The main goal of the thesis is to construct explicit, s-stage, strong-stability-preserving (SSP) Hermite–Birkhoff (HB) time discretization methods of order p with nonnegative coefficients for the integration of hyperbolic conservation laws. The Shu–Osher form and the canonical Shu–Osher form by means of the vector formulation for SSP Runge–Kutta (RK) methods are extended to SSP HB methods. The SSP coefficients of k-step, s-stage methods of order p, HB(k,s,p), as combinations of k-step methods of order (p − 3) with s-stage explicit RK methods of order 4, and k-step methods of order (p-4) with s-stage explicit RK methods of order 5, respectively, for s = 4, 5,..., 10 and p = 4, 5,..., 12, are constructed and compared with other methods. The good efficiency gains of the new, optimal, SSP HB methods over other SSP methods, such as Huang’s hybrid methods and RK methods, are numerically shown by means of their effective SSP coefficients and largest effective CFL numbers. The formulae of these new, optimal methods are presented in their Shu–Osher form.

New analytical and numerical solutions are developed to both the kinematic approximation to the St Venant equations and the Hairsine-Rose (HR) soil erosion model in order to gain a better physical understanding of soil erosion and sediment transport in shallow overland flow. The HR model is unique amongst physically based erosion models in that it is the only one that: considers the entire distribution of the soil s sediment size classes, considers the development of a layer of deposited non-cohesive sediment having different characteristics to the original underlying cohesive soil and considers separately the erosion processes of rainfall detachment, runoff entrainment and gravitational deposition. The method of characteristics and the method of lines were used to develop both the analytical and numerical solutions respectively. These solutions were obtained for boundary and initial conditions typical of those used in laboratory flume experiments along with physically realistic constant and time dependent excess rainfall rates. Depending on the boundary and initial conditions, interesting new solutions of the kinematic wave equation containing expansion waves, travelling shocks as well as solutions which split into an upslope and downslope drying profiles were found. Numerical solutions of the HR model were applied to the experimental flume data of Polyakov and Nearing (2003) obtained under flow conditions which periodically cycled between net erosion and net deposition conditions. While excellent agreement was found with suspended sediment data, the analysis suggested that an additional transport mechanisms, traditionally not included in soil erosion models, was occurring. While the inclusion of bed-load transport improved the ii overall model prediction, it was still not sufficient. Subsequent asymptotic analysis then showed that the interaction of the flow with an evolving bed morphology was in fact far more important than bed load transport. A very interesting finding from this work showed that the traditional criterion of validating sediment transport model based solely on suspended sediment data was not sufficient as reliable predictions could be obtained even when important transport mechanisms were neglected. Experimental plots of sediment discharge or suspended sediment concentration against water discharge in overland flow have been shown to contain significant hysteresis between the falling and rising limbs of the discharge hydrograph. In the final Chapter, the numerical solution developed for the complete system of soil erosion and kinematic flow was used to show that it was possible for the HR model to simulate three of the four hysteresis loops identified in the literature. Counter clock-wise loops, clock-wise loops and figure 8 loops could all be produced as a result of starting with different initial conditions, being mi(x; 0) = 0, mi(x; 0) = pimt and mi(x; 0) = 0:5pimt respectively. This is the first time that these types of hysteresis loops have been produced by any erosion model. The generation of these hysteresis loops are physically explainable in terms of sediment availability and is consistent with data obtained on the field scale.

