Mathematical models of isothermal chemical systems in reactor problems and Turing's theory of morphogenesis with an application in sea-shell patterning are studied. The reaction-diffusion systems describing these models are solved numerically. First- and second-order difference schemes are developed, which are economical and reliable in comparison to classical numerical methods. The linearization process decouples the reaction-diffusion equations thereby allowing the use of different time steps for each differential equation, which may be large due to the excellent stability properties of the methods. The methods avoid having to solve a non-linear algebraic system at each time step. The schemes are suitable for implementation on a parallel machine.

High order accurate finite difference methods for hyperbolic and parabolic initial boundary value problems (IBVPs) are considered. Particular focus is on time dependent wave propagating problems in complex domains. Typical applications are acoustic and electromagnetic wave propagation and fluid dynamics. To solve such problems efficiently a strictly stable, high order accurate method is required. Our recipe to obtain such schemes is to: i) Approximate the (first and second) derivatives of the IBVPs with central finite difference operators, that satisfy a summation by parts (SBP) formula. ii) Use specific procedures for implementation of boundary conditions, that preserve the SBP property. iii) Add artificial dissipation. iv) Employ a multi block structure. Stable schemes for weakly nonlinear IBVPs require artificial dissipation to absorb the energy of the unresolved modes. This led to the construction of accurate and efficient artificial dissipation operators of SBP type, that preserve the energy and error estimate of the original problem. To solve problems on complex geometries, the computational domain is broken up into a number of smooth and structured meshes, in a multi block fashion. A stable and high order accurate approximation is obtained by discretizing each subdomain using SBP operators and using the Simultaneous Approximation Term (SAT) procedure for both the (external) boundary and the (internal) interface conditions. Steady and transient aerodynamic calculations around an airfoil were performed, where the first derivative SBP operators and the new artificial dissipation operators were combined to construct high order accurate upwind schemes. The computations showed that for time dependent problems and fine structures, high order methods are necessary to accurately compute the solution, on reasonably fine grids. The construction of high order accurate SBP operators for the second derivative is one of the considerations in this thesis. It was shown that the second derivative operators could be closed with two order less accuracy at the boundaries and still yield design order of accuracy, if an energy estimate could be obtained.

Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 77-79). Available also in a digital version from Dissertation Abstracts.

This thesis analyses numerical methods used in finding solutions of diferential equations. Numerical methods are viewed as discrete dynamical systems that give useful information on continuous dynamical systems defined by systems of (ordinary) diferential equations. We analyse non-standard finite difference schemes that have no spurious fixed-points compared to the dynamical system under consideration, the linear stability/instability property of the fixed-points being the same for both the discrete and continuous systems. We obtain a sharper condition for the elementary stability of the schemes. For more complex dynamical systems which are dissipative, we design schemes that replicate this property. Furthermore, we investigate the impact of the above analysis on the numerical solution of partial differential equations. We specifically focus on reaction-diffusion equations that arise in many fields of engineering and applied sciences. Often their solutions enjoy the follow- ing essential properties: Stability/instability of the fixed points for the space independent equation, the conservation of energy for the stationary equation, and boundedness and positivity. We design new non-standard finite diference schemes which replicate these properties. Our construction make use of three strategies: the renormalization of the denominator of the discrete derivative, non-local approximation of the nonlinear terms and simple functional relation between step sizes. Numerical results that support the theory are provided. Copyright / Thesis (PhD)--University of Pretoria, 2009. / Mathematics and Applied Mathematics / unrestricted

We study noise from wind turbines while taking irregular terrain and non-constant atmosphere into consideration. We will show that simulating the distribution of 3D acoustic waves can be done by using only low frequencies, thus reducing the computational complexity significantly.

