Linear Transport Operator Research Articles

Solving nonlinear discretizations of [formula omitted] transport calculations

In this paper, we investigate iterative strategies to converge transport solutions using nonlinear discretization schemes, specifically the adaptive-weighted diamond-difference method and a set-to-zero fixup approach for the linear discontinuous method compatible with arbitrary polytopal cells. The nonlinear nature of these schemes precludes the use of linear methods, such as highly effective Krylov iterative solvers like GMRES. Instead, nonlinear Newton-like solvers such as Broyden’s method, nonlinear Krylov acceleration, and the Jacobian-free Newton-Krylov method are analyzed in conjunction with physics-based diffusion acceleration schemes. The Richardson method (i.e., source iteration) is also analyzed for comparison. Numerical experiments (both monoenergetic and multigroup) demonstrate effective reduction in transport work by the use of appropriately chosen iterative schemes, including choosing sufficiently consistent diffusion acceleration. These results are analogous to linear transport operators, and they can be used as a point of reference in determining solution strategies for other nonlinear transport discretization schemes.

Statistical Inversion of Surface Ocean Kinematics from Sea Surface Temperature Observations

AbstractThe sea surface temperature (SST) record provides a unique view of the surface ocean at high spatiotemporal resolution and holds useful information on the kinematics underlying the SST variability. To access this information, we develop a new local matrix inversion method that allows us to quantify the evolution of a given SST perturbation with a response function and to estimate velocity, diffusivity, and decay fields associated with it. The matrix inversion makes use of the stochastic climate model paradigm—we assume that SST variations are governed by a linear transport operator and a forcing that has a relatively short autocorrelation time scale compared to that of SST. We show that under these assumptions, the transport operator can be inverted from the covariance matrices of the underlying SST data. The accuracy of the results depends on the length of the time series, and in general the inverted properties depend on the spatial and time resolution of the SST data. Future studies could use the methodology to explore the interannual variability of SST anomalies; to estimate the scale dependency of ocean mixing; and to estimate anomaly propagation, both at the surface and in the interior. The methodology can be easily used with any gridded observations or model output with adequate time and spatial resolution, and it is not restricted to SST. The inversion code is written in Python and distributed as a MicroInverse package through GitHub and the Python Package Index.

Pointwise innovation–based state observation of exothermic tubular reactors

The problem of estimating the spatio-temporal temperature and concentration distribution in a tubular reactor with first order exothermic reaction and in-domain pointwise temperature measurement is considered. The problem is addressed using a simple point innovation structure, which imposes the measurement information at the sensor location in the simulator. Rigorous convergence conditions are presented in dependence of the sensor location and reactor parameters in terms of the dominant eigenvalues of the modified linear transport operators and the Lipschitz constant of the reaction rate. Numerical simulation results show that the proposed approach yields good performance even in the presence of unknown feed temperature perturbations.

Optimal adaptive ridgelet schemes for linear advection equations

In this paper we present a novel method for the numerical solution of linear advection equations, which is based on ridgelets. Such equations arise for instance in radiative transfer or in phase contrast imaging. Due to the fact that ridgelet systems are well adapted to the structure of linear transport operators, it can be shown that our scheme operates in optimal complexity, even if line singularities are present in the solution. The key to this is showing that the system matrix (with diagonal preconditioning) is uniformly well-conditioned and compressible – the proof for the latter represents the main part of the paper. We conclude with some numerical experiments about N -term approximations and how they are recovered by the solver, as well as localisation of singularities in the ridgelet frame.

Transport semigroup associated to positive boundary conditions of unit norm: A Dyson-Phillips approach

We revisit our study of general transport operator with general force field and general invariant measure by considering, in the $L^1$ setting, the linear transport operator $\mathcal{T}_H$ associated to a linear and positive boundary operator $H$ of unit norm. It is known that in this case an extension of $\mathcal{T}_H$ generates a substochastic (i.e. positive contraction) $C_0$-semigroup $(V_H(t))_{t\geq 0}$. We show here that $(V_H(t))_{t\geq 0}$ is the smallest substochastic $C_0$-semigroup with the above mentioned property and provides a representation of $(V_H(t))_{t \geq 0}$ as the sum of an expansion series similar to Dyson-Phillips series. We develop an honesty theory for such boundary perturbations that allows to consider the honesty of trajectories on subintervals $J \subseteq [0,\infty)$. New necessary and sufficient conditions for a trajectory to be honest are given in terms of the aforementioned series expansion.

The Criticality Eigenvalue Problem for the Transport Operator with General Boundary Conditions

The effective multiplication factor (keff) is the dominant eigenvalue for the linear particle transport operator. A proof for the existence of a simple dominant eigenvalue with a positive eigenfunction is established in the L1 setting in the presence of particle reflection at the domain boundary as characterized by a general integral albedo operator. By arguing that physics demands to work on an L1‐weighted space, a direct physical approach based on linearity and positivity allows us to circumvent some of the limitations encountered in the classical proofs.

On Hyperbolic Variational Inequalities of First Order and Some Applications

This paper presents further developments in the study of strong solutions and their coincidence sets to the obstacle problem for linear transport operators. Under natural mild assumptions, the strong convergence of the solutions and the characteristic functions of their coincidence sets is obtained in the passage of second order to first order problems. An application to a steady-state chemotaxis problem is given. The extension to the two obstacles problem of first order is also presented, having in view an application to a porous media model in presence of saturation arising in petroleum engineering.

