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This chapter discusses numerical simulations of the incompressible Navier-Stokes equations. Exact Navier-Stokes solutions are presented that are used as benchmarks to validate new codes and evaluate the accuracy of a particular discretization. Some aspects of direct numerical simulation (DNS) and large-eddy simulation (LES) are discussed. The issue of stabilization at high Reynolds number is then presented using the concepts of dynamic subgrid modelling, over-integration, and spectral vanishing viscosity. A new parallel paradigm based on multi-level parallelism is introduced that can help realize adaptive refinement more easily. The final section includes a heuristic refinement method for Navier-Stokes equations.

This chapter examines the effects of large variations in density with depth, that is, density stratification. It first describes anelastic models for 2D cartesian box and 2D cylindrical annulus geometries, using entropy and pressure as working thermodynamic variables or using temperature and pressure, for both convectively unstable and stable regions. In particular, it considers anelastic approximation and how to formulate the anelastic equations, as well as the anelastic form of mass conservation, momentum conservation with entropy as a variable, internal energy conservation with entropy as a variable, and temperature as a variable. It also discusses possible choices for a reference state, focusing on polytropes, before explaining modifications to the numerical method and presenting the numerical simulations using the anelastic model.

This chapter studies the evolution of perturbations in the nonlinear regime. It focuses for the most part on analytic models that shed light on the physical processes involved. The advent of computer technology has made numerical studies of nonlinear evolution almost routine, and many of today's theoretical calculations follow this path. The analytic approaches described here inform these calculations, but the numerical simulations help to sharpen this chapter's conclusions and predictions. The chapter discusses this synergy and describes “semianalytic” models that can be written analytically but whose ultimate justification lies in their good agreement with numerical simulations. It also describes the fundamental aspects of computational methods.

Chapter 7 discusses the role of numerical simulations in turbulence. The emphasis is on the new physical insights these numerical experiments have yielded, as well as some of the limitations of numerical methods.

One of the most important thing engineers and scientists do is to model natural phenomena. They develop conceptual and mathematical models to simulate physical events, whether they are aerospace, biological, chemical, geological, or mechanical. The mathematical models are developed using laws of physics and they are often described in terms of algebraic, differential, and/or integral equations relating various quantities of interest. A mathematical model of the motion of the pendulum can be constructed using the principle of conservation of linear momentum, that is, Newton's second law of motion. This chapter also discusses the traditional finite element method, nonlinear analysis, and classification of nonlinearities. The finite element method is a powerful method that can be used to perform numerical simulations of engineering problems.

This chapter discusses the effect of state perturbations on solutions to a hybrid system. It shows that state perturbations, of arbitrarily small size, can dramatically change the behavior of solutions. While such a phenomenon is also present in continuous-time and discrete-time dynamical systems, it is magnified in the hybrid setting, due to the flows and the jumps being constrained to the flow and the jump sets, respectively. Perturbations affecting the whole state of a hybrid system are usually considered. The resulting behaviors are quite representative of what may occur if perturbations come from state measurement error in a hybrid feedback control system or from errors present in numerical simulations of hybrid systems. The case of hybrid feedback control is given some attention in this chapter. Also, throughout the chapter, effects of perturbations are related to the regularity properties of the data of hybrid systems.

We describe the main features of the coarse-grained models that are typically useful in modelling soft interfaces, from force fields to the continuum descriptions involving density fields. We explain the theoretical basis of the main numerical methods that are used to explore the phase space associated with these models. Finally, three recent examples, illustrating the spirit in which relatively simple simulations can contribute to solving pending problems in soft matter physics, are briefly described. Clearly, a short series of lectures can offer, at best, a biased and restricted view of the available approaches. Our aim here will be to provide the reader with such an overview, with a focus on methods and descriptions that ‘bridge the scale’ between the molecular scale and the continuum or quasi-continuum one. The objective to present a guide to the relevant literature—which has now to a large extent appeared in the form of textbooks.

This chapter develops finite element formulations for flows of viscous incompressible fluids using high-order spectral/hp finite element technology and least-squares finite element formulation. The primary objective has been to present novel mathematical models and innovative discretization procedures in the numerical simulation of fluid mechanics problems, wherein the additional benefits of employing high-order spectral/hp finite element technology and least-squares formulations are pronounced. As a result, ad-hoc tricks (e.g., reduced integration and/or mixed interpolations) required to stabilize low-order finite element formulations are unnecessary.

Nanoscale scientific research suddenly arose during the 1980s and 1990s. It now constitutes an immense worldwide research domain. It is based on the invention of scanning probe microscopy and powerful numerical simulation, and on the capacity to manipulate individual molecules and atoms. The capacity of nanoscale research to synthesize artificial substances, often through epitaxy, such as fullerenes and low-dimensional objects, has fueled important transformations in scientific investigation and allowed the formulation of previously unimagined questions. The emergence of nanostructured materials and corresponding instrumentation has been fueled by cognitive, methodological, instrumental, and material combinatorials. “Combinatorial” refers to an association or interlocking of two or more components that give rise to a resulting novelty in the form of a synergy.

