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Publisher Summary This chapter provides an overview of shock waves and entropy. It describes systems of the first order partial differential equations in conservation form: ∂ t U + ∂ X F = 0, F = F(u). In many cases, all smooth solutions of the first order partial differential equations in conservation form satisfy an additional conservation law where U is a convex function of u. The chapter discusses that for all weak solutions of ∂ t u j +∂ x f j = 0, j=1,…, m, f j =f j (u 1 ,…, u m ), which are limits of solutions of modifications ∂ t u j +∂ x f j = 0, j=1,…, m, f j =f j (u 1 ,…, u m ) , by the introduction of various kinds of dissipation, satisfy the entropy inequality, that is, ∂ t U + ∂ x F≦ 0. The chapter also explains that for weak solutions, which contain discontinuities of moderate strength, ∂ t U + ∂ x F≦ 0 is equivalent to the usual shock condition involving the number of characteristics impinging on the shock. The chapter also describes all possible entropy conditions of ∂ t U + ∂ x F≦ 0 that can be associated to a given hyperbolic system of two conservation laws.

A critical analysis of well-known procedures for the computation of one-dimensional shocked flows is made, in order to show the inconveniences of computing finite differences across a discontinuity and to prove that the use of the equations of motion in conservation form does not make the results any more accurate. A technique is developed to treat one-dimensional inviscid problems and it is applied to the problem of an accelerating piston. Practical and safe ways to predict the formation of a shock and to follow it up in its evolution are given. (Author)

Purpose – The purpose of this paper is to apply the variational multi-scale framework to the finite element approximation of the compressible Navier-Stokes equations written in conservation form. Even though this formulation is relatively well known, some particular features that have been applied with great success in other flow problems are incorporated. Design/methodology/approach – The orthogonal subgrid scales, the non-linear tracking of these subscales, and their time evolution are applied. Moreover, a systematic way to design the matrix of algorithmic parameters from the perspective of a Fourier analysis is given, and the adjoint of the non-linear operator including the volumetric part of the convective term is defined. Because the subgrid stabilization method works in the streamline direction, an anisotropic shock capturing method that keeps the diffusion unaltered in the direction of the streamlines, but modifies the crosswind diffusion is implemented. The artificial shock capturing diffusivity is calculated by using the orthogonal projection onto the finite element space of the gradient of the solution, instead of the common residual definition. Temporal derivatives are integrated in an explicit fashion. Findings – Subsonic and supersonic numerical experiments show that including the orthogonal, dynamic, and the non-linear subscales improve the accuracy of the compressible formulation. The non-linearity introduced by the anisotropic shock capturing method has less effect in the convergence behavior to the steady state. Originality/value – A complete investigation of the stabilized formulation of the compressible problem is addressed.

We propose a modification for the tensor of artificial viscosity employable for generally comoving, curvilinear grids. We present a strong conservation form for the equations of radiation hydrodynamics for studying nonlinear pulsations of stars. However, the modification we propose is of general mathematical nature. We study a differential geometrically consistent artificial viscosity analytically and visualize a comparison of our approach to previous implementations by applying it to a simple self-similar velocity field which has a direct application in stars as the fundamental mode of pulsation is radial. We first give a general introduction to artificial viscosity and motivate its application in numerical computations. We then show how a tensor of artificial viscosity has to be designed when going beyond common static Eulerian or Lagrangian comoving rectangular grids. We derive and state the modified equations which include metrical terms that adjust the isotropic (pressure) part of the tensor of artificial viscosity.

The problem of simulating the scattering of a plane electromagnetic wave by a perfectly conducting obstacle in 3D is considered. The computational domain is represented by an unstructured assembly of linear tetrahedral elements and an edge based representation of the mesh is adopted. Maxwell's curl equations are expressed in conservation form and are discretized using a Galerkin finite element approximation in space and an explicit forward difference in time. The physical flux function is replaced by a numerical Lax-Wendroff flux function along each edge of the mesh. The numerical performance of the proposed approach is demonstrated by simulating scattering by a sphere and by a 3D cylindrical cavity. (Author)

An appropriate procedure to construct the symmetric conservative metrics is presented in the high-order conservative flux-reconstruction scheme. The present framework enables a direct discretization of the strong conservation form of governing equations without any errors in the freestream preservation and global conservation properties on threedimensionally moving and deforming grids. We show that a straightforward implementation of the symmetric conservative metrics often fails to construct metric polynomials with the same order of a solution polynomial, which severely degrades a numerical accuracy. On the other hand, the symmetric conservative metrics constructed by the appropriate procedure can preserve the freestream solution regardless of the order of shape functions; in addition, a convecting vortex is more accurately computed on deforming grids. The global conservation property is also demonstrated numerically for the convecting vortex on deforming grids.

We consider the Cauchy problem for systems of PDEs of the general form $$ u_t=P_0u + \eps_1P_1u +\eps_2 Q(u) + \sum_j D_j F_j(x,t), \quad u=u(x,t). $$ Here $P_0$ has constant coefficients. The terms $\eps_1 P_1u$ and $\eps_2 Q(u)$ describe linear and nonlinear perturbations, respectively, and $\textstyle\sum_j D_j F_j(x,t)$ is a forcing term, which decays to zero for $t\to \infty$. The perturbation terms are assumed to have conservation form. We call the system nonlinearly stable if the solution $u(x,t)$ with $u(x,0)=0$ remains smooth for all $t\geq 0$ and the maximum norm of u tends to zero for $t\to \infty$, provided that $\eps_1^2+\eps_2^2$ is sufficiently small. In the paper we give sufficient conditions for nonlinear stability. If the unperturbed system $u_t=P_0u$ is parabolic, then the Laplace transform technique is satisfactory to derive conditions for nonlinear stability. However, if $u_t=P_0u$ is hyperbolic or coupled parabolic-hyperbolic, then the Laplace transform technique fails if the perturb...

