General Source Term Research Articles

Error estimates on the random projection methods for hyperbolic conservation laws with stiff reaction terms

In this paper we give error estimates on the random projection methods, recently introduced by the authors, for numerical simulations of the hyperbolic conservation laws with stiff reaction terms: u t+f(u) x=− 1 ε (u−α) u 2−1 , −1<α<1. In this problem, the reaction time ε is small, making the problem numerically stiff. A classic spurious numerical phenomenon—the incorrect shock speed—occurs when the reaction time scale is not properly resolved numerically. The random projection method, a fractional step method that solves the homogeneous convection by any shock capturing method, followed by a random projection for the reaction term, was introduced in [J. Comput. Phys. 163 (2000) 216–248] to handle this numerical difficulty. In this paper, we prove that the random projection methods capture the correct shock speed with a first order accuracy, if a monotonicity-preserving method is used in the convection step. We also extend the random projection method for more general source term − 1 ε g(u) , which has finitely many simple zeroes and satisfying ug( u)>0 for large | u|.

Zero Reaction Limit for Hyperbolic Conservation Laws with Source Terms

In this paper we study the zero reaction limit of the hyperbolic conservation law with stiff source term∂tu+∂xf(u)=1εu(1−u2).For the Cauchy problem to the above equation, we prove that as ε→0, its solution converges to piecewise constant (±1) solution, where the two constants are the two stable local equilibria. The constants are separated by either shocks that travel with speed 12(f(1)−f(−1)), as determined by the Rankine-Hugoniot jump condition, or a non-shock discontinuity that moves with speed f′(0), where 0 is the unstable equilibrium. Our analytic tool is the method of generalized characteristics. Similar results for more general source term 1εg(u), having finitely many simple zeros and satisfying ug(u)<0 for large |u|, are also given.

Runge--Kutta Solutions of a Hyperbolic Conservation Law with Source Term

Spurious long-term solutions of a finite-difference method for a hyperbolic conservation law with a general nonlinear source term are studied. Results are contrasted with those that have been established for nonlinear ordinary differential equations. Various types of spurious behavior are examined, including spatially uniform equilibria that exist for arbitrarily small time-steps, nonsmooth steady states with profiles that jump between fixed levels, and solutions with oscillations that arise from nonnormality and exist only in finite precision arithmetic. It appears that spurious behavior is associated in general with insufficient spatial resolution. The potential for curbing spuriosity by using adaptivity in space or time is also considered.

The gravitomagnetic dynamo effect in accretion disks of rotating black holes

view Abstract Citations (23) References (20) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Gravitomagnetic Dynamo Effect in Accretion Disks of Rotating Black Holes Khanna, Ramon ; Camenzind, Max Abstract We present a previously unknown dynamo effect that arises from the coupling of the gravitomagnetic field of a Kerr black hole with electromagnetic fields. The axisymmetric dynamo equations are derived in the 3 + 1 split of Kerr spacetime. They are formally identical to their equivalents in flat space, augmented by a general relativistic source term for the poloidal magnetic field provided by the gravitomagnetic field. There is no need for any particular small-scale plasma motions. The effect does, however, require finite conductivity and is enhanced by anomalous or turbulent magnetic diffusivity. The results of our time-dependent numerical simulations of the mean field dyanmo in a turbulent accretion disk prove that the gravitomagnetic dynamo has growing modes in the vicinity of a rotating black hole. Publication: The Astrophysical Journal Pub Date: November 1994 DOI: 10.1086/187611 Bibcode: 1994ApJ...435L.129K Keywords: Accretion Disks; Black Holes (Astronomy); Dynamo Theory; Gravitational Effects; Magnetohydrodynamics; Stellar Magnetic Fields; Stellar Rotation; Computerized Simulation; Magnetic Effects; Numerical Analysis; Plasma Oscillations; Poloidal Flux; Relativity; Astrophysics; ACCRETION; ACCRETION DISKS; BLACK HOLE PHYSICS; MAGNETIC FIELDS; MAGNETOHYDRODYNAMICS: MHD; RELATIVITY full text sources ADS |

