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 surface distribution of stress is imposed on an ocean enclosed by two continental boundaries; the resulting transport circulation is studied between two latitudes of zero surface wind-stress curl, within which the curl reaches a single maximum. Under the assumption that turbulent transfer of relative vorticity has a minimum effect on the mean circulation, inviscid flow patterns are deduced in the limit of small transport Rossby number. Inertial currents, or naturally scaled regions of high relative vorticity, occur on both the eastern and the western continental coasts. Limits on the relative transports of the currents are obtained and found to depend on the direction of variation of the wind-stress curl with latitude, relative to that of the Coriolis accelerations. The most striking feature of the inviscid flow is a narrow inertial current the axis of which lies along the latitude of maximum wind-stress curl. All eastward flow occurs in this midlatitude jet.A feature of the flow which cannot remain essentially free of turbulent processes is the integrated vorticity relationship, since the imposed wind-stress distribution acts as a net source of vorticity for the ocean. Heuristic arguments are used together with this integral constraint to deduce the presence and strength of the turbulent diffusion which must occur in the region of the mid-latitude jet. It is further inferred that the turbulent meanders of the jet must effect a net meridional transport of relative vorticity.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112062000038

On the theory of the wind-driven ocean circulation

The steady one-dimensional isobaric combustion of a gaseous premixture of fuel and oxidant under a direct one-step irreversible Arrhenius-type exothermic chemical reaction is studied analytically for constant, but general, Lewis-Semenov number. Limit-process expansions are used to obtain solutions in the physically interesting limit of activation temperature large relative to the hot-boundary temperature. The eigenvalue or critical flow speed for an adiabatic system is established as a function of departure from stoichiometry. It is emphasized that, for relatively small departures from stoichiometry, the bimolecular system behaves as a monopropellant decomposition, to lowest order of approximation, because the richer reactant is effectively undepleted. The porous-disk-type flameholder for a flat-flame burner is modeled as a (nonadiabatic) heat source (supercritical flow speed) or heat sink (subcritical flow speed). The flame stand-off distance and the amount of departure of the hot-boundary tempe...

Stoichiometry and Flameholder Effects on a One-Dimensional Flame

On the Annular Damper for a Freely Precessing Gyroscope

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On the vibrations of the rotating ring

The phenomena which occur in a fluid contained in a rotating system are strongly influenced by the density gradients in that fluid and by certain features of the geometry. In this paper, we study several specific phenomena which involve stratification and/or irregular surface topography and some phenomena which arise because of an equatorial geometry. Although several of these studies are motivated by particular geophysical questions, we do not treat comprehensively any geophysical phenomenon. Nevertheless, the inferences we draw from these studies do much to clarify the roles of the mechanisms underlying various geophysical phenomena and these inferences should be of value when comprehensive geophysical investigations are attempted.

George F. Carrier

Papers

On the theory of the wind-driven ocean circulation

Stoichiometry and Flameholder Effects on a One-Dimensional Flame

On the Annular Damper for a Freely Precessing Gyroscope

On the vibrations of the rotating ring

Some effects of stratification and geometry in rotating fluids