This chapter presents the main ideas behind the application of LB methods to the simulation of turbulent flows. The attention is restricted to the case of direct numerical simulation, in which all scales of motion within the grid resolution are retained in the simulation. Turbulence modeling, in which the effect of unresolved scales on the resolved ones is taken into account by various forms of modeling, will be treated in a subsequent chapter.

The special cases considered in Chapter 4 yield insights, but within the framework of the assumptions made. How robust are these insights? What happens when we move away from quasi-linearity? On the basis of the first-order conditions it is possible to say relatively little about the general shape of the tax schedule. The optimal marginal tax rates in more general cases become considerably more difficult to interpret because labour supply can vary with skill and because of income effects. This chapter presents optimal tax schedules with alternative assumptions. Therefore, numerical calculations have proved useful in generating useful results. This chapter provides advanced numerical simulations of the extended Mirrlees models. Using numerical simulations, we study the role of different elements of the model in determining the pattern of optimal non-linear tax/transfer.

Over times shorter than that required for relaxation of enthalpy, a liquid can exhibit striking heterogeneities. The picture of these heterogeneities is complex with transient patches of rigidity, irregular yet persistent, intersected by tendrils of mobile particles, flickering intermittently into new spatial patterns of motion and arrest. The study of these dynamic heterogeneities has, over the last twenty years, allowed us to characterize cooperative dynamics, to identify new strategies in controlling kinetics in glass-forming liquids and to begin to systematically explore the relationship between dynamics and structure that underpins the behaviour of amorphous materials. Computer simulations of the dynamics in atomic and molecular liquids have played a dominant role in all of this progress. While some may be uneasy about this reliance on modelling, it is unavoidable, given the amount of microscopic detail needed to characterize the dynamic heterogeneities. The complexities revealed by these simulations have called for new conceptual tools. this chapter tries to provide the reader with a clear and complete account of how these tools have been developed in terms of the literature on kinetic length scales in molecular dynamics simulations. Through the `prism' of these length scales, this chapter addresses the question what have we learnt about dynamic heterogeneities from computer simulations?

When the collapse of massive stars is studied in some detail, it is discovered that visible or naked singularities do occur in gravitational collapse quite naturally once we use physically realistic assumptions. For example, a higher density at the center of the collapsing star, which decreases as we move outwards, gives rise to a naked singularity as the collapse final state. These conclusions and the wide variety of collapse models that have been investigated over the past years, including non-spherical collapse and numerical simulations, are discussed in this chapter. The chapter tries to understand and explores here in particular, why a naked singularity forms at all rather than a black hole, in gravitational collapse of a massive star. The chapter shows how the intricacies of Einstein gravity and various physical factors play a key role here so as to allow the super ultra-dense regions in the universe to become visible.

The introduction of scanning probe microscopy to scientific research sparked a “gold rush” over the next fifteen years in which hundreds and then thousands of publications were produced on nano instrumentation. Massive research also focused on the surface topography, properties, and behavior of single molecules. The capacity of nano instruments to control materials and to tailor them has led to the development of a new nanoscale specialty, epitaxy. In nanoscale scientific research, new horizons of collaborations functioned between epitaxy and metrological experimentation. For the first time, questions are not limited by the dictates of nature; artificial objects are designed and constructed in response to questions. In this effort, numerical simulation and metrological experimentation often work together—an instance of a combinatorial.

An overview of dielectric spectroscopy (DS) studies for erythrocytes, blood coagulation, and leukocytes is provided. Cell suspensions and whole blood show the main dielectric response in the MHz region due to interfacial polarization, analysis of which provides important information about the morphology and electric properties of the cells. At first, the dielectric signature of erythrocyte morphology is pointed out by comparing four types of erythrocytes with different shapes, such as discocytes, echinocytes, enlarged erythrocytes, and spherocytes. Subsequently, DS application to monitoring the quality of blood during its preservation is summarized. The glucose effects on the dielectric properties of erythrocytes are also presented. Next, studies of erythrocyte aggregation (rouleaux formation) and blood coagulation are introduced. Dielectric blood coagulometry is a very promising method for the evaluation of the blood coagulability. The chapter concludes with a summary of results on dielectric characterizations of normal and malignant leukocytes.

This chapter explains the development of mathematical models and numerical evaluation of the mathematical models using the finite element method. It also discusses nonlinear analysis and some mathematical preliminaries that are useful for the subsequent chapters. In particular, a review of ordinary vectors and tensors, index notation, and linear vector spaces is presented.

This chapter investigates the politico-economic determination of the welfare state generosity, along with the volume and skill composition of migration. The investigation examines the restricted- and free-migration policies. The native population of a host country selects the volume and skill composition of migration under the politico-economic process, along with the level of generosity of the welfare state. The characterization of the politico-economic equilibrium is demonstrated through numerical simulations under the investigation. The chapter considers another case where the volume and skill composition of migration are determined by the native population of the host country. It also studies the impact of changes in the productivity factor and the skill composition of the native population on the politico-economic equilibrium policy of the host country.