Design of asymptotic preserving finite volume schemes for the hyperbolic heat equation on unstructured meshes

We propose an asymptotic preserving nodal discretization of the hyperbolic heat equation, also known as the P ₁ equation, on unstructured meshes in 2-D. This method, in diffusive regime, overcomes the problem of the inconsistent limit with diffusion, of classical multidimensional extensions of 1-D asymptotic preserving schemes, based on edge formulation. We provide both theoretical and numerical results.     

Global existence to a reaction–diffusion equation through a self‐similar‐like upper solution

A demonstration method is presented, which will ensure the existence of positive global solutions in time to the reaction–diffusion equation ?u_t+Δu+u^p=0 in ?ⁿ×[0, ∞), for exponents p?3 and space dimensions n?3. This method does not require the initial value to have a specific uniform smallness condition, but rather to satisfy a bell‐like form. The method is based on a specific upper solution, which models the diffusion process of the heat equation. The upper solution is not self‐similar, but does have a self‐similar‐like form. After transforming the reaction–diffusion problem into an equivalent one, whose initial value is uniformly very small, a local solution is obtained in the time interval [0, 1] by the use of this upper solution. This local solution is then extended to [0, ∞) through an infinite sequence of extensions. At each step, an appropriate change of variables will transform the extension into a problem nearly identical to the local problem in [0, 1]. These transformations exploit the diffusive and self‐similar‐like nature of the upper solution. Copyright © 2009 John Wiley & Sons, Ltd.     

Asymptotic analysis of unsteady neutron transport equation

Consider the unsteady neutron transport equation with diffusive boundary condition in 2D convex domains. We establish the diffusive limit with both initial layer and boundary layer corrections. The major difficulty is the lack of regularity in the boundary layer with geometric correction. Our contribution relies on a detailed analysis of asymptotic expansions inspired by the compatibility condition and an intricate L^2m ? L^∞ framework, which yields stronger remainder estimates.     

Global dynamics of delayed reaction–diffusion equations in unbounded domains

We consider a nonlocal delayed reaction–diffusion equation in an unbounded domain that includes some special cases arising from population dynamics. Due to the non-compactness of the spatial domain, the solution semiflow is not compact. We first show that, with respect to the compact open topology for the natural phase space, the solutions induce a compact and continuous semiflow ${\Phi}$ on a bounded and positively invariant set Y in C ₊?=?C([?1, 0], X ₊) that attracts every solution of the equation, where X ₊ is the set of all bounded and uniformly continuous functions from ${\mathbb{R}}$ to [0, ∞). Then, to overcome the difficulty in describing the global dynamics, we establish a priori estimate for nontrivial solutions after describing the delicate asymptotic properties of the nonlocal delayed effect and the diffusion operator. The estimate enables us to show the permanence of the equation with respect to the compact open topology. With the help of the permanence, we can employ standard dynamical system theoretical arguments to establish the global attractivity of the nontrivial equilibrium. The main results are illustrated with the diffusive Nicholson’s blowfly equation and the diffusive Mackey–Glass equation.     

Critical exponents for large time asymptotics for fractal Hamilton-Jacobi-KPZ equations

Nonlinear and nonlocal evolution equations of the form u_t = ℒ︁u ± |∇u |^q, where ℒ︁ is a pseudodifferential operator representing the infinitesimal generator of a Lévy stochastic process, have been derived (see, [6]) as models for growing interfaces in the case when the continuous Brownian diffusion surface transport is augmented by a random hopping mechanism. The goal of this note is to report properties of solutions to this equation resulting from the interplay between the strengths of the “diffusive” linear and “hyperbolic” nonlinear terms, posed in the whole space R ^N , and supplemented with nonnegative, bounded, and sufficiently regular initial conditions. The full text of the paper, including complete proofs and other results, will appear in the Transactions of the American Mathematical Society. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Time dependent neutron transport theory in multigroup-P L-approximation

Summary A method is developed for solving the time dependent neutron transport equation in multigroupP _L approximation for one-dimensional geometries. The partial differential equations in time and space are solved by means of a power series expansion in the spatial variable. The resulting ordinary differential equations are solved up to theNth order, and the last spatial coefficients are used to satisfy the boundary conditions. Integration of the purely time dependent differential equations is carried out by means of Lie series.Numerical oscillations, appearing for high ordersN, are avoided by subdividing each zone into smaller subzones. Even and odd spatial moments must be developed in opposite directions in each subzone, and the stationary solution representing the initial condition for the time dependent calculation must be developed in the same manner.Results of two calculations in spherical geometry are presented. One is the start-up of a small experimental reactor usingP ₃ theory, the other is a demonstration of neutron waves inP ₁ theory.

