<p>U disertaciji su razvijeni kolokacioni postupci sa C¹- splajnovima&nbsp;proizvoljnog stepena za re&scaron;avanje singularno-perturbovanih problema&nbsp;reakcije-difuzije u jednoj i dve dimenzije. U 1D, pokazano je da kolokacioni&nbsp;postupak sa kvadratnim C¹-&nbsp;splajnom na modifikovanoj &Scaron;i&scaron;kinovoj mreži,&nbsp;konvergira uniformno, sa redom konvergencije skoro dva. Takođe, na gradiranim mrežama, ovaj metod ima red konvergencije dva &ndash; uniformno do na logaritamski faktor. Aposterirona ocena je postignuta za kolokacione postupke sa C¹- splajnovima proizvoljnog stepena na proizvoljnoj mreži. Ova ocena je iskori&scaron;ćena i za kreiranje adaptivnih mreža. Numerički rezultati povtrđuju dobijene ocene. U 2D su razmatrane kolokacije sa bikvadratnim splajnovima. Aposterirona ocena gre&scaron;ke je postignuta. Numerički rezultati potvrđuju dobijene teorijske rezultate.<br />&nbsp;</p> / <p>Collocations with arbitrary order C¹-splines for a singularly perturbed reaction-diffusion problem in one dimension and two dimensions are studied. In 1D, collocation with quadratic C¹-splines is shown to be almost second order accurate on modified Shishkin mesh in the maximum norm, uniformly in the perturbation parameter. Also, we establish a second-order maximum norm a priori estimate on recursively graded mesh uniformly up to a logarithmic factor in the singular perturbation parameter. A posteriori error bounds are derived for the collocation method with arbitrary order C¹-splines on arbitrary meshes. These bounds are used to drive an adaptivemeshmoving algorithm. An adaptive algorithm is devised&nbsp;to resolve the boundary layers. Numerical results are presented. In 2D, collocation with biquadratic C¹-spline is studied. Robust a posteriori error bounds are derived for the collocation method on arbitrary meshes. Numerical experiments completed our theoretical results.</p>

We present a novel dimensionally split Cartesian cut cell method to compute inviscid, viscous and turbulent flows around rigid geometries. On a cut cell mesh, the existence of arbitrarily small boundary cells severely restricts the stable time step for an explicit numerical scheme. We solve this `small cell problem' when computing solutions for hyperbolic conservation laws by combining wave speed and geometric information to develop a novel stabilised cut cell flux. The convergence and stability of the developed technique are proved for the one-dimensional linear advection equation, while its multi-dimensional numerical performance is investigated through the computation of solutions to a number of test problems for the linear advection and Euler equations. This work was recently published in the Journal of Computational Physics (Gokhale et al., 2018). Subsequently, we develop the method further to be able to compute solutions for the compressible Navier-Stokes equations. The method is globally second order accurate in the L1 norm, fully conservative, and allows the use of time steps determined by the regular grid spacing. We provide a full description of the three-dimensional implementation of the method and evaluate its numerical performance by computing solutions to a wide range of test problems ranging from the nearly incompressible to the highly compressible flow regimes. This work was recently published in the Journal of Computational Physics (Gokhale et al., 2018). It is the first presentation of a dimensionally split cut cell method for the compressible Navier-Stokes equations in the literature. Finally, we also present an extension of the cut cell method to solve high Reynolds number turbulent automotive flows using a wall-modelled Large Eddy Simulation (WMLES) approach. A full description is provided of the coupling between the (implicit) LES solution and an equilibrium wall function on the cut cell mesh. The combined methodology is used to compute results for the turbulent flow over a square cylinder, and for flow over the SAE Notchback and DrivAer reference automotive geometries. We intend to publish the promising results as part of a future publication, which would be the first assessment of a WMLES Cartesian cut cell approach for computing automotive flows to be presented in the literature.

General relativity, one of the pillars of our understanding of the universe, has been a remarkably successful theory. It has stood the test of time for more than 100 years and has passed all experimental tests so far. Most recently, the LIGO collaboration made the first-ever direct detection of gravitational waves, confirming a long-standing prediction of general relativity. Despite this, several fundamental mathematical questions remain unanswered, many of which relate to the global existence and the stability of solutions to Einstein's equations. This thesis presents our efforts to use numerical relativity to investigate some of these questions. We present a complete picture of the end points of black ring instabilities in five dimensions. Fat rings collapse to Myers-Perry black holes. For intermediate rings, we discover a previously unknown instability that stretches the ring without changing its thickness and causes it to collapse to a Myers-Perry black hole. Most importantly, however, we find that for very thin rings, the Gregory-Laflamme instability dominates and causes the ring to break. This provides the first concrete evidence that in higher dimensions, the weak cosmic censorship conjecture may be violated even in asymptotically flat spacetimes. For Myers-Perry black holes, we investigate instabilities in five and six dimensions. In six dimensions, we demonstrate that both axisymmetric and non-axisymmetric instabilities can cause the black hole to pinch off, and we study the approach to the naked singularity in detail. Another question that has attracted intense interest recently is the instability of anti-de Sitter space. In this thesis, we explore how breaking spherical symmetry in gravitational collapse in anti-de Sitter space affects black hole formation. These findings were made possible by our new open source general relativity code, GRChombo, whose adaptive mesh capabilities allow accurate simulations of phenomena in which new length scales are produced dynamically. In this thesis, we describe GRChombo in detail, and analyse its performance on the latest supercomputers. Furthermore, we outline numerical advances that were necessary for simulating higher dimensional black holes stably and efficiently.