Some thoughts on multipolar sources in a moving medium and the structure of the wave equation

The procedure of derivation of the wave equation is reviewed, it been pointed out that, while for a medium at rest there is a clear correspondence between point multipole sources originating from the expansion of a general source term and with multipoles resulting from singularities in the continuity and momentum equations, this is not so for a moving medium. Thus, a classification of source multipole order based on the expansion of a general source may lack physical significance. Starting from «natural» monopole and dipole operators, identified in the source function, different families of multipole operators are defined, based on the expansion of the appropriate terms. It is shown that the wave operator has a structure similar to that of the source function, being composed of particular multipole operators. Applications of these features in analyzing different formulations for the aerodynamical noise problem and in optimizing the choice of equivalent source terms are highlighted

Efficient yet accurate solution of the linear transport equation in the presence of internal sources: The exponential-linear-in-depth approximation

We present solutions to the linear transport equation valid for monoenergetic particles interacting with a multiple scattering and absorbing layered medium possessing a general anisotropic internal source term. A new exponential-linear approximation to the internal source as a function of scattering depth is introduced and compared to approximations that vary linearly and quadratically in scattering depth. The prime merit of this new approximation is to provide a very efficient yet accurate solution to the linear transport equation by substantially reducing the spatial mesh size. Numerical results pertaining to (1 ) an embedded thermal source and (2) a rapidly varying beam pseudosource demonstrate that the exponential-linear approximation is vastly superior both in speed and accuracy to a linear or quadratic approximation. Potential applications include neutral (photon, neutron) or charged (electron, ion) particle transport as well as linearized gas dynamics.

Particle re-emission during irradiation

A model of implanted gas concentration leading to surface blistering and re-emission of particles during irradiation is presented. The model assumes that particles diffuse from a Gaussian source which is independent of time and involves the solution of the time-dependent diffusion equation with a source term. The source is calculated assuming a Gaussian form with range and straggling obtained from Lindhard-Scharff-Schiøtt theory. The solution involves the Green's function for a general source term and finally a numerical integration. The re-emitted flux due to diffusion is also calculated as a function of time. The maximum concentration C m (x,t) of particles in the solid at distance x from the surface at any time t is the crucial parameter. When C m ⩾ p c NA anywhere in the implanted material, an interconnected region is assumed to form. N is the atomic density of the target material and A the area of implant. The critical concentration p c is taken to be the concentration required for the onset of percolation ( p c ≅ 0.24 for a b.c.c. lattice). An effective diffusion coefficient for He in Nb at ≈400° C (and Pd at −170°C) is extracted from experimental data and used to predict the dependence of the critical dose on the current density. The fractional release and rate of re-emission during irradiation are also calculated as a function of implanted dose.

Stochastic properties of nuclear reactors in a two-group model

The random processes of a zero-power reactor are studied using a point reactor model with two neutron groups. The probability balance equation with a general source term is written in terms of generating functions. Its solution is presented in terms of functions satisfying the sourceless Kolmogoroff equations. Recurrence formulas are stated for these functions and the criticality, the stationary and the extinction probabilities of an autonomous reactor are studied. Differential equations for the mean values, variances and covariances of the neutron populations are derived and stationarity solutions found for these. Autocorrelation and cross-correlation functions and power spectra are derived. The model suggests new experimental measurements, some of which are briefly described.

Markoff cascades with general source terms

The longitudinal development of the multiplicative cascade, considered as a generalized Markoff process, is presented for the case in which the source is not necessarily a δ-function of space or energy. Last collision diffusion equations are established and formally solved, and practical applications are discussed. In addition to its « traditional » application to cosmic radiation theory the Markoff cascade has important applications to the theory of nuclear reactors.