arXiv subjects

Jim E. Morel

Publications and source records attributed to Jim E. Morel.

5 recordsLinked to original sources

arXiv2026

Asymptotic constraints for 1D planar grey photon diffusion from linear transport with special-relativistic effects

We derive a grey linear diffusion equation for photons with respect to inertial (or lab-frame) space and time, using asymptotic analysis in 1D planar geometry. The solution of the equation is the comoving radiation energy density. Our analysis does not make use of assumptions about the magnitude of velocity; instead we derive an asymptotic scaling in the lab frame such that we avoid apparent non-physical pathologies that are encountered with the standard static-matter scaling. We permit the photon direction to be continuous (as opposed to constraining the analysis to discrete ordinates). The result is a drift-diffusion equation in the lab frame for comoving radiation energy density, with an adiabatic term that matches the standard semi-relativistic diffusion equation. Following a recent study for discrete directions, this equation reduces to a pure advection equation as the velocity approaches the speed of light. We perform preliminary numerical experiments comparing solutions to relativistic lab-frame Monte Carlo transport and to the well-known semi-relativistic diffusion equation.

astro-ph.HE↗

arXiv2024

A Second Moment Method for k-Eigenvalue Acceleration with Continuous Diffusion and Discontinuous Transport Discretizations

The second moment method is a linear acceleration technique which couples the transport equation to a diffusion equation with transport-dependent additive closures. The resulting low-order diffusion equation can be discretized independent of the transport discretization, unlike diffusion synthetic acceleration, and is symmetric positive definite, unlike quasi-diffusion. While this method has been shown to be comparable to quasi-diffusion in iterative performance for fixed source and time-dependent problems, it is largely unexplored as an eigenvalue problem acceleration scheme due to thought that the resulting inhomogeneous source makes the problem ill-posed. Recently, a preliminary feasibility study was performed on the second moment method for eigenvalue problems. The results suggested comparable performance to quasi-diffusion and more robust performance than diffusion synthetic acceleration. This work extends the initial study to more realistic reactor problems using state-of-the-art discretization techniques. Results in this paper show that the second moment method is more computationally efficient than its alternatives on complex reactor problems with unstructured meshes.

math.NA↗

arXiv2023

Reduced-Memory Methods for Linear Discontinuous Discretization of the Time-Dependent Boltzmann Transport Equation

In this paper, new implicit methods with reduced memory are developed for solving the time-dependent Boltzmann transport equation (BTE). One-group transport problems in 1D slab geometry are considered. The reduced-memory methods are formulated for the BTE discretized with the linear-discontinuous scheme in space and backward-Euler time integration method. Numerical results are presented to demonstrate performance of the proposed numerical methods.

math.NA↗

arXiv2019

Accelerating PDE-constrained Inverse Solutions with Deep Learning and Reduced Order Models

Inverse problems are pervasive mathematical methods in inferring knowledge from observational and experimental data by leveraging simulations and models. Unlike direct inference methods, inverse problem approaches typically require many forward model solves usually governed by Partial Differential Equations (PDEs). This a crucial bottleneck in determining the feasibility of such methods. While machine learning (ML) methods, such as deep neural networks (DNNs), can be employed to learn nonlinear forward models, designing a network architecture that preserves accuracy while generalizing to new parameter regimes is a daunting task. Furthermore, due to the computation-expensive nature of forward models, state-of-the-art black-box ML methods would require an unrealistic amount of work in order to obtain an accurate surrogate model. On the other hand, standard Reduced-Order Models (ROMs) accurately capture supposedly important physics of the forward model in the reduced subspaces, but otherwise could be inaccurate elsewhere. In this paper, we propose to enlarge the validity of ROMs and hence improve the accuracy outside the reduced subspaces by incorporating a data-driven ML technique. In particular, we focus on a goal-oriented approach that substantially improves the accuracy of reduced models by learning the error between the forward model and the ROM outputs. Once an ML-enhanced ROM is constructed it can accelerate the performance of solving many-query problems in parametrized forward and inverse problems. Numerical results for inverse problems governed by elliptic PDEs and parametrized neutron transport equations will be presented to support our approach.

physics.comp-ph↗

arXiv2017

An Accurate Globally Conservative Subdomain Discontinuous Least-squares Scheme for Solving Neutron Transport Problems

In this work, we present a subdomain discontinuous least-squares (SDLS) scheme for neutronics problems. Least-squares (LS) methods are known to be inaccurate for problems with sharp total-cross section interfaces. In addition, the least-squares scheme is known not to be globally conservative in heterogeneous problems. In problems where global conservation is important, e.g. k-eigenvalue problems, conservative treatment must be applied. We, in this study, proposed a SDLS method that retains global conservation. Such a method resembles the LS formulation in each subdomain without material interface and differs from LS in that an additional least-squares interface term appears for each interface. Scalar flux is continuous in each subdomain with continuous finite element method (CFEM) while discontinuous on interfaces for every pair of contiguous subdomains. SDLS numerical results are compared with those obtained from other numerical methods with test problems having material interfaces. High accuracy of scalar flux in fixed-source problems and $k_\mathrm{eff}$ in eigenvalue problems are demonstrated.

physics.comp-ph↗