Asymptotic Diffusion Research Articles

Asymptotic diffusion limit of cell temperature discretisation schemes for thermal radiation transport

This paper attempts to unify the asymptotic diffusion limit analysis of thermal radiation transport schemes, for a linear-discontinuous representation of the material temperature reconstructed from cell centred temperature unknowns, in a process known as ‘source tilting’. The asymptotic limits of both Monte Carlo (continuous in space) and deterministic approaches (based on linear-discontinuous finite elements) for solving the transport equation are investigated in slab geometry. The resulting discrete diffusion equations are found to have nonphysical terms that are proportional to any cell-edge discontinuity in the temperature representation. Based on this analysis it is possible to design accurate schemes for representing the material temperature, for coupling thermal radiation transport codes to a cell centred representation of internal energy favoured by ALE (arbitrary Lagrange–Eulerian) hydrodynamics schemes.

Rigorous Asymptotic and Moment-Preserving Diffusion Approximations for Generalized Linear Boltzmann Transport in Arbitrary Dimension

We derive new diffusion solutions to the monoenergetic generalized linear Boltzmann transport equation for the stationary collision density and scalar flux about an isotropic point source in an infinite d-dimensional absorbing medium with isotropic scattering. We consider both classical transport theory with exponentially distributed free paths in arbitrary dimensions as well as a number of nonclassical transport theories (nonexponential random flights) that describe a broader class of transport processes within partially correlated random media. New rigorous asymptotic diffusion approximations are derived where possible. We also generalize Grosjean’s moment-preserving approach of separating the first (or uncollided) distribution from the collided portion and approximating only the latter using diffusion. We find that for any spatial dimension and for many free-path distributions Grosjean’s approach produces compact, analytic approximations that are, overall, more accurate for high absorption and for small source-detector separations than either P1 diffusion or rigorous asymptotic diffusion. These diffusion-based approximations are exact in the first two even spatial moments, which we derive explicitly for various nonclassical transport types. We also discuss connections between the random-flight-theory derivation of the Green’s function and the discrete spectrum of the transport operator and report some new observations regarding the discrete eigenvalues of the transport operator for general dimensions and free-path distributions.

Drifter-derived estimates of lateral eddy diffusivity in the World Ocean with emphasis on the Indian Ocean and problems of parameterisation

The Global Drifter Program data set, collected in the period 1979–2011, is divided in drogue-on and drogue-off subsets which are then used to develop 5°×5° bin estimates of lateral eddy diffusivity in the World Ocean's mixed and surface layers, respectively, by means of a modification of the Davis (1987. Journal of Marine Research, 430 (45), 635–666.; 1991. Deep Sea Research. 38, 432 (1), S531–S571.) approach. The analysis is geographically focused on the Indian Ocean, the region where such calculations have been lacking. As the Indian Ocean circulation is strongly determined by monsoonal winds, an effort was made to calculate the diffusivity relative to monsoon-driven mean currents. The drifter-derived estimates of the asymptotic lateral diffusivity at large time lags K∞, were found to be below 1×104m2s−1 almost everywhere in the World Ocean, while the maximum diffusivity Kmax, equivalent to integrating the Lagrangian velocity autocorrelation to the first zero crossing, exceeds K∞ in strong ocean currents by a factor of 2–4. At mid latitudes (20°≤|ϕ|≤50°), the estimates of K∞ in the mixed and surface layers are shown to follow the frozen field regime parameterisation K∞~EKE×Lbc, where EKE is the eddy kinetic energy and Lbc is the baroclinic Rossby deformation radius. In contrast, at low latitudes (|ϕ|<29°) in the surface layer, the fixed float regime parameterisation Kmax,K∞=EKE×T0, where T0=1.9 days and 1.3 days is the typical value of the Lagrangian integral timescale in the cases of Kmax and K∞, respectively, is preferable.

Epidemic fronts in complex networks with metapopulation structure

Infection dynamics have been studied extensively on complex networks, yielding insight into the effects of heterogeneity in contact patterns on disease spread. Somewhat separately, metapopulations have provided a paradigm for modeling systems with spatially extended and "patchy" organization. In this paper we expand on the use of multitype networks for combining these paradigms, such that simple contagion models can include complexity in the agent interactions and multiscale structure. Using a generalization of the Miller-Volz mean-field approximation for susceptible-infected-recovered (SIR) dynamics on multitype networks, we study the special case of epidemic fronts propagating on a one-dimensional lattice of interconnected networks-representing a simple chain of coupled population centers-as a necessary first step in understanding how macroscale disease spread depends on microscale topology. Applying the formalism of front propagation into unstable states, we derive the effective transport coefficients of the linear spreading: asymptotic speed, characteristic wavelength, and diffusion coefficient for the leading edge of the pulled fronts, and analyze their dependence on the underlying graph structure. We also derive the epidemic threshold for the system and study the front profile for various network configurations.

