I Mathematical Theory- 1 Introduction- 11 Conservation laws- 12 Applications- 13 Mathematical difficulties- 14 Numerical difficulties- 15 Some references- 2 The Derivation of Conservation Laws- 21 Integral and differential forms- 22 Scalar equations- 23 Diffusion- 3 Scalar Conservation Laws- 31 The linear advection equation- 311 Domain of dependence- 312 Nonsmooth data- 32 Burgers' equation- 33 Shock formation- 34 Weak solutions- 35 The Riemann Problem- 36 Shock speed- 37 Manipulating conservation laws- 38 Entropy conditions- 381 Entropy functions- 4 Some Scalar Examples- 41 Traffic flow- 411 Characteristics and "sound speed"- 42 Two phase flow- 5 Some Nonlinear Systems- 51 The Euler equations- 511 Ideal gas- 512 Entropy- 52 Isentropic flow- 53 Isothermal flow- 54 The shallow water equations- 6 Linear Hyperbolic Systems 58- 61 Characteristic variables- 62 Simple waves- 63 The wave equation- 64 Linearization of nonlinear systems- 641 Sound waves- 65 The Riemann Problem- 651 The phase plane- 7 Shocks and the Hugoniot Locus- 71 The Hugoniot locus- 72 Solution of the Riemann problem- 721 Riemann problems with no solution- 73 Genuine nonlinearity- 74 The Lax entropy condition- 75 Linear degeneracy- 76 The Riemann problem- 8 Rarefaction Waves and Integral Curves- 81 Integral curves- 82 Rarefaction waves- 83 General solution of the Riemann problem- 84 Shock collisions- 9 The Riemann problem for the Euler equations- 91 Contact discontinuities- 92 Solution to the Riemann problem- II Numerical Methods- 10 Numerical Methods for Linear Equations- 101 The global error and convergence- 102 Norms- 103 Local truncation error- 104 Stability- 105 The Lax Equivalence Theorem- 106 The CFL condition- 107 Upwind methods- 11 Computing Discontinuous Solutions- 111 Modified equations- 1111 First order methods and diffusion- 1112 Second order methods and dispersion- 112 Accuracy- 12 Conservative Methods for Nonlinear Problems- 121 Conservative methods- 122 Consistency- 123 Discrete conservation- 124 The Lax-Wendroff Theorem- 125 The entropy condition- 13 Godunov's Method- 131 The Courant-Isaacson-Rees method- 132 Godunov's method- 133 Linear systems- 134 The entropy condition- 135 Scalar conservation laws- 14 Approximate Riemann Solvers- 141 General theory- 1411 The entropy condition- 1412 Modified conservation laws- 142 Roe's approximate Riemann solver- 1421 The numerical flux function for Roe's solver- 1422 A sonic entropy fix- 1423 The scalar case- 1424 A Roe matrix for isothermal flow- 15 Nonlinear Stability- 151 Convergence notions- 152 Compactness- 153 Total variation stability- 154 Total variation diminishing methods- 155 Monotonicity preserving methods- 156 l1-contracting numerical methods- 157 Monotone methods- 16 High Resolution Methods- 161 Artificial Viscosity- 162 Flux-limiter methods- 1621 Linear systems- 163 Slope-limiter methods- 1631 Linear Systems- 1632 Nonlinear scalar equations- 1633 Nonlinear Systems- 17 Semi-discrete Methods- 171 Evolution equations for the cell averages- 172 Spatial accuracy- 173 Reconstruction by primitive functions- 174 ENO schemes- 18 Multidimensional Problems- 181 Semi-discrete methods- 182 Splitting methods- 183 TVD Methods- 184 Multidimensional approaches

Numerical methods for conservation laws

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Inertial confinement fusion (ICF) is an approach to fusion that relies on the inertia of the fuel mass to provide confinement. To achieve conditions under which inertial confinement is sufficient for efficient thermonuclear burn, a capsule (generally a spherical shell) containing thermonuclear fuel is compressed in an implosion process to conditions of high density and temperature. ICF capsules rely on either electron conduction (direct drive) or x rays (indirect drive) for energy transport to drive an implosion. In direct drive, the laser beams (or charged particle beams) are aimed directly at a target. The laser energy is transferred to electrons by means of inverse bremsstrahlung or a variety of plasma collective processes. In indirect drive, the driver energy (from laser beams or ion beams) is first absorbed in a high‐Z enclosure (a hohlraum), which surrounds the capsule. The material heated by the driver emits x rays, which drive the capsule implosion. For optimally designed targets, 70%–80% of the d...

