The numerical solution of chemical reactions described at the meso-scale is the topic of this thesis. This description, the master equation of chemical reactions, is an accurate model of reactions where stochastic effects are crucial for explaining certain effects observed in real life. In particular, this general equation is needed when studying processes inside living cells where other macro-scale models fail to reproduce the actual behavior of the system considered. The main contribution of the thesis is the numerical investigation of two different methods for obtaining numerical solutions of the master equation. The first method produces statistical quantities of the solution and is a generalization of a frequently used macro-scale description. It is shown that the method is efficient while still being able to preserve stochastic effects. By contrast, the other method obtains the full solution of the master equation and gains efficiency by an accurate representation of the state space. The thesis contains necessary background material as well as directions for intended future research. An important conclusion of the thesis is that, depending on the setup of the problem, methods of highly different character are needed.

Obtaining accurate solutions to flows that involve discontinuous features still re- mains one of the most difficult tasks in computational fluid dynamics today. Some discontinuous features, such as shear waves and material interfaces, are quite deli- cate, yet they have a profound effect on the rest of the flow field. The accuracy of the numerical scheme and the quality of the grid discretisation of the flow domain, are both critical when computing multi-dimensional discontinuous solutions. Here, the second order WAF scheme is used in conjuction with an adaptive grid algorithm, which is able to automatically modify the grid in regions of discontinuous features and solid boundaries. The grid algorithm is a combination of two successful ap- proaches, namely Chimera and Cartesian grid Adaptive Mesh Refinement (AMR). The Chimera approach is able to accurately represent non-Cartesian boundaries, whilst the AMR approach yields significant savings in memory storage and cPu time. The combined algorithm has been thoroughly validated for convection test problems in gas dynamics. The computed solutions compare well with other numerical and experimental results. These tests have also been used to assess the efficiency of the grid adaption algorithms. Finally, the approach is applied to axi-symmetric, two- dimensional, two-phase, reactive flows in the context of internal ballistics problems. Again, the computed results are compared with other numerical and experimental results.

Effective and robust high resolution schemes are of vital importance for simulation of viscous and inviscid flows. Since second-order high resolution schemes in practice are inadquate for many applications, large efforts have been put towards developing higher- order accurate schemes in the past. Although some progress has been made, the efforts were frustrated by the lack of effective and robust new schemes. Therefore this thesis is aimed at challenging this difficult but very important issue. Some new theories and methodologies were established during this research, which covers the linear stability analysis for high-order numerical schemes; the fully discrete techniques for model equations; the formulation of conservative high-order schemes and the high-order Total Variation Diminishing (TVD) schemes. According to these theories arbitrary-order high resolution schemes can be developed. To illustrate the methodologies second-, third-, fourth-, and 20th-order schemes are presented. These high resolution schemes were tested and validated by solving some popular test problems for one and two dimensional Euler and incompressible Navier-Stokes equations. The efficiency and robustness are the features of these high-order schemes.

Deep neural networks (DNNs) have successfully been applied across various data intensive applications ranging from computer vision, language modeling, bioinformatics and search engines. Hyper-parameters of a DNN are defined as parameters that remain fixed during model training and heavily influence the DNN performance. Hence, regardless of application, the design-phase of constructing a DNN model becomes critical. Framing the selection and tuning of hyper-parameters as an expensive black-box optimization (BBO) problem, obstacles encountered in manual by-hand tuning could be addressed by taking instead an automated algorithmic approach. In this work, the following BBO algorithms: Nelder-Mead Algorithm (NM), ParticleSwarm Optmization (PSO), Bayesian Optimization with Gaussian Processes (BO-GP) and Tree-structured Parzen Estimator (TPE), are evaluated side-by-side for two hyper-parameter optimization problem instances. These instances are: Problem 1, incorporating a convolutionalneural network and Problem 2, incorporating a recurrent neural network. A simple Random Search (RS) algorithm acting as a baseline for performance comparison is also included in the experiments. Results in this work show that the TPE algorithm achieves the overall highest performance with respect to mean solution quality, speed ofimprovement and with a comparatively low trial-to-trial variability for both Problem 1 and Problem 2. The NM, PSO and BO-GP algorithms are shown capable of outperforming the RS baseline for Problem 1, but fails to do so in Problem 2.

<p> A fourth-order accurate finite volume method for curvilinear grids based on Hermitian interpolation and splines is here presented in one and two dimensions. The finite volume method is derived in detail starting in one dimension and then extended to two dimensions using isoparametric mapping. The method is applied to the quasi-one-dimensional Euler equations through a converging-diverging nozzle as well as the heat conduction equation through a body-fitted non-orthogonal grid. Comparisons are made between the methods presented here and similar techniques in the literature. Lastly, possible ways to improve the method&rsquo;s computational efficiency are discussed. </p><p>

Compact Riemann surfaces of genus greater than 1 can be realized as quotient spaces of the hyperbolic plane by the action of Fuchsian groups. The Teichmüller space is the set of all complex structures of Riemann surfaces and the moduli space the set of conformal equivalence classes of Riemann surfaces. For genus greater than two the branch locus of the covering of the moduli space by the Teichmüller space can be identified wi the set of Riemann surfaces admitting non-trivial automorphisms. Here we give the orbifold structure of the branch locus of surfaces of genus 5 by studying the equisymmetric stratification of the branch locus. This gives the orbifold structure of the moduli space. We also show that the strata corresponding to surfaces with automorphisms of order 2 and 3 belong to the same connected component for every genus. Further we show that the branch locus is connected with the exception of one isolated point for genera 5 and 6, it is connected for genus 7 and it is connected with the exception of two isolated points for genus 8.