Estimates of the Nash–Aronson type for degenerating parabolic equations

We consider second-order parabolic equations describing diffusion with degeneration and diffusion on singular and combined structures. We give a united definition of a solution of the Cauchy problem for such equations by means of semigroup theory in the space L 2 with a suitable measure. We establish some weight estimates for solutions of Cauchy problems. Estimates of Nash–Aronson type for the fundamental solution follow from them. We plan to apply these estimates to known asymptotic diffusion problems, namely, to the stabilization of solutions and to the “central limit theorem.”

Discrete-Time Semi-Markov Random Evolutions and their Applications

In this paper we introduce discrete-time semi-Markov random evolutions (DTSMREs) and study asymptotic properties, namely, averaging, diffusion approximation, and diffusion approximation with equilibrium by the martingale weak convergence method. The controlled DTSMREs are introduced and Hamilton–Jacobi–Bellman equations are derived for them. The applications here concern the additive functionals (AFs), geometric Markov renewal chains (GMRCs), and dynamical systems (DSs) in discrete time. The rates of convergence in the limit theorems for DTSMREs and AFs, GMRCs, and DSs are also presented.

Discrete-Time Semi-Markov Random Evolutions and their Applications

In this paper we introduce discrete-time semi-Markov random evolutions (DTSMREs) and study asymptotic properties, namely, averaging, diffusion approximation, and diffusion approximation with equilibrium by the martingale weak convergence method. The controlled DTSMREs are introduced and Hamilton–Jacobi–Bellman equations are derived for them. The applications here concern the additive functionals (AFs), geometric Markov renewal chains (GMRCs), and dynamical systems (DSs) in discrete time. The rates of convergence in the limit theorems for DTSMREs and AFs, GMRCs, and DSs are also presented.

MODEL OF THE FIELD LINE RANDOM WALK EVOLUTION AND APPROACH TO ASYMPTOTIC DIFFUSION IN MAGNETIC TURBULENCE

The turbulent random walk of magnetic field lines plays an important role in the transport of plasmas and energetic particles in a wide variety of astrophysical situations, but most theoretical work has concentrated on determination of the asymptotic field line diffusion coefficient. Here we consider the evolution with distance of the field line random walk using a general ordinary differential equation (ODE), which for most cases of interest in astrophysics describes a transition from free streaming to asymptotic diffusion. By challenging theories of asymptotic diffusion to also describe the evolution, one gains insight on how accurately they describe the random walk process. Previous theoretical work has effectively involved closure of the ODE, often by assuming Corrsin's hypothesis and a Gaussian displacement distribution. Approaches that use quasilinear theory and prescribe the mean squared displacement Δx 2 according to free streaming (random ballistic decorrelation, RBD) or asymptotic diffusion (diffusive decorrelation, DD) can match computer simulation results, but only over specific parameter ranges, with no obvious marker of the range of validity. Here we make use of a unified description in which the ODE determines Δx 2 self-consistently, providing a natural transition between the assumptions of RBD and DD. We find that the minimum kurtosis of the displacement distribution provides a good indicator of whether the self-consistent ODE is applicable, i.e., inaccuracy of the self-consistent ODE is associated with non-Gaussian displacement distributions.

Time scales of turbulent relative dispersion

Tracers in a turbulent flow separate according to the celebrated t3/2 Richardson-Obukhov law, which is usually explained by a scale-dependent effective diffusivity. Here, supported by state-of-the-art numerics, we revisit this argument. The Lagrangian correlation time of velocity differences increases too quickly for validating this approach, but acceleration differences decorrelate on dissipative time scales. Phenomenological arguments are used to relate the behavior of separations to that of a "local energy dissipation," defined as the average ratio between the cube of the longitudinal velocity difference and the distance between the two tracers. This quantity is shown to stabilize on short time scales and this results in an asymptotic diffusion ∝t1/2 of velocity differences. The time of convergence to this regime is shown to be that of deviations from Batchelor's initial ballistic regime, given by a scale-dependent energy dissipation time rather than the usual turnover time. It is finally demonstrated that the fluid flow intermittency should not affect this long-time behavior of the relative motion.

Time-Fractional Telegrapher’s Equation (P1) Approximation for the Transport Equation

In this technical note we develop a new approximation from the solution of the time-dependent Boltzmann equation, which includes a fractional constitutive equation of the neutron current density, for a general medium. The fractional constitutive equation in combination with the conservation law that governs the particle collision and reaction processes (P1) approximation for the transport equation gives a time-fractional telegrapher's equation (TFTE). The wave velocity found with this approximation is 3-γ/2 for γ < 1. The numerical results are compared with the exact solution and Heizler's approximation. We found that the TFTE gives the best estimate for a purely absorbing medium, where most approximations fail. The asymptotic diffusion coefficient was applied for a heterogeneous medium, and the results show that the behavior of the TFTE improves.