Most realistic financial derivatives models are too complex to allow explicit analytic solutions. The computational techniques used to implement those models fall into two broad categories: finite difference methods for the solution of partial differential equations (PDEs) and Monte Carlo simulation. Accordingly, the course consists of two sections. The first half of the course focuses on finite difference methods. The following topics are discussed; Parabolic PDEs, Black-Scholes PDE for European and American options; binomial and trinomial trees; explicit, implicit and Crank- Nicholson finite difference methods; far boundary conditions, convergence, stability, variance bias; early exercise and free boundary conditions; parabolic PDEs arising from fixed income derivatives; implied trees for exotic derivatives, adapted trees for interest rate derivatives. The second half of the course focuses on Monte Carlo. The following topics are discussed; Random number generation and testing; evaluation of expected payoff by Monte Carlo simulation; variance reduction techniques�antithetic variables, importance sampling, martingale control variables; stratification, low-discrepancy sequences and quasi-Monte Carlo methods; efficient evaluation of sensitivity measures; methods suitable for multifactor and term-structure dependent models. Computational Methods of Financial Mathematics is taught by Marcel Blais, a professor at Worcester Polytechnic Institute.

A stable boundary treatment of the dynamic beam equation (DBE) with two different sets of boundary conditions has been conducted using the summation-by-parts-simultaneous-approximation-term (SBP-SAT) method. As the DBE involves a fourth derivative in space the numerical boundary treatment is highly non-trivial. Using SBP-SAT operators together with suitable time integration schemes the DBE has been simulated and a convergence study has been made. The results show that the SBP-SAT method produces a stable discretistation that is accurate enough to capture the dispersive nature of the dynamic beam equation. In additions simulations were made presenting the importance of a stable boundary treatment showing that the numerical solutions diverge when the boundaries were not handled correctly.

Magister Scientiae - MSc / This thesis deals with the design and implementation of some novel numerical methods for non-linear singular pertubations problems (NSPPs). It provide a survey of asymptotic and numerical methods for some NSPPs in the past decade. By considering two test problems, rigorous asymptotic analysis is carried out. Based on this analysis, suitable numerical methods are designed, analyzed and implemented in order to have some relevant results of physical importance. Since the asymptotic analysis provides only qualitative information, the focus is more on the numerical analysis of the problem which provides the quantitative information. / South Africa

>Magister Scientiae - MSc / This thesis deals with the design and implementation of some novel numerical methods for nonlinear singular perturbations problems (NSPPs). We provide a survey of asymptotic and numerical methods for some NSPPs in past decade. By considering two test problems, rigorous asymptotic analysis is carried out. Based on this analysis, suitable numerical methods are designed, analyzed and implemented in order to have some relevant results of physical importance. Since the asymptotic analysis provides only qualitative information, the focus is more on the numerical analysis of the problem which provides the quantitative information.

This thesis concerns the analytical and practical aspects of applying the Closest Point Method to solve elliptic partial differential equations (PDEs) on smooth surfaces and domains with smooth boundaries. A new numerical scheme is proposed to solve surface elliptic PDEs and a novel geometric multigrid solver is constructed to solve the resulting linear system. The method is also applied to coupled bulk-surface problems. A new embedding equation in a narrow band surrounding the surface is formulated so that it agrees with the original surface PDE on the surface and has a unique solution which is constant along the normals to the surface. The embedding equation is then discretized using standard finite difference scheme and barycentric Lagrange interpolation. The resulting scheme has 2nd-order accuracy in practice and is provably 2nd-order convergent for curves without boundary embedded in ℝ². To apply the method to solve elliptic equations on surfaces and domains with boundaries, the "ghost" point approach is adopted to handle Dirichlet, Neumann and Robin boundary conditions. A systematic method is proposed to represent values of ghost points by values of interior points according to boundary conditions. A novel geometric multigrid method based on the closest point representation of the surface is constructed to solve the resulting large sparse linear systems. Multigrid solvers are designed for surfaces with or without boundaries and domains with smooth boundaries. Numerical results indicate that the convergence rate of the multigrid solver stays roughly the same as we refine the mesh, as is desired of a multigrid algorithm. Finally the above methods are combined to solve coupled bulk-surface PDEs with some applications to biology.