The aim of the thesis is to find a method, in conjunction with the ordinary differential equation (ODE) based method of lines (MOL) solution of Richards’ equation, to model the steep wetting front infiltration in very dry soils, accurately and efficiently. Due to the steep pressure head or steep water volumetric content gradients, highly nonlinear soil hydraulic properties and the rapid movement of the wetting front, accurate solutions for infiltration into a dry soil are usually difficult to obtain. Additionally, such problems often require very small time steps and large computation times. As an enhancement to the used ODE/MOL approach, Higher Order Finite Differencing, Varying Order Finite Differencing, Vertical Scaling, Adaptive Schemes and Non-uniform Stretching Techniques have been implemented and tested in this thesis. Success has been found in the ability of Vertical Scaling to simulate very steep moving front solution for the Burgers’ equation. Unfortunately, the results also show that Vertical Scaling needs significant research and improvement before their full potential in routine applications for difficult nonlinear problems, such as Richard’s equation with very steep moving front solution, can be realized. However, we have also shown that the use of the composed form of RE and a 2nd order finite differencing for the first order derivative approximation is conducive for modelling steep moving front problem in a very dry soil. Additionally, with the combination of an optimal influx value at the edges of the inlet, the ODE/MOL approach is able to model a 2-D infiltration in very dry soils, effectively and accurately. Furthermore, one of the strengths of this thesis is the use of a MATLAB PDE template. Implementing the ODE/MOL approach via a MATLAB PDE template has shown to be most suitable for modelling of partial differential equations. The plug and play mode of modifying the PDE template for solving time-dependent partial differential equations is user-friendly and easy, as compared to more conventional approaches using Pascal, Fortran, C or C++. The template offers greater modularity, flexibility, versatility, and efficiency for solving PDE problems in both 1-D and 2-D spatial dimensions. Moreover, the 2-D PDE template has been extended for irregular shaped domains.

There is the need for applying numerical methods to problems that cannot be solved analytically and as the spatial dimension of the problem is increased the need for computational recourses increase exponentially, a phenomenon known as the “curse of dimensionality”. In the Black-Scholes-Merton framework the American option pricing problem has no closed form solution and a numerical procedure has to be employed for solving a PDE. The multi-asset American option introduces challenging computational problems, since for every added asset the dimension of the PDE is increased by one. One way to deal with the curse of dimensionality is threw parallelism. Here the finite element method-of-lines is used for pricing a multi-asset American option dependent on up to four assets in a parallel environment. The problem is also solved with the PSOR method giving a accurate benchmark used for comparison. In finance the put option is one of the most fundamental derivatives since it is basically asset-value insurance and a lot of research is done in the field of quantitative finance on accurate and fast pricing techniques for the multi-dimensional case. “What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.” Norbert Wiener “As soon as an Analytical Engine exists, it will necessarily guide the future course of the science. Whenever any result is sought by its aid, the question will then arise – by what course of calculation can these results be arrived at by the machine in the shortest time?” Charles Babbage

A non-iterative pressure based algorithm which consists of splitting the solution of momentum energy and species equations into a sequence of predictor-corrector stages was developed for the simulation of transient reacting radiating flows. A semi-discrete approach called the Method of Lines (MOL) which enables implicit time-integration at all splitting stages was used for the solution of conservation equations. The solution of elliptic pressure equation for the determination of pressure field was performed by a multi-grid solver (MUDPACK package). Radiation calculations were carried out by coupling previously developed gray and non-gray radiation models with the algorithm. A first order (global) reaction mechanism was employed to account for the chemistry. The predictions of the algorithm for the following test cases: i) non-isothermal turbulent pipe flow and ii) laminar methane-air diffusion flame / were benchmarked against experimental data and numerical solutions available in the literature and the capability of the code to predict transient solutions was demonstrated on these test cases. Favorable agreements were obtained for both test cases. The effect of radiation and non-gray treatment of the radiative properties were investigated on the second test case. It was found that incorporation of radiation has significant effect on Temeprature and velocity fields but its effect is limited in species predictions. Executions with both radiation models revealed that the non-gray radiation model considered in the present study produces similar results with the gray model at a considerably higher computational cost. The algorithm developed was found to be an efficient and versatile tool for the timedependent simulation of different flow scenarios constitutes the initial steps towards the computation of transient turbulent combustion.

A radiation code based on method of lines (MOL) solution of discrete ordinates method (DOM) for the prediction of radiative heat transfer in nongray absorbing-emitting media was developed by incorporation of two different gas spectral radiative property models, namely wide band correlated-k (WBCK) and spectral line-based weighted sum of gray gases (SLW) models. Predictive accuracy and computational efficiency of the developed code were assessed by applying it to the predictions of source term distributions and net wall radiative heat fluxes in several one- and two-dimensional test problems including isothermal/non-isothermal and homogeneous/non-homogeneous media of water vapor, carbon dioxide or mixture of both, and benchmarking its steady-state predictions against line-by-line (LBL) solutions and measurements available in the literature. In order to demonstrate the improvements brought about by these two spectral models over and above the ones obtained by gray gas approximation, predictions obtained by these spectral models were also compared with those of gray gas model. Comparisons reveal that MOL solution of DOM with SLW model produces the most accurate results for radiative heat fluxes and source terms at the expense of computation time when compared with MOL solution of DOM with WBCK and gray gas models. In an attempt to gain an insight into the conditions under which the source term predictions obtained with gray gas model produce acceptable accuracy for engineering applications when compared with those of gas spectral radiative property models, a parametric study was also performed. Comparisons reveal reasonable agreement for problems containing low concentration of absorbing-emitting media at low temperatures. Overall evaluation of the performance of the radiation code developed in this study points out that it provides accurate solutions with SLW model and can be used with confidence in conjunction with computational fluid dynamics (CFD) codes based on the same approach.

