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A spatially second-order or higher method is typically employed in practical simulations of FVM. To achieve at least second-order accuracy, the cell-internal distribution of variables should be reconstructed. There are various reconstruction methods such as MUSCL, (Weighted) Least Squares, Green-Gauss, and hybridizations of Green-Gauss/Least Squares, … but which is the best way? The answer depends on the employed numerical setup such as cell geometries and data structure (e.g., whether a structured or unstructured grid is used). In addition to this, a slope limiter such as the minmod limiter or Venkatakrishnan limiter is required to prevent spurious oscillations at discontinuities. This chapter will summarize those reconstruction methods and slope limiters, as well as our recent proposals, including the second limiter and the post limiter.

This chapter will describe the extensions of the AUSM-family fluxes (specifically, SLAU2) to real fluids of multiphase flows, supercritical fluids, and magnetohydrodynamics (MHD), where the governing equations and/or their discretizations differ from those for the perfect gas. These fluids are of great importance to physics and industries, but call for special care due to the changes in the equations. Some readers may wonder why the author appears to have a preference for the AUSM family. Here is a list of primary reasons: 1. They are robust and accurate for resolving shock waves at high speeds. 2. All-speed variants (e.g., AUSM+-up, SLAU2) are available that are applicable to low speeds. 3. No differentiation of a flux function or its Eigenstructure is required, which allows its straightforward application to the complex equation-of-state (EoS) of multiphase flows, supercritical fluids, or MHD.

There are many flux functions available today to obtain the inviscid flux through the cell interface. The idea of treating the cell interface was first proposed by Godunov, who regarded the variable jump there as a Riemann problem. Later, many Godunov-type methods were invented, such as an approximate Riemann solver by Roe, flux vector splitting (FVS) by Steger and Warming or van Leer, its simplified version AUSM (Advection Upstream Splitting Method), and the further modified SLAU (Simple Low-dissipation AUSM) or SLAU2. The Roe flux successfully avoided the iterations required in Godunov’s exact Riemann solver with a reasonable approximation of discontinuities with the “Roe average.” The FVS’s feature is simplicity: it splits waves traveling in the positive (right) and negative (left) directions. Liou further simplified the FVS by treating the mass flux (= convective field) and pressure flux (= acoustic field) separately in AUSM. SLAU is one of its variants extended to low speeds, and SLAU2 greatly enhanced its robustness at high speeds. There is also an HLL (Harten–Lax–van Leer) family which is also popular partly due to its conceptual clarity. In addition, combinations of these methods are, of course, possible. In the following, the flux functions will be explained in two-dimensional forms, unless otherwise stated.

In spite of the (assumed) maturity of compressible CFD technologies today, simulations of a hypersonic flow and a low-speed flow (nearly incompressible flow) are still challenging. The former suffers from anomalous solutions arising from shock capturing (e.g., the carbuncle phenomena). This can be mitigated at least partly by carefully selecting a numerical flux function, a slope limiter, and a computational grid. However, there is no universal scheme that is completely free of those shock anomalies. The latter, the low-speed flow problem, is divided into two subproblems: very slow convergence, known as the “stiffness problem,” and a physically incorrect solution due to huge numerical dissipation. Fortunately, these are known to be avoided if an appropriate combination is adopted for a time integration method and a numerical flux function. In this chapter‚ the current status of these problems is introduced with some actual numerical examples and the author’s explanations based on his experience. The detailed formulations of the numerical algorithms are deferred to the next chapter or simply referred to external sources.

Computational fluid dynamics (CFD) is sometimes regarded as a mature technology. In the author’s opinion, this is only partly correct. The fundamental ideas of CFD such as the finite volume method (FVM) are indeed well established, whereas some classes of problems such as hypersonic heating or real fluid simulations are still challenging, as will be discussed in the subsequent chapters. Thus, we begin this book by reviewing those basics in this chapter, specifically by describing the governing equations (i.e., the Navier–Stokes equations, or equations of mass, momentum, and energy conservation) for compressible flows. They are discretized, for computational purposes, by the FVM, which is the conventional procedure. Through this approximation, the equations are closed along with the equation of state and are valid for every computational element (=cell). In this chapter, only a brief explanation is given of how those equations are numerically solved in the framework of the FVM, using methods such as space reconstruction, slope limiting, flux computation, and time integration. Interested readers, including who have just entered the CFD world, are encouraged to refer to the standard books mentioned in the Preface.