. Mathematicians are aware of the need to bridge the gap between highly specialised mathematical research and the high demand for innovation from industry. In this presentation we discuss the experiences at the Mathematics in Industry Study Group-South Africa (MISGSA) which has become an annual event. Case studies of

Numerical Mathematics and Advanced Applications, 2004

Nonlinear Analysis: Real World Applications, 2009

We present a high-order accurate relaxed non-oscillatory scheme for solving magnetohydrodynamic (... more We present a high-order accurate relaxed non-oscillatory scheme for solving magnetohydrodynamic (MHD) equations. The computations reported here demonstrate the remarkable simplicity and versatility of semi-discrete relaxation schemes as solvers for ideal MHD equations. Simulations will be presented for two prototype MHD problems, the one-dimensional Brio–Wu shock tube problems. A qualitative comparison reveals an excellent agreement with previous results based on

Journal of Computational and Applied Mathematics, 2005

We present a relaxed scheme with more precise information about local speeds of propagation and a... more We present a relaxed scheme with more precise information about local speeds of propagation and a multidimensional construction of the cell averages. The physical domain of dependence is simulated correctly and high resolution is maintained. Relaxation schemes have advantages that include high resolution, simplicity and explicitly no (approximate) Riemann solvers and no characteristic decomposition is necessary. Performance of the scheme is illustrated by tests on two-dimensional Euler equations of gas dynamics.

International Journal For Numerical Methods in Fluids 2007 Vol 55 Pp 673 692 Peer Reviewed Journal, Nov 1, 2007

We apply the lattice Boltzmann (LB) method for solving the shallow water equations with source te... more We apply the lattice Boltzmann (LB) method for solving the shallow water equations with source terms such as the bed slope and bed friction. Our aim is to use a simple and accurate representation of the 9 source terms in order to simulate practical shallow water flows without relying on upwind discretization or Riemann problem solvers. We validate the algorithm on problems where analytical solutions are available.

Siam Journal on Scientific Computing, 2006

Journal of Scientific Computing, 2016

Journal of Numerical Mathematics, 2016

Mathematics of Computation, 2008

A relaxation system based on a Lattice-Boltzmann type discrete velocity model is considered in th... more A relaxation system based on a Lattice-Boltzmann type discrete velocity model is considered in the low Mach number limit. A third order relaxation scheme is developed working uniformly for all ranges of the mean free path and Mach number. In the incompressible Navier-Stokes limit the scheme reduces to an explicit high order finite difference scheme for the incompressible Navier-Stokes equations based on nonoscillatory upwind discretization. Numerical results and comparisons with other approaches are presented for several test cases in one and two space dimensions.

Siam Journal on Scientific Computing, 2009

We derive a Godunov-type relaxation scheme for turbulent flows with heat transfer. The building b... more We derive a Godunov-type relaxation scheme for turbulent flows with heat transfer. The building block of this approach is a kinetic Boltzmann-type formulation for a model of turbulent thermal flows based on large-eddy simulation modelling. Discrete-velocity moments are taken in order to derive a relaxation system which is in turn solved by a relaxation scheme. High-order schemes are obtained by discretizing the relaxation system spatially with weighted non-oscillatory reconstruction and temporally using an implicit-explicit discretization. In the limit of small Mach and small Knudsen numbers one obtains a high-order relaxation scheme for large-eddy simulation of turbulent thermal flows. The efficiency, accuracy and high-resolution properties of the models are demonstrated in a variety of test examples.

Journal of Computational Physics, May 28, 2007

The convection-radiation effects in thermal fluid flows are studied based on the lattice-Boltzman... more The convection-radiation effects in thermal fluid flows are studied based on the lattice-Boltzmann method. Nine-velocity flow and temperature distributions are used to obtain the mass, momentum and energy equations in thermal incompressible flows by studying equivalent moment systems. The radiative heat flux in the energy equation is obtained using the discrete-ordinates solution of the radiative transfer equation. A non-oscillatory relaxation scheme is used to solve the coupled moment equations. Such schemes have the advantage of being simple and easy to implement. Numerical results are presented for two test examples on coupled convection-radiation flows in two dimensional enclosures. Detailed simulation results at different flow and radiative regimes, as well as benchmark solutions, are presented and discussed.