Climate models are the primary tools used for understanding past climate variations and for future projections. The atmospheric radiation is the key component of these models. Accurate modeling of atmosphere necessitates reliable evaluation of the medium radiative properties and accurate solution of the radiative transfer equation in conjunction with the time-dependent multi-dimensional governing equations of atmospheric models. Due to difficulty in solving the equations of atmospheric and radiation models simultaneously, radiation equations have been solved when input data such as concentration, temperature etc. were made available upon solution of equations of atmospheric models. Generally, time step of conservation equations are 10-30 minutes but radiative transfer equation is called only once every 1-3 hours. However, there is inaccuracy due to the fixed radiation fluxes over the intervening time steps. To overcome this problem, the equations of atmospheric and radiation models have to be solved simultaneously and the solution methods have to be compatible. For this purpose, a radiative-convective model with radiation model based on method of lines (MOL) solution of discrete ordinate method (DOM) with wide band correlated-k (WBCK) was developed. To achieve this objective, a previously developed MOL solution of DOM with WBCK model was adapted to 1-D longwave clear sky atmosphere and its predictive accuracy and computational efficiency was examined on the test problem by using benchmark solution obtained from Line-by-line Radiative Transfer Model (LBLRTM). The radiation code was then coupled with radiative-convective model and the predictive accuracy of this model was examined for several coupling intervals. Comparisons reveal that as coupling interval increases, although the computation time of the model decreases, the predicted temperature profiles diverge from the one obtained when equations of radiative-convective model and the radiation model are solved simultaneously and percentage relative error in temperature increases an order of magnitude when coupling time between radiative-convective model and the radiation model increases from 2 to 10 hours. Therefore, it can be concluded that the equations of the radiation model have to be solved simultaneously with the equations of the climate model. Overall evaluation of the performance of the radiation model used in this study points out that it provides accurate and computationally efficient solutions and can be used with confidence in conjunction with the climate models for simultaneous solution of governing equations with radiation transfer equation.

Title: Numerical analysis of approximation of nonpolygonal domains for discon- tinuous Galerkin method Author: Filip Klouda Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Vít Dolejší, Ph.D., DSc., KNM MFF UK Abstract: In this work we use the discontinuous Galerkin finite element method for the semidiscretization of a nonlinear nonstationary convection-diffusion pro- blem defined on a nonpolygonal two-dimensional domain. Using so called appro- ximating curved elements we define a piecewise polynomial approximation of the boundary of the domain and a space on which we search for a solution. We study the convergence of the method considering a symmetric as well as nonsymmetric discretization of diffusion terms and with the interior and boundary penalty. The obtained results allow us to derive an error estimate for the Discontinuous Galer- kin method employing the approximating curved elements. This estimate depends on the order of the approximation of the solution and also on the order of the approximation of the boundary. We describe one possibility of the construction of the approximating curved elements with the aid of a polynomial mapping given by an interpolation of points on the boundary. We present numerical experiments. Keywords: nonlinear convection-diffusion equation, discontinuous...

This work deals with techniques of time-domain analysis of interconnects. After a studying crucial issue of time-domain analysis of interconnects methods of modeling and simulation simple interconnects in electrotechnics are described. For transient effect analysis two elementary methods can be used: the state variable method and the FDTD (Finite - Difference Time - Domain) method. The FDTD method can be used to solve partial differential equations in time domain, for instance equations of transmission lines. The method is very effective and delivers satisfactory results in case of linear and non-linear lines with a single “live” conductor. The method can be easily programmed in Matlab.