David L. Book

Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works

Abstract This paper describes a class of explicit, Eulerian finite-difference algorithms for solving the continuity equation which are built around a technique called “flux correction.” These flux-corrected transport algorithms are of indeterminate order but yield realistic, accurate results. In addition to the mass-conserving property of most conventional algorithms, the FCT algorithms strictly maintain the positivity of actual mass densities so steep gradients and inviscid shocks are handled particularly well. This first paper concentrates on a simple one-dimensional version of FCT utilizing SHASTA, a new transport algorithm for the continuity equation, which is described in detail.

Flux-corrected transport II: Generalizations of the method

Abstract The recently developed method of Flux-Corrected Transport (FCT) can be applied to many of the finite-difference transport schemes presently in use. The result is a class of improved algorithms which add to the usual desirable properties of such schemes—conservation, stability, second-order (in some cases) accuracy, etc.—the property of maintaining the intrinsic positivity of quantities like density, energy density, and pressure. Illustrations are given for algorithms of the Lax-Wendroff, leapfrog, and upstreaming types. The errors introduced by the flux-correction process which lies at the heart of the method are cataloged and their effect described. Phoenical FCT, a refinement which minimizes residual diffusive errors, is analyzed. Applications of FCT to general fluid systems, multidimensions, and curvilinear geometry are described. The results of computer tests are shown in which the various types of FCT are compared with one another and with some conventional algorithms.

Flux-corrected transport. III. Minimal-error FCT algorithms

Abstract This paper presents an error analysis of numerical algorithms for solving the convective continuity equation using flux-corrected transport (FCT) techniques. The nature of numerical errors in Eulerian finite-difference solutions to the continuity equation is analyzed. The properties and intrinsic errors of an “optimal” algorithm are discussed and a flux-corrected form of such an algorithm is demonstrated for a restricted class of problems. This optimal FCT algorithm is applied to a model test problem and the error is monitored for comparison with more generally applicable algorithms. Several improved FCT algorithms are developed and judged against both standard flux-uncorrected transport algorithms and the optimal algorithm. These improved FCT algorithms are found to be four to eight times more accurate than standard non-FCT algorithms, nearly twice as accurate as the original SHASTA FCT algorithm, and approach the accuracy of the optimal algorithm.

Effect of compressibility on the Rayleigh–Taylor instability

Eigenfrequencies are calculated for infinitesimal perturbations of the system consisting of two semi‐infinite regions, each filled with a constant‐temperature ideal polytrope stratified exponentially against gravity. The linear growth rate for the Rayleigh–Taylor instability which occurs when the density above the interface exceeds that below it is shown in the model to vary linearly with wavenumber k as k→0. The incompressible fluid result is obtained when the adiabatic index γ approaches ∞. For finite γ (i.e., compressible fluids), the growth rates are in general larger than in the incompressible case. Numerical results and limiting cases are described which illustrate this conclusion.

Stability of imploding shocks in the CCW approximation

Analytical and computational techniques are developed to investigate the stability of converging shock Avaves in cylindrical and spherical geometry. The linearized Chester-Chisnell-Whitham (CCW) equations describing the evolution of an arbitrary perturbation about an imploding shock wave in an ideal fluid are solved exactly in the strong-shock limit for a density profile ρ( r ) ∼ r − q . All modes are found to be relatively unstable (i.e. the ratio of perturbation amplitude to shock radius diverges as the latter goes to zero), provided that q is not too large. The nonlinear CCW equations are solved numerically for both moderate and strong shocks. The small-amplitude limit agrees with the analytical results, but some forms of perturbation which are stable at small amplitude become unstable in the nonlinear regime. The results are related to the problem of pellet compression in experiments on inertial confinement fusion.

PLASMA WAVE REGENERATION IN INHOMOGENEOUS MEDIA

The one‐dimensional problem of electrons confined in a quadratic potential is solved in the WKB approximation. The unique feature of the problem is that, since particle bounce time is independent of energy, resonant particles that normally cause Landau damping periodically interact with the wave and regenerate all phase information. Because of the efficiency of the regeneration mechanism, the dielectric properties governing the system change and give rise to “regenerative modes” associated with harmonics at twice the bounce frequency. The eigenvalues calculated in this WKB approximation are in agreement with previous numerical calculations.

Ion Sound Instability in a Collisionless Shock Wave

The nonlinear development of an ion sound instability is calculated for a case in which a diamagnetic current of electrons propagates across a magnetic field, and in which ion gyration in the magnetic field can be neglected. The energy content of the mode is calculated; the parameter range is appropriate to experiments on collisionless low‐amplitude magnetic shock waves.

Model Equations for Mode Coupling Saturation in Unstable Plasmas.

Some simple one‐dimensional model equations for instability saturation by resonant mode coupling which are of the general form ∂u/∂t; + σ1 u ∂u/∂x + M[u] = O, where M[u] = i/2π∫ d k ω(k)∫ d x ′ u(x ) exp [k(x − x′)] are examined theoretically and numerically. For nondispersive ω(k), it is found the temporal evolution is characterized by the formation and merging of shocks. Also, the level of saturation can depend on the physical size of the unstable system. If disperisive effects are strong, the solution is characterized by the formation of solitons.

Hydrodynamic Stability of a Rotating Liner.

The Rayleigh‐Taylor instability is investigated for a nonsteady basic state. A model of a magnetically imploded cylindrical metallic liner compressing an axial magnetic field is constructed and used as the basis of a linear stability analysis. The liner, idealized to be without energy loss mechanisms, can be given an initial rotation about its axis. Analytic and numerical techniques are used to study the stability of flutelike (∼eimφ) irrotational perturbations about this state. Stability is quantified in terms of the tendency of the liner to disrupt or to encroach toward the axis, and is determined as a function of mode number m, the form of initial disturbance, liner thickness and the amount of rotation. It is shown that thickening the liner tends to stabilize against both encroachment and disruption, while increasing rotational velocity tends to stabilize against encroachment. Implications for experimental designs are discussed, in particular for experiments with deep compressions (large ratio of initi...

Nonlinear evolution of the sausage instability

Sausage instabilities of an incompressible, uniform, perfectly conducting Z pinch are studied in the nonlinear regime. In the long wavelength limit (analogous to the ’’shallow water theory’’ of hydrodynamics), a simplified set of universal fluid equations is derived, with no radial dependence, and with all parameters scaled out. Analytic and numerical solutions of these one‐dimensional equations show that an initially sinusoidal perturbation grows into a ’’spindle’’ or cylindrical ’’spike and bubble’’ shape, with sharp radial maxima. In the short wavelength limit, the problem is shown to be mathematically equivalent to the planar semi‐infinite Rayleigh–Taylor instability, which also grows into a spike‐and‐bubble shape. Since the spindle shape is common to both limits, it is concluded that it probably obtains in all cases. The results are in agreement with dense plasma focus experiments.