Applied Mathematics and Computation

We investigate the performance of a meshless method for the numerical simulation of depth-average... more We investigate the performance of a meshless method for the numerical simulation of depth-averaged turbulence flows. The governing equations are shallow water equations obtained by depth averaging of the full Reynolds equations including bed frictions, eddy viscosity, wind shear stresses, and Coriolis forces. As a double-phase closure turbulence model, we consider the depth-averaged kmodel. A truly meshless numerical method based on radial basis functions is employed to obtain an accurate approximation to the solution of the model. We validate the algorithm on a linear shallow water problem where analytical solutions are available. Numerical results are also compared with experimental data for a backward-facing flow problem. Furthermore, we test the method on a practical problem by simulating tidal flows in the Strait of Gibraltar. The main focus is to examine the performance of the meshless method for complex geometries with irregular bathymetry. The obtained results demonstrate its ability to capture the main flow features. Y. ALHURI ET AL. than water waves. For instance, the shallow water equations have applications in environmental and hydraulic engineering such as tidal flows in an estuary or coastal regions, rivers, reservoir, and open channel flows. Such practical flow problems are not trivial to simulate because the geometry can be complex, and the topography tends to be irregular. In addition, most of water free-surface flows encountered in engineering practice are turbulent characterized by: (i) highly unsteady features such that time series of the flow field at any point in the computational domain would appear random to an observer unfamiliar with these flows; (ii) appearance of a great deal of vorticity in which vortex stretching is one of the principal mechanisms by which the intensity of turbulence is increased; (iii) reduction of the velocity gradients due to the action of viscosity, which reduces the kinetic energy of the flow, that is, mixing is a dissipative process, and the lost energy is irreversibly converted into internal energy of the water; (iv) existence of repeatable coherent structures and essentially deterministic events that are responsible for a large part of the mixing; and (v) turbulent flows fluctuate on a broad range of length and time scales. This property makes direct numerical simulation of turbulent flows very difficult. It should also be stressed that turbulent flows contain variations on a much wider range of length and time scales than laminar flows. However, the random component of turbulent flows causes these events to differ from each other in size, strength, and time interval between occurrences, making their analysis very difficult. Despite that they are similar to the laminar flow, the equations describing turbulent flows are usually much more difficult and computationally demanding to solve. Considering the usual length scales in engineering practice, and the small kinematic viscosity of water, in most cases, the Reynolds number is large enough in order to consider the flow as fully turbulent. Even in the simplest situations of river flow, we can observe small eddies that appear and disappear with an apparently chaotic movement, showing the complexity of turbulent motion. In coastal regions, large eddies often occur because of the separation of the flow past a headland, a breakwater, or an island. These eddies are very important in environmental engineering, and they may influence the solute and sediment trapping.

Journal of Computational and Applied Mathematics, 2015

In this paper a general drift-flux model describing a subsonic and isentropic multi-phase fluid i... more In this paper a general drift-flux model describing a subsonic and isentropic multi-phase fluid in connected pipes is considered. Each phase is assumed to be isentropic with its own sonic speed. The components are gamma-law gases with γ > 1. For such, a computational challenge at a junction is the computation of rarefaction waves which do not have a readily available analytical form. Firstly, the well-posedness of the Riemann problem at the junction is discussed. It is suggested that rarefaction waves should be linearized in order to obtain a more efficient numerical method for coupling such multi-component flow. Some computational results on the dynamics of the multi-phase gas in the pipes demonstrate the qualitative behavior of this approach.

Mathematical Control and Related Fields, 2013

ABSTRACT Suitable numerical discretizations for boundary control problems of systems of nonlinear... more ABSTRACT Suitable numerical discretizations for boundary control problems of systems of nonlinear hyperbolic partial differential equations are presented. Using a discrete Lyapunov function, exponential decay of the discrete solutions of a system of hyperbolic equations for a family of first-order finite volume schemes is proved. The decay rates are explicitly stated. The theoretical results are accompanied by computational results.

Theory, Numerics and Applications(In 2 Volumes), 2012