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Zusammenfassung Es wird eine Methode zur Lösung der zeitabhängigen Neutronentransportgleichung in der Multigruppen-P _L-Näherung für eindimensionale Geometrien entwickelt. Für die Lösung der partiellen Differentialgleichungen in Ort und Zeit wird eine Potenzreihe im Ort angesetzt. Das sich ergebende gewöhnliche Differentialgleichungssystem wird bis zu einer gewählten OrdnungN erfüllt, und die letzten Koeffizienten der Ortsentwicklungen dienen zur Befriedigung der Randbedingungen. Die Integration in der Zeit erfolgt dann mit der Lie-Reihen-Methode.Zur Vermeidung der Oszillationen, die bei hohen OrdnungenN auftreten, werden die einzelnen Zonen in kleinere Abschnitte unterteilt. Die geraden und ungeraden Momente müssen in den Abschnitten in entgegengesetzter Richtung entwickelt werden. Die stationäre Lösung, die als Anfangsbedingung für die zeitabhängige Rechnung dient, muss mit demselben Entwicklungsschema ermittelt werden.Die Methode wird angewendet zur Lösung derP ₁- undP ₃-Gleichungen in sphärischen Reaktoren. Zwei Beispiele werden damit berechnet: Das Anfahren eines kleinen Versuchsreaktors inP ₃-Näherung und die Nachweisung von Neutronenwellen inP ₁-Theorie.

     

Exponential estimate for reaction diffusion models

Summary. We consider the superposition of a speeded up symmetric simple exclusion process with a Glauber dynamics, which leads to a reaction diffusion equation. Using a method introduced in [Y] based on the study of the time evolution of the H ₋₁ norm, we prove that the mean density of particles on microscopic boxes of size N ^α , for any 12/13<α<1, converges to the solution of the hydrodynamic equation for times up to exponential order in N, provided the initial state is in the basin of attraction of some stable equilibrium of the reaction–diffusion equation. From this result we obtain a lower bound for the escape time of a domain in the basin of attraction of the stable equilibrium point. Received: 3 March 1995 / In revised form: 2 February 1996     

On the analytic solutions of the monoenergetic neutron transport equation for slab geometry

The classical initial value problem for the monoenergetic neutron transport equation in slab geometry is solved, using the power series method. A general formula for analytic solutions of this equation is presented. It is shown that a polynomial solution exists only forc=1 and is linear inx and . Other analytic solutions are given in a closed form. The Taylor series expansion method is compared with the spherical harmonic approach.

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Zusammenfassung Das klassische Eigenwertproblem für die monoenergetische Neutronentransportgleichung in ebener Geometrie wird mit Hilfe der Taylorreihe gelöst und eine allgemeine Formel für analytische Lösungen dieser Gleichung angegeben. Es wird gezeigt, dass die Polynomiallösung nur fürc=1 exisitiert und linear ist in undx. Obwohl die analytischen Lösungen nur beschränkt praktische Anwendung haben zeigt sich, dass die abgebrochene Taylorreihe für praktische Probleme verwendbar ist. Rechnungen mit der abgebrochenen Taylorreihe wirden mitP _N -Rechnungen verglichen.

