This chapter is concerned with the main topic of the monograph, namely, the solution of the GRP for quasi-1-D, inviscid, compressible, nonisentropic, time-dependent flow. In Section 5.1 we formulate the problem and study its solution in the Lagrangian and Eulerian frames. In particular, we state and prove the main ingredient in the GRP method, Theorem 5.7. A weaker form of this theorem leads to the “acoustic approximation” (Proposition 5.9). Summary 5.24 gives a step-by-step description of the GRP analysis. In Section 5.2 we present the GRP methodology for the construction of second-order, high-resolution finite-difference (or finite-volume) schemes. Starting out from the (first-order) Godunov scheme, we present the basic (E1) GRP scheme. It is based on the acoustic approximation and constitutes the simplest second-order extension of Godunov's scheme. This is followed by a presentation of the full array of GRP schemes (as well as MUSCL). Generally speaking, the presentation in this chapter follows closely the GRP papers [7] and [10].

The GRP for Quasi-1-D, Compressible, Inviscid Flow

In Section 4.2 we studied the Euler equations (4.45) governing the quasi-1-D flow in a duct of variable cross section. We emphasized in particular the role of the Riemann problem (“shock tube problem”), namely, the IVP subject to initial data (4.100). As we shall see in this chapter, the solution to the Riemann problem is a basic ingredient in the numerical resolution of the flow.

In Definition 2.15 we gave the most practical version of the entropy condition. It limits admissible shocks to those obtained by the intersection of “forward-moving” characteristics. These are therefore discontinuities that “cannot be avoided” or replaced by a rarefaction wave. In this Appendix we give some further insight into this concept of an “entropy satisfying” weak solution to (2.1), (2.2).

Our starting point is the physical notion of a “vanishing viscosity solution.” In general terms, an equation leading to discontinuous solutions [such as (2.1)] is supplemented by “dissipative terms” (also referred to as “viscous terms”). In analogy to the physical situation, such terms have a “smoothing effect” on solutions with large gradients, thus replacing discontinuities by “transition zones” where the solution varies smoothly, albeit rapidly. As the viscous effects are diminished, those transition zones shrink to surfaces of zero width, across which the solution has a sharp jump. Mathematically speaking, the additional viscous terms are often represented by second-order derivatives with a small (“vanishing”) coefficient.

To illustrate the situation, we consider the “moving step” problem for Burgers' equation (Example 2.12).

This chapter addresses one of the most central issues of computational fluid dynamics, namely, the simulation of flows under complex geometric settings. The diversity of these issues is briefly outlined in Section 8.1, which points out the role played by the present extensions: the (1-D) “singularity tracking” and the (2-D) “moving boundary tracking” (MBT) schemes. Section 8.2 deals with the first extension, and Section 8.3 is devoted to an outline of the second one. In the former we present the scheme methodology and refer to GRP papers for examples. In the latter, the basic principles of the method are presented, and we refer to [39] for more algorithmic details. Finally, an illustrative example of the MBT method shows how an oval disk is “kicked-off” by a shock wave.

Grids That Move in Time

In Part I of this monograph we dealt with finite-difference approximations to the quasi-1-D hydrodynamic conservation laws, where the underlying grid was fixed and equally spaced in the majority of cases. In our two-dimensional numerical extension (Section 7.3) we restricted the treatment to a Cartesian (rectangular) grid. Naturally, finite-difference approximations assume their simplest form on such grids, and the motivation for seeking geometric extensions comes primarily from physical applications.

Computational fluid dynamics (CFD) is a relatively young branch of fluid dynamics, the other two being the experimental and the theoretical disciplines. Its rapid development was enabled by the spectacular progress in high power computers, as well as by a matching progress in numerical schemes.

The starting point for the formulation of CFD schemes is the governing equations. In fact, the term “fluid dynamical equations” is much too general and indeed ambivalent. In practice there exist numerous models of such equations. They reflect a variety of stipulations on the nature of the flow, such as compressibility, viscosity, or elasticity. They also involve various effects such as heat conduction or chemical reactions. A large portion of these models do not fall, mathematically speaking, under the category of “hyperbolic conservation laws,” which is the subject matter of this monograph. We refer the reader to the book by Landau and Lifshitz [75] for a general survey of fluid dynamical models.

In this monograph we are concerned with time-dependent, inviscid, compressible flow, which is studied primarily in the “quasi-one-dimensional” geometric setting. This leads to a system of partial differential equations expressing the conservation of mass, momentum, and energy. There are various approaches to the numerical resolution of this system, such as the classical method of characteristics or the “artificial viscosity” scheme.

In this chapter we consider the system of equations governing compressible reacting flow. The fluid is a homogeneous mixture of two species. The evolution of the flow under the mechanical conservation laws of mass, momentum and energy is coupled to the (continuous or abrupt) conversion of the “unburnt” species to the “burnt” one. We take the simplest model of such a reaction, namely, an irreversible exothermic process. The equation of state of the fluid depends on its chemical composition. The resulting (augmented) system is still nonlinear hyperbolic (in the sense of Chapter 4) and is amenable to the GRP methodology. The basic hypotheses are presented in Section 9.1, leading to the derivation of the characteristic relations and jump conditions. In Section 9.2 we describe the classical Chapman—Jouguet model of deflagrations and detonations, and the Zeldovich—von Neumann—Döring (Z—N—D) solution is presented in Section 9.3. In Section 9.4 we study the generalized Riemann problem for the system of reacting flow. The treatment here is close to that of the basic GRP case (Section 5.1), but there are significant differences because of the reaction equation. In Section 9.5 we outline briefly the resulting GRP numerical scheme and study a physical problem of ozone decomposition.

