Introduction

From previous discussions, it is apparent that it is possible to solve sets of differential equations numerically for practical parameter ranges, by solving PDEs with stringent requirements of resolving wide ranges of spatial and temporal scales. In the literature, one comes across a number of efforts which solve Navier–Stokes equation for turbulent flows at moderate Re. Also, large RAM machines are now available at desk-top, allowing one to solve problems with large number of grid points. This encourages search for faster methods; faster than the methods discussed in previous chapters. Higher accuracy methods are expected to fulfill this, as these allow solving the same problem with fewer grid points. Conversely, same methods allow solving problems at higher Reynolds number with the same grid. This chapter deals with developments and analyses of higher accuracy methods.

Effectiveness of CFD schemes is generally ranked by the leading order term of the Taylor series expansion used to define the truncation error of any scheme. It is expected that higher the order, better will be the quality of the scheme. While this may appear as a correct expectation, discussions in Chapters 8 and 10 will convince the readers that classification of schemes simply based on order of truncation error is incomplete. In Chapter 10, schemes were assessed based on their behaviour in the spectral plane. The Fourier spectral method provides maximum accuracy and it is seen that higher the order of explicit schemes, better is the representation.

Introduction

We have discussed classical developments, as well as, recent higher accuracy computing methods so far. We have compared various discrete computing methods with the same framework of spectral analysis. In this chapter, we include topics of more recent origin, which have bearing with the central theme of high accuracy computing of flows and wave phenomena.

In Sections 4.7–4.10, we have discussed about length- and time-scales excited in flows from the perspective of waves as building blocks. While future computational activities would be predominantly for DNS, there are many issues to be sorted out. For DNS, error must be the major concern. In Chapters 8–11, we have identified various sources of error with the help of developed error dynamics in Eqn. (8.31) for convection dominated problems. These sources of error are due to numerical stability, phase and dispersion errors. In subsequent chapters, we have added to this list by considering spurious upstream propagating waves (q-waves), the Gibbs' phenomenon, aliasing error etc. We also discuss about better time discretization methods developed with DRP analysis of Chapter 8. This helps solving space–time dependent problems by optimized time integration method which reduces error. This has been presented for high accuracy two-time level, multi-stage Runge–Kutta methods.

Introduction

A beginner in the field of computing is hardly aware of the importance of formulations responsible for the accuracy in scientific computing. In theoretical fluid mechanics it is not important as to which form of Navier–Stokes equation is solved. But, in computing this is paramount, and there is always ongoing debate among practitioners about the superiority of different formulations and numerical methods employed by different schools of thought.

Governing differential equations are obtained by considering a control volume and balancing fluxes of quantities of interest, obtained in the limit of vanishing size of the control volume. This provides a point-by-point description of intrinsic properties of interest. In many methods of computing, this point description of conservation principle is integrated over a finite control volume. Obtaining governing differential equation is described in this way here, for the conservation of mass, momentum and energy. This is followed by discussion on desirability of casting differential equations in conservation form, which is found to be impervious to details of discretization to a great extent, as compared to non-conservation form.

Here, we also note that often one requires to investigate the problem by formulating the governing equation in non-inertial frame. Readers will have no difficulty in appreciating the need for it in weather forecasting. It is also appropriate and convenient for many engineering flows, where one part of the body is in relative motion with respect to other parts, as in problems of aeroelasticity. This is also needed for rigid bodies of arbitrary shape executing time-dependent motion.

Introduction

Initial developments in computing were dominated by two classes of problems: (i) the jury or the boundary value problems – typically classified as elliptic partial differential equation in Chapter 3; (ii) the evolution or the initial–boundary value problems which are represented by parabolic and hyperbolic partial differential equations. In fact, the solution methods for heat equation (a parabolic partial differential equation) were central to the early development of the subject. These classical approaches are discussed in this chapter, with additional insight brought through spectral analysis of the schemes. It is noted that the stability analysis of numerical schemes was developed with respect to heat equation by von Neumann, as described in [41, 53]. This was considered a major milestone in the development of the subject. But, the readers' attention is also drawn to the correct analysis advanced recently, as described in [259] and Chapter 8, with respect to 1D convection equation.

In fluid dynamics, a major milestone was the introduction of boundary layer concept by Ludwig Prandtl in 1904, which dominated fluid dynamics studies. Readers are referred to [209] for details of the development. Boundary layer equation is an example of parabolic partial differential equation.

There are many new developments in scientific computing, in its application to fluid flows and wave phenomena, which warrant their consolidation in a single source, covering some of the key developments. I have been convinced by many students and peers that there is a definitive need for a single source book which deals with topics covered here. I would like to acknowledge their inspiration. My main motivation in writing this manuscript is to communicate something new and powerful as opposed to conventional derivatives of products churned out by existing schools of thought.

However, this book also provides general introduction to computational fluid dynamics (CFD), using well tested classical methods of solving partial differential equations (PDEs) for the sake of completeness. These are to be found in Chaps. 1 to 9 and 13, but re-interpreted using the spectral analysis method introduced in Chaps. 4, 8 and 10. This provides an unity of approach in understanding numerical methods for parabolic, elliptic and hyperbolic PDEs. The spectral analysis tool has been refined in recent years by the author's group, with which disparate methods can be easily compared.

Introduction

Waves are usually associated with hyperbolic partial differential equations. What waves have to do in general with CFD? It is noted earlier that many problems governed by parabolic or elliptic PDEs require numerical algorithms which treat the problems as belonging to hyperbolic system. It is explained later that the numerical characteristics of the equivalent hyperbolic system must include the single characteristic of a parabolic system for the method to be stable and accurate. Similarly, solving an elliptic PDE by iterative methods is equivalent to solving an equivalent time-evolution problem. In fact, it is in this context the eigenvalues and eigenvectors of associated linear algebraic system provide physical justification. Thus, this chapter contains expository material in understanding wave-attributes of many problems which support waves directly as in hyperbolic system or any other system that supports dispersive waves.

Wave motions have the characteristic property that after a signal (information) is observed at one point, a closely related signal (information) may later be observed at a different point. Thus, waves are the means by which information travel in space and time, without significant movement of the medium. Quite often, what is perceived as motion is related to movement of phase and energy. These two motions are characterized by phase speed and group velocity which will be described shortly.