     

On Existence and Weak Stability of Matrix Refinable Functions

We consider the existence of distributional (or L ₂ ) solutions of the matrix refinement equation where P is an r×r matrix with trigonometric polynomial entries. One of the main results of this paper is that the above matrix refinement equation has a compactly supported distributional solution if and only if the matrix P (0) has an eigenvalue of the form 2 ⁿ , . A characterization of the existence of L ₂ -solutions of the above matrix refinement equation in terms of the mask is also given. A concept of L ₂ -weak stability of a (finite) sequence of function vectors is introduced. In the case when the function vectors are solutions of a matrix refinement equation, we characterize this weak stability in terms of the mask. August 1, 1996. Date revised: July 28, 1997. Date accepted: August 12, 1997.     

Boltzmann diffusive limit beyond the Navier‐Stokes approximation

Given a normalized Maxwellian μ and n ≥ 1, we establish the global‐in‐time validity of a diffusive expansion for a solution F^ε to the rescaled Boltzmann equation (diffusive scaling) inside a periodic box ??³. We assume that in the initial expansion (0.1) at t = 0, the fluid parts of these f_m(0,x,v) have arbitrary divergence‐free velocity fields as well as temperature fields for all 1 ≤ m ≤ n while f₁(0,x,v) has small amplitude in H². For m ≥ 2, these f_m(t,x,v) are determined by a sequence of linear Navier‐Stokes‐Fourier systems iteratively. More importantly, the remainder f(t,x,v) is proven to decay in time uniformly in ε via a unified nonlinear energy method. In particular, our results lead to an error estimate for f₁(t,x,v), the well‐known Navier‐Stokes‐Fourier approximation, and beyond. The collision kernel Q includes hard‐sphere, the cutoff inverse‐power, as well as the Coulomb interactions. © 2005 Wiley Periodicals, Inc.     

Quintuplets of orthoprojectors associated by a linear relation

We consider the equation α₁ P ₁ + α₂ P ₂ + … α _n P _n = I over orthoprojectors P ₁, … ,P _n in a Hilbert space. We show that the set of real parameters (α₁, … α _n ) for which there exists a solution of this equation in orthoprojectors contains an open set from ℝ⁵.     

Critical exponents in a degenerate parabolic equation with weighted source

This article deals with the Fujita-type theorems to the Cauchy problem of degenerate parabolic equation not in divergence form with weighted source u _t ?=?u ^p Δu?+?a(x)u ^q in ? ⁿ ?×?(0,?T), where p?≥?1, q?>?1, and the positive weight function a(x) is of the order |x| ^m with m?>??2. It was known that for the degenerate diffusion equation in divergence form, the weight function affects both of the critical Fujita exponent and the second critical exponent (describing the critical smallness of initial data required by global solutions via the decay rates of the initial data at space-infinity). Contrarily, it is interesting to prove that the weight function in the present model with degenerate diffusion not in divergence form influences the second critical exponent only, without changing the critical Fujita exponent.     

The integrable nonlinear degenerate diffusion equationu t=[f(u)u x −1 x and its relatives

The nonlinear diffusion equationu _t=[f(u)g(u _x )] arises in recent models of turbulent transport and of stress dissipation in rock blasting. A Lie point symmetry analysis produces many similarity reductions of exponential and power-law forms, and reveals that for all choices off the equation is always integrable wheng(u _x )=1/u _x . We identify the functionsf(u) which guarantee equivalence to the linear heat equation. For all other choices off, the linear canonical form leads to a self-adjoint differential equation by separation of variablesx andt. We construct a number of explicit solutions with simple boundary conditions, which illustrate behavior in the vicinity of the degenerate region withu _x =. If zero flux and constant concentration are maintained on free boundaries, then steep concentration gradients may evolve from smooth initial conditions. For other boundary conditions, unlike the examples of strong degeneracy, smoothing will occur at initial step discontinuities.     

