On the convergence of the multigroup approximations for multidimensional media

Summary In this paper, we investigate the stability, convergence, and consistency properties of steady-state multigroup models for submultiplying media with spatial dimensions greater than one. We define these concepts in a Banach space whose norm measures the collision density integrated over phase space. Stability and consistency occur under the conditions that the maximum fluctuations in the total cross section, and in the expected number of secondary particles arising from each energy level, tend to zero as the energy mesh becomes finer. A concluding example and discussion deal with pathologies of the multigroup model in situations where these fluctuations do not tend to zero as the norms of the energy partitions. The results in this paper complement the time-dependent results of both Belleni-Morante and Busoni for isotropic slabs and of Yang Mingzhu and Zhu Guangtian for bounded, three-dimensional media. This work directly extends the steady-state results of Paul Nelson, Jr. and H. D. Victory, Jr. for slab media to steady-state transport in multidimensional media.This research was partially supported by the Alexander von Humboldt Foundation and by a Faculty Development Leave from Texas Tech University during the academic year 1982–1983. Partial support was also provided in the initial stages of this work by the U.S. National Science Foundation under Grant CPE 8007396.     

On the convergence of the multigroup approximations for submultiplying slab media

The stability, convergence, and consistency properties of the steady-state multigroup model are investigated for submultiplying slab media. These concepts are defined in a Banach space setting in which the norm of the angular flux is the collision density integrated over phase space. It is shown that the multigroup approximations are stable and are both consistent with, and convergent to, the transport equation under the conditions that the maximum fluctuations in the total cross section and in the expected number of secondary particles, arising from each energy level, tend to zero as the energy mesh becomes finer. A concluding discussion deals with pathologies of the multigroup approximation for situations in which these fluctuations do not tend to zero as the norms of the energy partitions. The results in this paper complement the time-dependent results of Belleni-Morante and Busoni for isotropic slabs and the results of Nelson for steady-state rod media.     

Convergence of the multigroup approximations for subcritical slab media with applications to shielding calculations

The stability, convergence, and consistency properties of the steady-state multigroup model are investigated for subcritical slab media. These concepts are defined in a Banach space setting in which the norm of the angular flux yields the maximum value, over the position and angular variables, of the energy-integrated angular flux. It is shown that the multigroup approximations are stable, and are both consistent with, and convergent to, the transport equation under the conditions that the maximum fluctuations in the total cross section, and in the number of secondary particles arising from each energy level, tend to zero as the energy mesh becomes finer. These results in this particular Banach space setting extend to slab transport those results obtained by P. Nelson in a shielding norm setting for multigroup transport in submultiplying rod media.     

On the convergence region for decomposition solutions

The author's decomposition method using his A_n polynomials for the nonlinearities has been shown to apply to wide classes of nonlinear (or nonlinear stochastic) operator equations providing a computable, accurate solution which converges rapidly. In computation the above is sufficient for a rapid test of convergence region.     

Exact solutions for two-phase colloidal-suspension transport in porous media

Two-phase transport of colloids and suspensions occurs in numerous areas of chemical, environmental, geo-, and petroleum engineering. The main effects are particle capture by the rock and altering the flux by changing the suspended and retained concentrations. Multiple mechanisms of suspended particle capture are discussed. The mathematical model for m independent particle-capture mechanisms is considered, resulting in an (m + 2) × (m + 2) system of partial differential equations. Using the stream-function as an independent variable instead of time splits the system into an (m + 1) × (m + 1) auxiliary system, containing only concentrations and one lifting hydrodynamic equation for an unknown phase saturation. Introduction of the concentration potential linked with retention concentrations yields an exact solution of the auxiliary problem. The exact formulae allow for predicting the profiles and breakthrough histories for the suspended and retained concentrations, and phase saturations. The solution shows that for small retained concentrations, the suspended concentration is in a steady-state behind the concentration front, where all the retained concentrations are proportional to the mass of suspended particles that passed via a given reservoir cross-section. The maximum penetration depths for suspended and retained particles are the same and are equal to those for a single-phase flow.     

Oscillatory radial solutions for subcritical biharmonic equations

It is well known that the biharmonic equation Δ²u=u|u|^p−1 with p∈(1,∞) has positive solutions on Rn if and only if the growth of the nonlinearity is critical or supercritical. We close a gap in the existing literature by proving the existence and uniqueness, up to scaling and symmetry, of oscillatory radial solutions on Rn in the subcritical case. Analyzing the nodal properties of these solutions, we also obtain precise information about sign-changing large radial solutions and radial solutions of the Dirichlet problem on a ball.     