Here the fluid dynamical theory and GRP schemes of Chapters 4 and 5 are applied to one-dimensional test cases. The problems are aimed primarily at demonstrating the capabilities of the scheme, but they are also revealing of nontrivial fluid dynamical phenomena that arise even at the relatively simple one-dimensional settings considered here. In Section 6.1 we treat a shock tube problem, using several scheme options to solve it. An interesting class of fluid dynamical problems is that of wave interactions, to which Section 6.2 is devoted. We selected four different cases in this class, shock—shock, shock—contact, shock—rarefaction, and rarefaction—contact interactions. In each case the GRP solution is compared to either an exact one or to a solution of a Riemann problem that approximates the exact one in some “asymptotic” sense. In the remainder of the chapter we employ the quasi-one-dimensional (“duct flow”) scheme, solving three different problems, comparing each numerical solution to the corresponding exact one. Section 6.3 treats a spherically converging flow of cold gas, and Section 6.4 is devoted to the flow induced by an expanding sphere. Finally, in Section 6.5 we present a detailed treatment of the steady flow in a converging—diverging nozzle, obtained numerically as a large-time solution by the GRP scheme.

The notation for the fluid dynamical variables here is identical to that of Chapters 4 and 5.

This chapter introduces the GRP method in the context of the scalar conservation law ut + f(u)x = 0. We start in Section 3.1 with the classical first-order (conservative) “Godunov Scheme,” which leads naturally to its second-order GRP extension. Section 3.2 contains a number of numerical (one-dimensional) examples, for linear and nonlinear equations, illustrating the improved resolution obtained by the GRP method. In Section 3.3 we extend the GRP methodology to the two-dimensional scalar conservation law ut + f(u)x + g(u)y = 0. Analytical and numerical results are compared for simple as well as complex wave interactions.

From Godunov to the GRP Method

In this section we discuss the GRP method, aimed at a high-resolution numerical approximation of the solution to a conservation law of the form (2.1). We always assume that f(u) is strictly convex:f″(u) ≥ μ > 0. It is shown that this method is a natural analytic extension to the Godunov (upwind) scheme. This latter scheme has been extensively studied in Section 2.2, in the context of the linear convection equation. We start here by studying this scheme in the nonlinear case.

As in Section 2.2, we take a uniform spatial grid xj = jΔx, − ∞ < j < ∞, and uniformly spaced time levels tn+1 = tn+k, t0 = 0.

This monograph deals with the generalized Riemann problem (GRP) of mathematical fluid dynamics and its application to computational fluid dynamics. It shows how the solution to this problem serves as a basic tool in the construction of a robust numerical scheme that can be successfully implemented in a wide variety of fluid dynamical topics. The flows covered by this exposition may be quite different in nature, yet they share some common features; they all belong to the class of compressible, inviscid, time-dependent flows. Fluid dynamical phenomena of this type often contain a number of smooth flow regions separated by singularities such as shock fronts, detonation waves, interfaces, and centered rarefaction waves. One must then address various computational issues related to this class of fluid dynamical problems, notably the “capturing” of discontinuities such as shock fronts, detonation waves, or interfaces; resolution of centered rarefaction waves where flow gradients are unbounded; and evaluation of flow variables in irregular computational cells at the intersection of a moving boundary surface with an underlying mesh.

From the mathematical point of view, the various systems of equations governing compressible, inviscid, time-dependent flow phenomena may all be characterized as systems of “(nonlinear) hyperbolic conservation laws.”

Hyperbolic conservation laws (in one space variable) are systems of time-dependent partial differential equations.

The GRP algorithm was developed in Part I and tested against a variety of analytical examples. The next challenge is to adapt it to the needs of “Scientific Computing,” that is, to implement it in the simulation of problems of physical significance. This is a formidable task for any numerical algorithm. The problems encountered in applications usually are multidimensional and involve complex geometries and additional physical phenomena. In this chapter we first describe in more detail some of these problems, as a general background for the following chapters. We then proceed to the main goal of the chapter, namely, the upgrading of the GRP algorithm (which is essentially one dimensional) to a two-dimensional, second-order scheme. To this end we recall (Section 7.2) Strang's method of “operator splitting” and then (Section 7.3) apply it to the (planar) fluid-dynamical case. Although the method can be further extended to the three-dimensional case, we confine our presentation to the two-dimensional setup, which serves in our numerical examples (Chapters 8 and 10).

General Discussion

In Part I we studied the mathematical basis of the GRP method. We also considered a variety of numerical examples for which analytic (or at least asymptotic) solutions were available. This provided us with some measure of the accuracy of the computational results, and helped identifiy potential difficulties (such as the resolution of contact discontinuities).