Multiple stream superposition of a two-stream combined analytical numerical initial value problem solver for aerodynamic mixing

In spite of the rapid advances in both scalar and parallel computational flow simulation tools, the large number and breadth of variables involved in both design and inverse problems make the use of complex fluid flow models impractical. With this restriction, it may be concluded that an important family of methods for mathematical/computational development are reduced or approximate models. An approximate model for two stream mix problems utilizing a combined perturbation/numerical modelling methodology has been developed [1]. The numerical portion of the model uses a compact finite difference scheme, while analytical solutions are used to resolve singular behavior that is inherent to this flow. Approximate representation of the flow in terms of flux variables yields a linear transport operator, thus facilitating the additive decomposition of the solution into numerical and analytical portions. Additionally, linearity permits superposition of the basic two-stream initial value problem to construct multiple stream mixing problems. Multiple stream results are presented to illustrate the efficiency of this methodology.

On the spectrum of the linear transport operator in a slab geometry with generalized periodic boundary conditions

The spectrum of the transport operator A in a nonhomogeneous slab with continuous energy is discussed in consideration of anisotropic scattering and fission with generalized periodic boundary conditions. We show the relative compactness of the streaming operator B, investigate the spectrum of A, especially obtain the dominant and the strictly dominant eigenvalue in L1 space which is the most natural space for the transport problem.

Duality, Green's functions and all that

The basic duality relation between the direct and adjoint linear transport operators is analyzed and used to derive a duality relation between the direct and adjoint Green's functions with general albedo conditions. Convolution relations are derived for Green's functions defined on domains with non empty intersection. The duality framework is used to derive known results such as the reciprocity theorem, integrals equations and Placzek's lemma with applications to the construction of Green's functions with factored albedos and boundary perturbation problems. A new equation for the source function is also given.

Linear stability of the spatially homogeneous equilibria of the Vlasov-Poisson system with collisions

In this work we study the stability of the spatially homogeneous solutions of the Vlasov-Poisson system (Vlasov equilibria) when a collision term, in the form of a BGK operator with velocity-dependent collision frequency, is added to the Vlasov equation. Generalizing earlier results, obtained for the same collision model with a constant collision frequency, we find the spectrum and the eigenfunctions of the linear transport operator and derive a new linear dispersion relation for the linearized kinetic equations. Finally, we present some numerical results.

Some notes on the spectral properties of Co semigroups generated by linear transport operators

The main aim of this paper is to investigate some aspects of the spectral properties of strongly continuous semigroups generated by linear transport operator. These properties are closely related to the large time behavior of time dependent solutions of transport equations. Our attention is focused on the transport equations with reentry boundary conditions. A class of abstract kinetic systems is studied, and, as applications, some transport equations are investigated.

The spectrum of the linear transport operator with a force in a torus

The spectrum of the linear transport operator with a force in a torus

The generalization and applications of Kato's perturbation theorem

In this paper we generalize Kato's perturbation theory to weakly compact perturbations. As applications of this theory, we discuss the distribution of the essential spectrum of linear transport operators.

Real criticality eigenvalues of the one-speed linear transport operator

Abstract The criticality eigenvalue problem has been studied for the one-speed neutron transport equation. Convex bodies of arbitrary shape and with vacuum boundary conditions have been considered. The cross sections may be space dependent, and the scattering is assumed to be anisotropic. Several new conditions have been derived which ensure that a point spectrum of eigenvalues exists and that all the eigenvalues are real. The most general such condition is that the even order coefficients in the development of the scattering function have a different sign than the odd order ones. As a consequence, for linearly anisotropic scattering the eigenvalues are all real if the average cosine of the scattering angle is negative. Numerical results computed for homogeneous bodies in the form of spheres and infinite slabs and cylinders confirm the theoretical considerations.

Compactness properties for linear transport operators with abstract boundary conditions in slab geometry

Abstract The purpose of this paper is to give a systematic analysis of the compactness for one-dimensional transport equations, with general boundary conditions including all known classical boundary conditions. We give necessary and sufficient conditions for some power of (λ-T)-1 K to be compact where T is the streaming operator and K the scattering operator. We also give necessary conditions for some remainder term Rn(t) of the Dyson-Phillips expansion to be compact. Finally, an open problem is indicated.

Dependence of the time eigenvalue of linear transport operator on the system size and other parameters -an application of the Perron-Erobenius theorem

Abstract It is shown that the fundamental time eigenvalue of the linear transport operator increases with the size of the system. This follows from the increase in the largest eigenvalue of a non-negative irreducible matrix whenever any matrix element is increased. This result of matrix analysis is generalised to more general Krein-Rutman operators that leave a cone of vectors invariant.

Linear transport equation with specular reflection boundary condition

In this paper, the time dependent linear transport equation with anisotropic, energy dependent, non-uniform bounded convex media and specular reflection boundary condition is considered. The existence, uniqueness and stability of its solution for initial value are proved. The spectral properties of the time independent linear transport operator are studied.

A spectral, mapping theorem for the exponential function in ltnear transport theory

Abstract The relation between the spectra of the linear transport operator and the C0-semigroup generated by it is studied by means of a spectral mapping theorem for the exponential function for positive operators on Banach lattices.

Spectral properties of the linear multiple scattering operator in L1-Banach lattices

In this article the spectrum of the linear transport operator for a scattering system is studied. Because only positive solutions of the corresponding transport equation are physically meaningful, the concepts of Banach lattices and positive operators are used. The spectrum contains information about the asymptotic behavior for large times of the transport system.