A posteriori error analysis for nonconforming approximations of an anisotropic elliptic problem

We develop in this article an a posteriori error estimator for the P₁‐nonconforming finite element approximation, for a diffusion‐reaction equation. We adopt the error in a constitutive law approach in two and three dimensional space, for not necessary piecewise constant data of problems. The efficiency and the reliability of our estimators are proved, neither Helmholtz decomposition of the error nor saturation assumption. The constants are explicitly given, which prove the robustness of these estimators. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 950–976, 2015     

Large deviations for the symmetric simple exclusion process in dimensions d≥ 3

We consider symmetric simple exclusion processes with L=&ρmacr;N ^d particles in a periodic d-dimensional lattice of width N. We perform the diffusive hydrodynamic scaling of space and time. The initial condition is arbitrary and is typically far away form equilibrium. It specifies in the scaling limit a density profile on the d-dimensional torus. We are interested in the large deviations of the empirical process, N ^{− d} [∑ ^L ₁δ _{xi (·)}] as random variables taking values in the space of measures on D[0.1]. We prove a large deviation principle, with a rate function that is more or less universal, involving explicity besides the initial profile, only such canonical objects as bulk and self diffusion coefficients. Received: 7 September 1997 / Revised version: 15 May 1998     

A characterization of some q-orthogonal polynomials

We show that the only orthogonal polynomials satisfying a q-difference equation of the form π(x)D _q P _n (x) = (α _n x + β _n )P _n (x) + γ _n P _n−1(x) where π(x) is a polynomial of degree 2, are the Al-Salam Carlitz 1, little and big q-Laguerre, the little and big q-Jacobi, and the q-Bessel polynomials. This is a q-analog of the work carried out in [1]. 2000 Mathematics Subject Classification Primary—33C45, 33D45     

Waiting-Time Behavior in a Nonlinear Diffusion Equation

We study the nonlinear diffusion equation u_t*=(uⁿu_x)_x, which occurs in the study of a number of problems. Using singular-perturbation techniques, we construct approximate solutions of this equation in the limit of small n. These approximate solutions reveal simply the consequences of this variable diffusion coefficient, such as the finite propagation speed of interfaces and waiting-time behavior (when interfaces wait a finite time before beginning to move), and allow us to extend previous results for this equation.     

Second‐order backward stochastic differential equations and fully nonlinear parabolic PDEs

For a d‐dimensional diffusion of the form dX_t = μ(X_t)dt + σ(X_t)dW_t and continuous functions f and g, we study the existence and uniqueness of adapted processes Y, Z, Γ, and A solving the second‐order backward stochastic differential equation (2BSDE) If the associated PDE has a sufficiently regular solution, then it follows directly from Itô's formula that the processes solve the 2BSDE, where ?? is the Dynkin operator of X without the drift term. The main result of the paper shows that if f is Lipschitz in Y as well as decreasing in Γ and the PDE satisfies a comparison principle as in the theory of viscosity solutions, then the existence of a solution (Y, Z,Γ, A) to the 2BSDE implies that the associated PDE has a unique continuous viscosity solution v and the process Y is of the form Y_t = v(t, X_t), t ∈ [0, T]. In particular, the 2BSDE has at most one solution. This provides a stochastic representation for solutions of fully nonlinear parabolic PDEs. As a consequence, the numerical treatment of such PDEs can now be approached by Monte Carlo methods. © 2006 Wiley Periodicals, Inc.     

On Nonexistence of any λ0-invariand positive harmonic function,a counter example to stroock' conjecture

Consider a sub-Markovian semigroup such that λ₀, the border number between recrrence and transience, equals zero. In 1982, D. W. Stroock conjectured that under general hypotheses on the semi-group the corresponding process always admits an invariant measure.

In this paper we present an example of a second order elliptic operator P with a generalized principal eigenvalue λ₀ which equals zero such that the parabolic equation does not admit any positive invariant P—harmonic function and also any invariant measure. This gives a counter example to Stroock's conjecture for diffusion processes. We also present an example of a complete Riemannian manifold M which does not admit any positive invariant harmonic function while λ₀, the bottom of the spectrum of M, is zero. This gives a partial answer to a question of Stroock and Sullivan.     

Diffusion limit for the linear boltzmann equation of the neutron transport theory

In this paper we present the asymptotic analysis of the linear Boltzmann equation for neutrons with a small positive parameter ? related to the mean free path, based upon the Chapman–Enskog procedure of the kinetic theory. We prove that if proper initial conditions derived by considering initial layer solutions are used, the diffusion equation gives the uniform approximation to the neutron density function with the O(?²) accuracy.