Partially ionized plasmas are ubiquitous in both nature and the laboratory, and their behaviour is best described by models which take into account the interactions between the neutral and charged species. We present a new non-linear, 3-dimensional, finite difference Gas-MHD Interactions Code designed to solve simultaneously the time evolution of fluid equations of both species in the conservation form as well as collisional interactions between them via appropriate choices of source term; in particular, we present results from this code in simulating Alfven ionization in a partially ionized plasma. In this fashion, larger changes in the ionization fraction than were addressable in the linear limit are possible. Alfven ionization is shown to impart plasmas with an inherent resistance to rapid recombination, where the recombination itself is significant enough to drive relative motion between the ionised and neutral species at speeds in excess of the critical velocity.

Many of the differential equations arising in fluid dynamics may be stated in conservation-law form. A number of investigations have been conducted with the aim to derive the conservation-law form of the Navier-Stokes equations in general nonsteady coordinate systems. The present note has the objective to illustrate a mathematical methodology with which such forms of the equations may be derived in an easier and more general fashion. For numerical applications, the scalar form of the equations is eventually provided. Attention is given to the conservation form of equations in curvilinear coordinates and numerical considerations. 6 references.

THE transonic full potential equation in strong conservation form is solved on a body-fitted coordinate system using a strongly implicit procedure enhanced by the multigrid procedure. Weak viscous effects are accounted for by computing the displacement thickness using the NashMcDonald integral boundary-layer equation. The convergence speed of the multigrid procedure is found to be comparable to the state-of-the-art multigrid-alternating direction implicit procedures on identical grids. Several numerical cases, some involving weak viscous effects, are presented and compared with published results. Contents Recently the author and his co-workers developed a relaxation procedure for the solution of the steady transonic full potential equation in a body-fitted coordinate system.1 This procedure has the following steps. The governing equation is linearized at each step of the iteration by lagging the density by one iteration. The density is biased in supersonic regions using the artificial compressibility concept.2'3 Standard central differences are used to discretize the governing equation and the transformation metrics. The above steps lead to the following system of equations

A new formalism that brings out the wave character of flame propagation of one-dimensional flames has been explicitly presented for simple as well as complex kinetics and realistic diffusion. While the combustion literature has accepted the idea of a combustion wave, this is the first time that the equations expressing this feature mathematically have been presented. The governing equations are shown to have a strong conservation form suitable for numerical computation. They are solved for a single-step reaction using the two-step MacCormack explicit scheme and are shown to be very fast compared to the unsteady computational technique. Compared to the unsteady technique, application of the same numerical scheme to complex kinetics seems to be numerically less efficient.

A computer code has been developed to analyze the inviscid flow field in a supersonic combustion ramjet (scramjet) inlet. The code uses the three-dimensional Euler equations in full conservation form to describe the inlet flow. An algebraic numerical coordinate transformation is used to generate a set of boundary-fitted curvilinear coordinates. The governing equations are solved by a time-asymptotic, unsplit, two-step, finite-difference method. This method is highly efficient on the vector processing computers for which the current code is written. Detailed results are presented for two scramjet inlet configurations over a range of Mach numbers. The calculated results are compared with the available experimental and theoretical results.

In this paper a set of approximate equations is derived which is applicable to very nonadiabatic, nondissipative, buoyant flows of a perfect gas. The flows are assumed to be generated by a heat source in which the heat is added slowly. The study is motivated by the occurrence of such flows in fires. There, the time scale associated with the fire growth and resultant fluid motion is usually long compared with the transit time of an acoustic signal (based on the temperature derived from the heat added) across the spatial extent of the fire. The approximate equations are characterized by a spatially uniform mean pressure appearing in both the energy equation and the equation of state with the spatially nonuniform portion of the pressure only appearing in the momentum equation. Therefore, the pressure remains almost constant in space while significant density and temperature variations, such as might occur in a fire, are allowed. The approximate equations are shown to reduce to the Boussinesq equations when the heat addition is mild. These equations are also shown in general to admit internal-wave motions while “filtering out” high-frequency, acoustic waves. In addition, they are shown to be expressible in conservation form, the pressure satisfying an elliptic equation whose homogeneous terms are derivable from the wave equation by letting the sound speed become infinite. An equation for the mean pressure is also obtained. For the special case of a room heated at a uniform rate with a small leak to the outside, an approximate solution for the mean pressure is determined explicitly.

This paper presents the underlying symbolic manipulation techniques used by the authors to produce FORTRAN code with the artificial intelligence code MACSYMA. The FORTRAN codes produced use finite difference methods to solve the governing equations for numerical grid generation, including elliptic and variational methods, and to solve the hosted equations, including fluid dynamics, heat transfer, and electro‐statics. Consideration will be given to writing the hosted equations in strong conservation form in general nonorthogonal coordinates; high order and conditional differencing schemes; colocated variables versus staggered grids; and code validation procedures.