A block-based adaptive mesh refinement (AMR) finite-volume scheme is developed for solution of hyperbolic conservation laws on two-dimensional hybrid multi-block meshes. A Godunov-type upwind finite-volume spatial-discretization scheme, with piecewise limited linear reconstruction and Riemann-solver based flux functions, is applied to the quadrilateral cells of a hybrid multi-block mesh and these computational cells are embedded in either body-fitted structured or general unstructured grid partitions of the hybrid grid. A hierarchical quadtree data structure is used to allow local refinement of the individual subdomains based on heuristic physics-based refinement criteria. An efficient and scalable parallel implementation of the proposed algorithm is achieved via domain decomposition. The performance of the proposed scheme is demonstrated through application to solution of the compressible Euler equations for a number of flow configurations and regimes in two space dimensions. The efficiency of the AMR procedure and accuracy, robustness, and scalability of the hybrid mesh scheme are assessed.

A block-based adaptive mesh refinement (AMR) finite-volume scheme is developed for solution of hyperbolic conservation laws on two-dimensional hybrid multi-block meshes. A Godunov-type upwind finite-volume spatial-discretization scheme, with piecewise limited linear reconstruction and Riemann-solver based flux functions, is applied to the quadrilateral cells of a hybrid multi-block mesh and these computational cells are embedded in either body-fitted structured or general unstructured grid partitions of the hybrid grid. A hierarchical quadtree data structure is used to allow local refinement of the individual subdomains based on heuristic physics-based refinement criteria. An efficient and scalable parallel implementation of the proposed algorithm is achieved via domain decomposition. The performance of the proposed scheme is demonstrated through application to solution of the compressible Euler equations for a number of flow configurations and regimes in two space dimensions. The efficiency of the AMR procedure and accuracy, robustness, and scalability of the hybrid mesh scheme are assessed.

This study presents an improved ghost-cell immersed boundary approach to represent a solid body in compressible flow simulations. In contrast to the commonly used approaches, in the present work ghost cells are mirrored through the boundary described using a level-set method to farther image points, incorporating a higher-order extra/interpolation scheme for the ghost cell values. In addition, a shock sensor is in- troduced to deal with image points near the discontinuities in the flow field. Adaptive mesh refinement (AMR) is used to improve the representation of the geometry efficiently. The improved ghost-cell method is validated against five test cases: (a) double Mach reflections on a ramp, (b) supersonic flows in a wind tunnel with a forward- facing step, (c) supersonic flows over a circular cylinder, (d) smooth Prandtl-Meyer expansion flows, and (e) steady shock-induced combustion over a wedge. It is demonstrated that the improved ghost-cell method can reach the accuracy of second order in L1 norm and higher than first order in L∞ norm. Direct comparisons against the cut-cell method demonstrate that the improved ghost-cell method is almost equally accurate with better efficiency for boundary representation in high-fidelity compressible flow simulations. Implementation of the improved ghost-cell method in reacting Euler flows further validates its general applicability for compressible flow simulations.

This dissertation presents a multilevel algorithm to solve constant and variable coeffcient elliptic boundary value problems on adaptively refined structured meshes in 2D and 3D. Cacheaware algorithms for optimizing the operations to exploit the cache memory subsystem areshown. Keywords: Multigrid, Cache Aware, Adaptive Mesh Refinement, Partial Differential Equations, Numerical Solution.

