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This chapter considers two ways of employing a spatial resolution that varies with position within a finite-difference method: using a nonuniform grid and mapping to a new coordinate variable. It first provides an overview of nonuniform grids before discussing coordinate mapping as an alternative way of achieving spatial discretization. It then describes an approach for treating both the vertical and horizontal directions with simple finite-difference methods: defining a streamfunction, which automatically satisfies mass conservation, and solving for vorticity via the curl of the momentum conservation equation. It also explains the use of the Chebyshev–Fourier method to simulate the convection or gravity wave problem by employing spectral methods in both the horizontal and vertical directions. Finally, it looks at the basic ideas and some issues that need to be addressed with respect to parallel processing as well as choices that need to be made when designing a parallel code.

Introduces some of the methods and underlying ideas behind computational fluid dynamics—in particular, the use is discussed of finite‐difference methods for the simulation of dynamic economies. A standard stochastic dynamic programming model is considered of a macroeconomy. Finite‐difference methods are applied to this problem (model), resulting in a second‐order nonlinear partial differential equation that has some features in common with the governing equations of fluid dynamics; the idea is also introduced of ‘upwind’ or solution‐dependent differencing methods, and the stability of these is discussed through the analysis of model problems. An implicit solution to the nonlinear dynamic programming problem is then developed and tested, with the motivation of reducing the computer time required to solve it. Finally, the extension of the finite‐difference method to a two‐state dynamic programming problem is considered.

Most people in the computing disciplines know of the famous difference engine designed by Charles Babbage, but many are unaware of several others that were produced after Babbage's efforts and even one mention of the concept that predates him. Difference methods were once of fundamental importance in the construction of tables but they have fallen into disuse since the invention of the digital computer. Indeed, they were already of lesser importance during the era of the massive electro-mechanical machines developed by, among others, Bell Laboratories and Harvard University in the 1940s. Because of their relative obscurity, this chapter explains these methods and then examines the machines that used them. These include Müller's machine, Babbage machine, Scheutz machines, Wiberg machine, and Grant's difference engine.

This chapter describes a numerical method for solving equations of thermal convection on a computer. It begins by introducing the vorticity-streamfunction formulation as a means of conserving mass. The approach involves updating for the vorticity first and then solving for the fluid velocity each time step. The chapter continues with a discussion of two very different spatial discretizations, whereby the vertical derivatives are approximated with a finite-difference method and the horizontal derivatives with a spectral method. The nonlinear terms are computed in spectral space. The chapter also considers the Adams-Bashforth time integration scheme and explains how the Poisson equation can be solved at each time step for the updated streamfunction given the updated vorticity.

Shrublands pose particularly difficult challenges for primary production measurement because of the complex structure of the vegetation. Classical approaches are described, including dimension analysis and the difference method both of which rely on destructive sampling and allometric relationships. A new spatially-explicit sampling approach and other procedures for optimizing sampling intensity are explained. Additional methodological considerations, including leaf habit and herbivory, also must be carefully evaluated to obtain reliable primary production estimates in shrubland ecosystems.

This chapter presents a detailed development of several numerical methods for calculating the band structure of semiconductor quantum wells and superlattices. These include the transfer-matrix method, the finite-difference method, and the reciprocal-space approach. The relative merits and drawbacks of each approach are briefly considered. It is pointed out that real-space methods often introduce spurious states for the most common forms of the Hamiltonian. The chapter also discusses how the tight-binding and pseudopotential methods can be applied to model quantum structures.

This chapter introduces non-Monte Carlo methods of MBS valuation including closed-form solution of the OAS equation. Authors consider different specifications of prepayment function and illustrate how the solution can be designed. The Active-Passive Decomposition model is introduced in a backward inducting scheme over a probability tree, or a finite difference grid. Finite difference methods in single- and multi-dimensional spaces of factors are described. The chapter discusses the issue of modeling mortgage market (“current-coupon”) rate. It compares statistical methods (an empirical function of other reference rates) to those connecting to the OAS valuation principle. Finally, it introduces valuation with a universal refinancing S-curve defined as a function of loan price rather than interest rates.

Chapter 10 introduces the main idea of the account of the function of modal thinking defended in the book. Modal concepts originate in the practice of supporting claims about causal and other explanatory relationships by counterfactual reasoning, i.e. by showing that the explanandum depends counterfactually on the alleged explainer. This routine is an extension of a form of causal reasoning that John Stuart Mill called the method of difference, which is widely used both in ordinary life and in scientific experiments. Everyday applications of this procedure can be reconstructed as resting on what may be called the “determination idea.” Applied to deterministic causation, this amounts to the thesis that the causes nomically determine their effect. The use of counterfactual reasoning to establish explanatory claims. In applying this method, we can often draw on the same knowledge and capacities that we employ to make predictions.

The main purpose of this chapter is to show how Ptolemy combined all three approaches discussed in the previous chapter in order to create a theory of sight that accounts broadly not only for visual perception per se but also for visual misperception: i.e., visual illusions, which Ptolemy attributed to improper physical conditions, interference with the normal functioning of the visual system, or psychological deception. Two types of illusion—image-displacement and deformation according to reflection and refraction—were of paramount concern to Ptolemy, whose analysis of reflection and refraction in books 3-5 of the Optics remained more or less canonical until the seventeenth century. In fact, as claimed at the end of the chapter, Ptolemy set the agenda for optics as it developed until the early modern period, much as his Almagest set the agenda for astronomy.

This chapter offers a peek at the vast literature on numerical methods for partial differential equations. The focus is on finite difference methods (FDM): approximating differential operators by functions of difference operators. Padé approximants (Fornberg) give a unifying principle for deriving the various stencils used by numericists. Boundary value problems for the Poisson equation and initial value problems for the diffusion equation are solved using FDM. Numerical instability of explicit schemes are explained physically and implicit schemes introduced. A discrete version of theClebsch formulation of incompressible Euler equations is proposed. The chapter concludes with the radial basis function method and its application to a discrete version of the Lagrangian formulation of Navier–Stokes.

Chapter 4 describes the mix of qualitative comparative methods used to analyse the case study data. First, a diachronic, most similar systems’ design is outlined in the form of a quasi-experimental set-up. The UK is analysed longitudinally—albeit over a relatively short period of time—to assess the impact of the devolution measures introduced in 1998–9 (Experiment I) and of the change of government in Edinburgh in May 2007 (Experiment II), while preference intensity configurations are alternatively modified across policy areas. Second, a synchronic, most similar systems’ design is outlined in the form of comparative statics where, in a series of ‘conceptual experiments’ (Lake and Powell, 1999), the impact of devolution, preference intensity, and party political configurations is formally assessed. The case studies selected to cover all combinations of values on the independent variables are Scotland, Salzburg, Rhône-Alpes, and Alsace.