RANDOM BALLISTIC INTERPRETATION OF NONLINEAR GUIDING CENTER THEORY

Nonlinear guiding center (NLGC) theory has been used to explain the asymptotic perpendicular diffusion coefficient κ⊥ of energetic charged particles in a turbulent magnetic field, which can be applied to better understand cosmic ray transport. Here we re-derive NLGC, replacing the assumption of diffusive decorrelation with random ballistic decorrelation (RBD), which yields an explicit formula for κ⊥. We note that scattering processes can cause a reversal of the guiding center motion along the field line, i.e., backtracking, leading to partial cancellation of contributions to κ⊥, especially for low-wavenumber components of the magnetic turbulence. We therefore include a heuristic backtracking correction (BC) that can be used in combination with RBD. In comparison with computer simulation results for various cases, NLGC with RBD and BC provides a substantially improved characterization of the perpendicular diffusion coefficient for a fluctuation amplitude less than or equal to the large-scale magnetic field.

Asymptotic diffusion coefficients and anomalous diffusion in a meandering jet flow under environmental fluctuations

The nontrivial dependence of the asymptotic diffusion on noise intensity has been studied for a Hamiltonian flow mimicking the Gulf Jet Stream. Three different diffusion regimes have been observed depending on the noise intensity. For intermediate noise the asymptotic diffusion decreases with noise intensity at a rate which is linearly dependent to the flow's meander amplitude. Increasing the noise the fluid transport passes through a superdiffusive regime and finally becomes diffusive again at large noise intensities. The presence of inner circulation regimes in the flow has been found to be determinant to explain the observed behavior.

Discrete-Time Semi-Markov Random Evolutions – Average and Diffusion Approximation of Difference Equations and Additive Functionals

This article introduces discrete-time semi-Markov random evolution and studies asymptotic properties, namely, averaging and diffusion approximation by weak convergence methods. The cases considered here concern difference equations and additive functionals with application in estimation of stationary probability of semi-Markov chains.

Structure–transport correlation for the diffusive tortuosity of bulk, monodisperse, random sphere packings

The mass transport properties of bulk random sphere packings depend primarily on the bed (external) porosity ɛ, but also on the packing microstructure. We investigate the influence of the packing microstructure on the diffusive tortuosity τ = D m/ D eff, which relates the bulk diffusion coefficient ( D m) to the effective (asymptotic) diffusion coefficient in a porous medium ( D eff), by numerical simulations of diffusion in a set of computer-generated, monodisperse, hard-sphere packings. Variation of packing generation algorithm and protocol yielded four Jodrey–Tory and two Monte Carlo packing types with systematically varied degrees of microstructural heterogeneity in the range between the random-close and the random-loose packing limit ( ɛ = 0.366–0.46). The distinctive tortuosity–porosity scaling of the packing types is influenced by the extent to which the structural environment of individual pores varies in a packing, and to quantify this influence we propose a measure based on Delaunay tessellation. We demonstrate that the ratio of the minimum to the maximum void face area of a Delaunay tetrahedron around a pore between four adjacent spheres, ( A min/ A max) D, is a measure for the structural heterogeneity in the direct environment of this pore, and that the standard deviation σ of the ( A min/ A max) D-distribution considering all pores in a packing mimics the tortuosity–porosity scaling of the generated packing types. Thus, σ( A min/ A max) D provides a structure–transport correlation for diffusion in bulk, monodisperse, random sphere packings.

Asymptotic Telegrapher’s Equation (P1) Approximation for the Transport Equation

The diffusion approximation for the Boltzmann (transport) equation suffers from several disadvantages. First, the diffusion approximation succeeds in describing the particle density only if it is isotropic, or close to isotropic. This feature causes the diffusion approximation to be quite accurate for highly isotropically scattering media but to yield poor agreement with the exact solution for the particle density in the case of nonisotropic behavior. To handle general media, the asymptotic diffusion approximation was first developed in the 1950s. The second disadvantage is that the parabolic nature of the diffusion equation predicts that particles will have an infinite velocity; particles at the tail of the distribution function will show up at infinite distance from a source in time t = 0+. The classical P1 approximation (which gives rise to the Telegrapher’s equation) has a finite particle velocity but with the wrong value, namely, υ/√3. In this paper we develop a new approximation from the asymptotic solution of the time-dependent Boltzmann equation, which includes the correct eigenvalue of the asymptotic diffusion approximation and the (almost) correct time behavior (such as the particle velocity), for a general medium.

