arXiv subjects

Kimberly Matsuda

Publications and source records attributed to Kimberly Matsuda.

2 recordsLinked to original sources

arXiv2026

Multi-Scale Kinetic Simulation: Asymptotic Preserving IMEX-BDF-DG Schemes with Three Implicit-Explicit Partitionings

Kinetic transport models are mesoscopic mathematical descriptions of the transport of particles as well as their interactions with the background media or among themselves, and they have wide applications in many areas of mathematical physics such as nuclear and biomedical engineering, rarefied gas dynamics, and plasma physics. They are often multi-scale, with different characteristics (e.g. hyperbolic, diffusive) depending on the material properties. As our continuing effort to design and analyze numerical methods for accurate and robust simulation of the multi-scale kinetic transport models, in this work, we consider a linear kinetic transport model, a simplified radiative transfer equation, in a diffusive scaling, and propose and analyze three families of asymptotic preserving (AP) methods. Numerical methods with the AP property, that is to preserve the asymptotic behavior of the models at the discrete level on under-resolved meshes, can work uniformly well to simulate multi-scale models across a wide range of scales. The proposed methods start from the micro-macro decomposition of the model, and involve discontinuous Galerkin (DG) methods in space, the discrete ordinates method (i.e. $S_N$ method) in velocity, and implicit-explicit (IMEX) BDF methods in time, with three different IMEX partitionings. A systematic study, both analytically and computationally, is presented regarding their difference in stability, accuracy, computational complexity and AP property. These methods, with multi-step time integrators, are also compared in terms of their accuracy and efficiency with the ones that only differ in using certain IMEX Runge-Kutta methods in time. Together with our previous developments, the present work further contributes to high order DG AP methods for multi-scale kinetic simulation, especially by utilizing the structure of the micro-macro decomposition of the models.

math.NA↗

arXiv2025

Reduced Basis Methods for Parametric Steady-State Radiative Transfer Equation

The radiative transfer equation (RTE) is a fundamental mathematical model to describe physical phenomena involving the propagation of radiation and its interactions with the host medium. Deterministic methods can produce accurate solutions without any statistical noise, yet often at a price of expensive computational costs originating from the intrinsic high dimensionality of the model. With this work, we present the first systematic investigation of projection-based reduced order models (ROMs) following the reduced basis method (RBM) framework to simulate the parametric steady-state RTE with isotropic scattering and one energy group. Four ROMs are designed, with each defining a nested family of reduced surrogate solvers of different resolution/fidelity. They are based on either a Galerkin or least-squares Petrov-Galerkin projection and utilize either an $L_1$ or residual-based importance/error indicator. Two of the proposed ROMs are certified in the setting when the absorption cross section is positively bounded below uniformly. One technical focus and contribution lie in the proposed implementation strategies under the affine assumption of the parameter dependence of the model. These well-crafted broadly applicable strategies not only ensure the efficiency and accuracy of the offline training stage and the online prediction of reduced surrogate solvers, they also take into account the conditioning of the reduced systems as well as the stagnation-free residual evaluation for numerical robustness. Computational complexities are derived for both the offline training and online prediction stages of the proposed model order reduction strategies, and they are demonstrated numerically along with the accuracy and robustness of the reduced surrogate solvers. Numerically we observe four to six orders of magnitude speedup of our ROMs compared to full order models for some 2D2v examples.

math.NA↗