The frequent presence of ripples on the free surface of water. on both thin film flows and ponds or lakes motivates this theoretical investigation into the propagation of ripples on gravity waves. These ripples are treated as "slowly-varying" waves in a reference frame where the gravity wave flow is steady. The methods used are those of the averaged Lagrangian (Whitham 1965,1967,1974) and the averaged equations of motion (Phillips 1966) which are shown to be equivalent. The capillary wave modulation is taken to be steady in the reference frame which brings the gravity wave, or gravity driven flow, to rest. Firstly the motion over ponds or lakes is considered. Linear capillary-gravity waves are examined in order to set the scene. Crapper's (1957) exact finite-amplitude waves are examined next to show the actual behaviour of the flow field. The underlying gravity driven flow is that of pure gravity waves over an' "infinite" depth liquid. These gravity waves are modelled with "numerically exact" solutions for periodic plane-waves. The initial studies are inviscid and show that steep gravity waves either "absorb" or "sweep-up" a range of capillary waves or, alternatively, cause them to break in the vicinity of gravity wave crests. Improvements on the theory are made by including viscous dissipation of wave energy. This leads to a number of solutions approaching "stopping velocities" or the "stopped waves solution". In addition to these effects "higher-order dispersion" is introduced for weakly nonlinear waves near linear caustics. This clarifies aspects of the dissipation results and shows that wave reflection sometimes occurs. Secondly, waves on thin film flows are considered. Linear capillary-gravity waves are again examined in order to set the scene. Kinnersley's (1957) exact finite-amplitude waves are examined next to show the actual behaviour of the flow field. The underlying gravity driven flow is given by shallow water gravity waves. No modelling of these is necessary simply because they are included within Whitham's or Phillips' equations ab initio. This study is inviscid and shows the unexpected presence of critical velocities at which pairs of solution branches originate. iii

This thesis deals with fast and efficient methods for electrostatic calculations with application in molecular dynamics simulations. The electrostatic calculations are often the most expensive part of MD simulations of charged particles. Therefore, fast and efficient algorithms are required to accelerate these calculations. In this thesis, two types of methods have been considered: FFT-based methods and fast multipole methods (FMM). The major part of this thesis deals with fast N.log(N) and spectrally accurate methods for accelerating the computation of pairwise interactions with arbitrary periodicity. These methods are based on the Ewald decomposition and have been previously introduced for triply and doubly periodic problems under the name of Spectral Ewald (SE) method. We extend the method for problems with singly periodic boundary conditions, in which one of three dimensions is periodic. By introducing an adaptive fast Fourier transform, we reduce the cost of upsampling in the non periodic directions and show that the total cost of computation is comparable with the triply periodic counterpart. Using an FFT-based technique for solving free-space harmonic problems, we are able to unify the treatment of zero and nonzero Fourier modes for the doubly and singly periodic problems. Applying the same technique, we extend the SE method for cases with free-space boundary conditions, i.e. without any periodicity. This thesis is also concerned with the fast multipole method (FMM) for electrostatic calculations. The FMM is very efficient for parallel processing but it introduces irregularities in the electrostatic potential and force, which can cause an energy drift in MD simulations. In this part of the thesis we introduce a regularized version of the FMM, useful for MD simulations, which approximately conserves energy over a long time period and even for low accuracy requirements. The method introduces a smooth transition over the boundary of boxes in the FMM tree and therefore it removes the discontinuity at the error level inherent in the FMM. / <p>QC 20171213</p>

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District heating is today, in Sweden, the most common method used for heating buildings in cities. More than half of all the buildings, both commercial and residential, are heated using district heating. The load on the district heating networks are affected by, among other things, the time of the day and different external conditions, such as temperature differences. One has to be able to simulate the heat and pressure losses in the network in order to deliver the amount of heat demanded by the customers. Expansions of district heating networks and disrupted pipes also demand good simulations of the networks. To cope with this, energy companies use simulation software. These software need to contain numerical methods that provide accurate and stable results and at the same time be fast and efficient. At the moment there are available software packages that works but these have some limitations. Among other things you may need to divide the whole network into smaller loops or try to guess how the distribution of pressure and flow in the network looks like. The development in recent years makes it possible to use better and more efficient algorithms for these types of problems. The purpose of this report is therefore to introduce a better and more efficient method than that used in the current situation. This work is the first step in order to replace a current method used in a simulation software provided by Vitec energy. Therefore, we will in this report, stick to computing pressure and flow in the network. The method we will introduce in this report is called the gradient method and it is based on the Newton Raphson method. Unlike with older methods like Hardy Cross which is a relaxation method, you do not have to divide the network into loops. Instead you create a matrix representation of the network that is used in the computations. The idea is also that you should not need to make good initial guesses to get the method to converge quickly. We performed a number of test simulations in order to examine how the method performs. We tested how different initial guesses and how different sizes of the networks affected the number of iterations. The results shows that the model is capable of solving large networks within a reasonable number of iterations. The results also show that the initial guesses have little impact on the number of iterations. Changing the initial guess on the pressure does not affect the number at all but it turns out that changing the initial guess on the flow can affect the number of iterations a little, but not much.