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Development of the indirect‐drive approach to inertial confinement fusion and the target physics basis for ignition and gain

This paper calls attention to some of the most important and easily recognizable forms of surface wave, pointing out that their essential common characteristic is the evanescent field structure over the wave front, as represented by equiphase planes. The problems of launching and supporting surface waves must, in general, be distinguished from one another and it does not necessarily follow that because a particular surface is capable of supporting a surface wave that a given aperture distribution of radiation, e.g. a vertical dipole, can excite such a wave. The paper concludes with a discussion of the behavior of surface waves and their applications.

http://ieeexplore.ieee.org/iel5/10933/4065481/04065500.pdf

Surface Waves

We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman's path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families. This review was completed in February 1972.

https://iopscience.iop.org/article/10.1088/0034-4885/35/1/306/pdf

Semiclassical approximations in wave mechanics

A spherical perfectly catalytic conductor with negative potential bias lies in a quiescent slightly ionized gas whose density is large enough to warrant the use of continuum theory. The Debye number e, the ratio of the Debye length to the conductor radius, is small. Principal attention is given to the case, amenable to asymptotic analysis, of probe potential ψp > 0(ln e). The relation between the current collected by the probe and the electrode potential is sought; the current virtually saturates (increases very slowly relative to increases in applied potential difference) for ψp ≫ 0(ln e). A reconstruction of the chemically frozen case with three novel features is furnished: a transformation of variables that renders the transition‐zone equations almost uniformly valid; a very accurate, easily obtained closed‐form approximation to the solution of the resulting boundary‐value problem; and a current‐potential relationship that is derived on the basis of the gross behavior of this approximate closed‐form so...

Electrostatic probe in a reacting gas.

Applied Mathematics what is Needed in Research and Education

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The thermal-stress and body-force problems of the infinite orthotropic solid

The uniform steady flow of an incompressible, viscous, electrically conducting fluid is distorted by the presence of a symmetrically oriented semi-infinite flat plate. The ambient magnetic field is coincident with the ambient velocity field. The description of the resulting fields depends on the physical co-ordinates measured in units of Reynolds number and on the two parameters e = ωμν and β = μ H 2 /ρ v 2 . This description of the fields is approximated in three different ways and essentially covers the full range of e and β. In particular, when β [Gt ] 1, no steady flow which is uniform at large distances from the plate exists.

Magnetohydrodynamic Flow Past a Flat Plate

Steady laminar isobaric two-dimensional propagation of a fuel-lean large-activation-temperature premixed flame between plane parallel (isothermal noncatalytic) walls is examined theoretically. A Shvab-Zeldovich engineering-type formulation, with a treatment of convective transport of species and heat in the manner of Oseen, is integrated numerically by quasilinearization/alternating-direction-implicit techniques. Of particular interest in this transcendentally nonlinear elliptical (convective-diffusive-reactive) boundary-value problem is how the flame speed (an eigenvalue) decreases as the wall separation decreases, for different wall temperatures. At a finite wall separation the quenching distance is achieved, such that the flame speed goes abruptly to zero; this critical quenching distance is found to be smaller for hotter walls. This investigation seeks to lend credibility of principle to one possible strategy for reducing the level of unburned-hydrocarbon species exhausted from Otto-cycle-type cylinders. The primary source for hydrocarbon emissions is now generally acknowledged to be the failure of flame to propagate into the narrow piston crevice owing to heat loss to cold walls. Higher wall temperature at the piston crown and rings would seem to abet flame propagation and assist burn-up during the combustion event; also, persistent fuel vapor would be discharged at higher temperature into the hot, oxygen-rich bulk cylinder contents during the powerstroke expansion.

George F. Carrier

Papers

Electrostatic probe in a reacting gas.

Applied Mathematics what is Needed in Research and Education

The thermal-stress and body-force problems of the infinite orthotropic solid

Magnetohydrodynamic Flow Past a Flat Plate

Laminar flame propagation/quench for a parallel-wall duct