Manufactured Solutions in the Thick Diffusion Limit

Spatially analytic SN solutions currently exist only under very limited circumstances. For cases in which analytical solutions may not be available, one can turn to manufactured solutions to test the properties of spatial transport discretization schemes. In particular, we show it is possible to use a manufactured solution to conduct such tests in the thick diffusion limit, even though the computed solution is independent of the problem characteristics. We show that a diffusion limit scaling with a manufactured solution source term results in an expression that is valid in the diffusion limit, though it is not of the standard form used in asymptotic diffusion limit analysis. We then derive a necessary, but not sufficient, condition that must be satisfied in order for a spatial discretization of the transport equation to preserve the thick diffusion limit. This condition is stated in terms of the difference between a numerically computed scalar flux solution compared against a known scalar flux. For a sufficiently diffusive problem and optically thick mesh cells, the necessary condition states that if a spatial discretization of the SN equations has the thick diffusion limit, the norm of the difference in the two solutions must converge to zero with decreasing mesh cell spacing. Based on the first observation that the diffusion limit holds for a manufactured solution source term, the known solution can conveniently be taken to be a manufactured solution in a mesh refinement numerical experiment to check whether a spatial discretization satisfies this condition. We present computational examples that verify our analysis and illustrate the expediency of this approach.

Nonlinear Weighted Flux Methods for Particle Transport Problems in Two-Dimensional Cartesian Geometry

A family of nonlinear weighted flux (NWF) methods for solving the transport equation in two-dimensional (2-D) Cartesian geometry is considered. The low-order equations of these methods are defined by means of special linear-fractional factors that are determined by the high-order transport solution. An asymptotic diffusion limit analysis is performed on methods with a general weight function. The analysis revealed conditions on the weight necessary for an accurate approximation of the diffusion equation in this limit. We study methods with weights defined by linear and bilinear functions of directional cosines. As a result, we developed 2-D NWF methods formulated with the low-order equations that give rise to the diffusion equation in optically thick diffusive regions if their factors are calculated by means of the leading-order transport solution. The inherent asymptotic boundary conditions for the NWF methods are analyzed. Numerical results are presented to confirm theoretical results and demonstrate performance of the proposed methods.

A tempered multiscaling stable model to simulate transport in regional-scale fractured media

[1] Accurate and efficient simulation of contaminant transport in fractured rock is practically important, yet current approaches suffer from numerical constraints that limit the full inclusion of fracture network properties at the regional scale. We propose a multidimensional transport model, nonlocal in space and time, which describes complex transport behavior in fractured media at regional scales, without introducing the computational burden of explicitly incorporating individual rock fractures. The model utilizes a direction-dependent tempered stable process to describe the transition of multiscaling anomalous diffusion to asymptotic diffusion limits. Applications show that the model can capture efficiently the transient superdiffusion found in two-dimensional synthetic fracture networks, and suggest that the truncation of leading plume edges is related to fracture network properties.

Asymptotic Diffusion-Limit Accuracy of Sn Angular Differencing Schemes

In a previous paper, Morel and Montry used a Galerkin-based diffusion analysis to define a particular weighted diamond angular discretization for Sn calculations in curvilinear geometries. The weighting factors were chosen to ensure that the Galerkin diffusion approximation was preserved, which eliminated the discrete ordinates flux dip. It was also shown that the step and diamond angular differencing schemes, which both suffer from the flux dip, do not preserve the diffusion approximation in the Galerkin sense. In this paper we re-derive the Morel and Montry weighted diamond scheme using a formal asymptotic diffusion-limit analysis. The asymptotic analysis yields more information than the Galerkin analysis and demonstrates that the step and diamond schemes do in fact formally preserve the diffusion limit to leading order, while the Morel and Montry weighted diamond scheme preserves it to first order, which is required for full consistency in this limit. Nonetheless, the fact that the step and diamond differencing schemes preserve the diffusion limit to leading order suggests that the flux dip should disappear as the diffusion limit is approached for these schemes. Computational results are presented that confirm this conjecture. We further conjecture that preserving the Galerkin diffusion approximation is equivalent to preserving the asymptotic diffusion limit to first order.

Moments for Tempered Fractional Advection-Diffusion Equations

This paper develops moment formulas for exponentially tempered, fractional advection-diffusion equations (TFADEs) that transition from anomalous to asymptotic diffusion limits over time. Exact analytical expressions or series representations for spatial moments up to the fourth order are derived by integral transform or asymptotic expansion approach. A fully Lagrangian solver, cross verified by an implicit Eulerian approach, is also developed to calculate numerically the complete evolution of moments for the TFADEs with complex initial and boundary conditions. Moment analysis identifies the diffusion equation that attracts the tempered anomalous diffusion in the long time limit. Fitting of moments measured at two end members of alluvial systems checks the applicability of moment analysis in understanding real diffusion.