The design of time-integration methods for steady state problems is discussed in this chapter, including a variety of methods for accelerating convergence to a steady state. A variety of convergence acceleration techniques are discussed, including ways in which the time integration process itself may be optimized for steady state calculations. In this case, the accuracy of the time integration scheme is no longer a consideration. This enables the use of modified RK schemes of reduced computational complexity. Moreover, the schemes may be tailored to increase the allowable time step, thereby promoting more rapid convergence to a steady state. Moreover, they can be tailored to drive multigrid acceleration.

The turbulence models most widely used in aerodynamics are outlined in this appendix.

The equations governing the motion of a fluid modeled as a continuum are developed in integral and differential forms. The physical phenomena at play are briefly discussed, and the mathematical structure of the resulting conservation laws is analyzed in detail.

Basic concepts of polynomial interpolation that are foundational for all numerical analysis are reviewed in this appendix.

The well known concepts of Euclidean space can be generalized to n-dimensional vectors and also to functions. These concepts are extremely useful in the analysis of numerical methods and are briefly reviewed in this appendix.

Successful numerical methods for the solution of the transonic potential equation are presented in this appendix for both historical and practical reasons as they are still widely used for aerodynamic analysis in the early stages of a new design.

The basic elements of approximation theory are reviewed in this appendix.

The numerical methods that have been widely used for the solution of partial differential equations (PDEs), both in fluid dynamics and in other disciplines, fall into three main branches: finite difference methods, finite element methods, and finite volume methods. These methides are reviewed in this chapter together with basic theory of spectral methods.

This chapter examines the stability of difference schemes for initial value problems defined by ordinary or partial differential equations. Three simple examples are examined first: an ordinary differential equation, the linear advection equation, which is the prototype for hyperbolic equations, and the diffusion equations.

This appendix contains additional material complementing the 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, and alternative 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 reconstruction discretizations.

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 methods may 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.