This appendix contains additional materialcomplementingthe stability theory presented in the main body of the book.

Next, the general definitions of consistency, convergence, and stability are introduced in terms of an arbitrary norm, leading to the Lax equivalence theorem that consistent and stable schemes must converge to the true solution in the limit as the mesh interval and time step are reduced toward zero. Then stability in the Euclidean norm is examined, and the von Neumann stability test is introduced as a convenient way to deduce the stability of any linear scheme.

Nonlinear conservation laws generally admit solutions containing discontinuities, such as shock waves in a fluid flow. This motivates the need for difference schemes in conservation form, andalternative measures of stability are needed, leading to the introduction of concepts such as total variation diminishing (TVD) and local extremum diminishing (LED) schemes.

In this chapter,issues related to the calculation of viscous flows are addressed. The flows of interest in aeronautical science are generally characterized by high Reynolds numbers, of the order of 5–100 million in the case of flows over aircraft wings, and accordingly the emphasis here will be on such a range.

This chapter surveys some of the principal developments of computational aerodynamics, with a focus on aeronautical applications. It is written with the perspective that computational mathematics is a natural extension of classical methods of applied mathematics, which has enabled the treatment of more complex, in particular nonlinear, mathematical models, and also the calculation of solutions in very complex geometric domains, not amenable to classical techniques such as the separation of variables.

This chapter introduces the prevailing strategies that allow the development of high resolution schemes on unstructured meshes. These includes discontinuous Galerkin, spectral differences, and flux reconstructiondiscretizations.

This chapter reviews the formulation and application of optimization techniques based on control theory for aerodynamic shape design in both inviscid and viscous compressible flow. The theory is applied to a system defined by the partial differential equations of the flow, with the boundary shape acting as the control. The Frechet derivative of the cost function is determined via the solution of an adjoint partial differential equation, and the boundary shape is then modified in a direction of descent. This process is repeated until an optimal solution is approached. Each design cycle requires the numerical solution of both the flow and the adjoint equations, leading to a computational cost roughly equal to the cost of two flow solutions. Representative results are presented for viscous optimization of transonic wing–body combinations.

Higher order methodsmay be needed for the solution of multiscale flow problems. This chapter introduces the building blocks for higher order discretization, including compact finite differences and other methods suitable for building high resolution schemes on structured grids.

This chapter extendsshock capturing to systems of conservation laws.

The use of energy estimates to analyze the stability of nonlinear conservation laws is explored in this chapter.

Herethe author first discussesthe formulation of nonoscillatory schemes for scalar conservation laws in one or more space dimensions, and illustrates the construction of schemes that yield second order accuracy in the bulk of the flow but are locally limited to first order accuracy at extrema. In the following chapters, hediscussesthe formulation of finite volume schemes for systems of equations such as the Euler equations of gas dynamics, and analyzesthe construction of interface flux formulas with favorable properties such as sharp resolution of discontinuities and assurance of positivity of the pressure and density. The combination of these two ingredients leads to a variety of schemes that have proved successful in practice.

In this chapter, the author examines the properties of some general classes of time integration methods, derived from the well establishedtheory of integration for ODEs.

Industrial applications require both the development of techniques to generate appropriate computational meshes and the development of discretization schemes compatible with whatever type of mesh is chosen. The principal alternatives are Cartesian meshes, body-fitted curvilinear meshes, and unstructured tetrahedral meshes. Each of these approaches has some advantages that have led to their use. This chapter addresses the development of methods suitable for topologically complex domains.

This chapter addresses issues arising in the time accurate simulation of unsteady flows. In order to enable accurate simulations of time dependent flows with moving shocks and contact discontinuities, there is a need for higher order accurate time discretization schemes that can preserve the TVD property. Additionally, time dependent calculations are needed for a number of important applications, such as flutter analysis or the analysis of the flow past a helicopter rotor, in which the stability limit of an explicit scheme forces the use of much smaller time steps than would be needed for an accurate simulation. This motivates the “dual time stepping” scheme, in which a multigrid explicit scheme can be used in an inner iteration to solve the equations of a fully implicit time stepping scheme. Such schemes are developed in this chapter.

Ship-shaped offshore installations that are operated in shallow water (e.g., at depths of 10 m deep or less) are used for various purposes, such as oil terminals, floating storage and regasification units (FSRUs), power plants and bunkering. These usually remain afloat in operation, with a gap between the seabed and the bottom of the hull. In other situations, such as those for ship-shaped offshore power plant facilities containing nuclear reactors, hull bottoms are touched down onto the seabed by using heavy ballasting materials, such as concrete or sand. However, offshore installations are not fixed to the seabed and move under the effects of environmental actions, but may be moored.