Four nontrivial solutions for subcritical exponential equations

We show that a semilinear Dirichlet problem in bounded domains of in presence of subcritical exponential nonlinearities has four nontrivial solutions near resonance. Research supported by the Italian National Project Metodi Variazionali ed Equazioni Differenziali Non Lineari.     

On convergence of solutions to equilibria for quasilinear parabolic problems

We show convergence of solutions to equilibria for quasilinear parabolic evolution equations in situations where the set of equilibria is non-discrete, but forms a finite-dimensional C¹-manifold which is normally hyperbolic. Our results do not depend on the presence of an appropriate Lyapunov functional as in the ?ojasiewicz-Simon approach, but are of local nature.     

On the convergence of the Galerkin method for nonsmooth solutions of integral equations

Summary It is shown that the stability region of the Galerkin method includes solutions not lying in the conventional energy space. Optimal order error estimates for these nonsmooth solutions are derived. The new result is compared with the classical statement by means of the basic potential problem.     

On convergence of the solutions of a functional differential equation

A sufficient condition is given for the solutions of a functionally perturbed linear system of ordinary differential equations to have limits at ± ∞.     

On the convergence of solutions of some evolution differential equations

Two conjectures concerning the asymptotic behaviour of solutions of two types of evolution differential equations are presented.     

On the convergence of solutions of the enskog equation to solutions of the boltzmann equation

For the Enskog equation with a symmetrized kernel in a box an existence theorem is proved for initial data with finite mass, energy and entropy. Then by letting the diameter of the molecules go to zero we prove the weak convergence of solutions of the Enskog equation to solutions of the Boltzmann equation.     

On the convergence of the sequence of solutions for a family of eigenvalue problems

The asymptotic behavior of the sequence {u _n } of positive first eigenfunctions for a class of eigenvalue problems is studied in a bounded domain with smooth boundary ? Ω. We prove , where δ is the distance function to ? Ω. Our study complements some earlier results by Payne and Philippin, Bhattacharya, DiBenedetto, and Manfredi, and Kawohl obtained in relation with the “torsional creep problem .”     

Convergence of two-dimensional Nyström discrete-ordinates in solving the linear transport equation

Summary In the discrete-ordinates approximation to the linear transport equation, the integration over the directional variable is replaced by a numerical quadrature rule involving a weighted sum over functional values at selected directions. The purpose of this paper is to show that the Nyström technique of defining the angular flux in directions other than the quadrature points, as outlined by P.M. Anselone and A. Gibbs and utilized by P. Nelson for anisotropically scattering slabs, produces an approximation scheme which is stable, consistent with, and convergent to the transport equation in two-dimensional geometry.     

Chaos,transport and mesh convergence for fluid mixing

Chaotic mixing of distinct fluids produces a convoluted structure to the interface separating these fluids. For miscible fluids （as considered here）, this interface is defined as a 50% mass concentration isosurface. For shock wave induced （Richtmyer-Meshkov） instabilities, we find the interface to be increasingly complex as the computational mesh is refined. This interfacial chaos is cut off by viscosity, or by the computational mesh if the Kolmogorov scale is small relative to the mesh. In a regime of converged interface statistics, we then examine mixing, i.e. concentration statistics, regularized by mass diffusion. For Schmidt numbers significantly larger than unity, typical of a liquid or dense plasma, additional mesh refinement is normally needed to overcome numerical mass diffusion and to achieve a converged solution of the mixing problem. However, with the benefit of front tracking and with an algorithm that allows limited interface diffusion, we can assure convergence uniformly in the Schmidt number. We show that different solutions result from variation of the Schmidt number. We propose subgrid viscosity and mass diffusion parameterizations which might allow converged solutions at realistic grid levels.     

On the convergence of solutions of certain inhomogeneous fourth-order differential equations

The main purpose of this paper is to give sufficient conditions for the convergence of solutions of a certain class of fourth-order nonlinear differential equations using Lyapunov’s second method. Nonlinear functions involved are not necessarily differentiable, but a certain incrementary ratio for a function h lies in a closed subinterval of the Routh–Hurwitz interval.