Transient multi-phase flow problems in porous media are ubiquitous in engineering and environmental systems and processes; examples include heat exchangers, reservoir simulation, environmental remediation, magma flow in the earth crust and water management in porous electrodes of PEM fuel cells. This thesis focuses on the development of accurate and computationally efficient numerical models to simulate such flows. The research challenges addressed in this work fall in two areas. For a numerical standpoint, conventional numerical methods including Newton-Raphson linearization and a simple upwind scheme do not always provide the required computational efficiency or sufficiently accurate resolution of the flow field. From a modelling perspective, closure schemes required in volume-averaged formulations, such as the generalized Leverett J function for capillary pressure, are specific to certain media (e.g. lithologic media) and are not valid for fibrous porous media, which are of central interest in fuel cells. This thesis presents a set of algorithms that are integrated efficiently to achieve computations that are more than two orders of magnitude faster compared to traditional techniques. The method uses an adaptive operator splitting method based on an a posteriori criterion to separate the flow from the transport equations which eliminates unnecessary and costly solution of the implicit pressure-velocity term at every time step; adaptive meshing to reduce the size of the discretized problem; efficient block preconditioned solver techniques for fast solution of the discrete equations; and a recently developed artificial diffusion strategy to stabilize the numerical solution of the transport equation. The significant improvements in accuracy and efficiency of the approach is demosntrated using numerical experiments in 2D and 3D. The method is also extended to advection-dominated problems to specifically investigate two-phase flow in heterogeneous porous media involving capillary transport. Both hydrophilic and hydrophobic media are considered, and insights relevant to fuel cell electrodes are discussed.

Accurate numerical solution of time-dependent, high-dimensional partial differential equations (PDEs) usually requires efficient numerical techniques and massive-scale parallel computing. In this thesis, we implement and evaluate discretization schemes suited for PDEs of higher dimensionality, focusing on high order of accuracy and low computational cost. Spatial discretization is particularly challenging in higher dimensions. The memory requirements for uniform grids quickly grow out of reach even on large-scale parallel computers. We utilize high-order discretization schemes and implement adaptive mesh refinement on structured hyperrectangular domains in order to reduce the required number of grid points and computational work. We allow for anisotropic (non-uniform) refinement by recursive bisection and show how to construct, manage and load balance such grids efficiently. In our numerical examples, we use finite difference schemes to discretize the PDEs. In the adaptive case we show how a stable discretization can be constructed using SBP-SAT operators. However, our adaptive mesh framework is general and other methods of discretization are viable. For integration in time, we implement exponential integrators based on the Lanczos/Arnoldi iterative schemes for eigenvalue approximations. Using adaptive time stepping and a truncated Magnus expansion, we attain high levels of accuracy in the solution at low computational cost. We further investigate alternative implementations of the Lanczos algorithm with reduced communication costs. As an example application problem, we have considered the time-dependent Schrödinger equation (TDSE). We present solvers and results for the solution of the TDSE on equidistant as well as adaptively refined Cartesian grids. / eSSENCE

For many years, terrorism has threatened life, property and business. Targets are largely in urban areas where there is a greater density of life and economic value. Governments, insurers and engineers have sought to mitigate these threats through understanding the effects of urban bombings, increasing the resilience of buildings and improving estimates of financial loss for insurance purposes. This has led to a desire for an improved approach to the prediction of blast propagation in urban cityscapes. Urban geometry has a significant impact on blast wave propagation. Presently, only computational fluid dynamics (CFD) methods adequately simulate these effects. However, for large-scale urban domains, these methods are both challenging to use and are computationally expensive. Adaptive mesh refinement (AMR) methods alleviate the problem, but are difficult to use for the non-expert and require significant tuning. We aim to make CFD urban blast simulation a primary choice for governments, insurers and engineers through improvements to AMR and by studying the performance of CFD in relation to other methods used by the industry. We present a new AMR flagging approach based on a second derivative error norm for compressive shocks (ENCS). This is compared with existing methods and is shown to lead to a reduction in overall refinement without affecting solution quality. Significant improvements to feature tracking over long distances are demonstrated, making the method easier to tune and less obtuse to non-experts. In the chapter that follows, we consider blast damage in urban areas. We begin with a validation and a numerical study, investigating the effects of simple street geometry on blast resultants. We then investigate the sensitivity of their distribution to the location of the charge. We find that moving the charge by a small distance can lead to a significant change in peak overpressures and creates a highly localised damage field due to interactions between the blast wave and the geometry. We then extend the investigation to the prediction of insured losses following a large-scale bombing in London. A CFD loss model is presented and compared with simpler approaches that do not account for urban geometry. We find that the simpler models lead to significant over-predictions of loss, equivalent to several hundred million pounds for the scenario considered. We use these findings to argue for increased uptake of CFD methods by the insurance industry. In the final chapter, we investigate the influence of urban geometry on the propagation of blast waves. An earlier study on the confinement effects of narrow streets is repeated at a converged resolution and we corroborate the findings. We repeat the study, this time introducing a variable porosity into the building facade. We observe that the effect of this porosity is as significant as the confinement effect, and we recommend to engineers that they consider porosity effects in certain cases. We conclude the study by investigating how alterations to building window layout can improve the protective effects of a facade. Maintaining the window surface area constant, we consider a range of layouts and observe how some result in significant reductions to blast strength inside the building.