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We present physics-based preconditioning and a time-stepping strategy for a momentbased scale-bridging algorithm applied to the thermal radiative transfer equation. Our goal is to obtain (asymptotically) second-order time accurate and consistent solutions without nonlinear iterations between the high-order (HO) transport equation and the low-order (LO) continuum system within a time step. Modified equation analysis shows that this can be achieved via a simple predictor-corrector time stepping that requires one inversion of the transport operator per time step. We propose a physics-based preconditioning based on a combination of the nonlinear elimination technique and Fleck-Cummings linearization. As a result, the LO system can be solved efficiently via a multigrid preconditioned Jacobian-free Newton-Krylov method. For a set of numerical test problems, the physics-based preconditioner reduces the number of GMRES iterations by a factor of 3∼4 as compared to a standard preconditioner for advection-diffusion. Furthermore, the performance of the proposed physics-based preconditioner is insensitive to the time-step size.1. Introduction. Accurate modeling of neutral particle transport behavior is of great importance in many science and engineering applications [8,22]. Many numerical transport algorithms suffer from slow convergence in optically thick regions where particles undergo a large number of scattering and/or absorption and re-emission events. A moment-based scale-bridging algorithm for thermal radiative transfer (TRT) problems has shown great potential for accelerating the solution of the Boltzmann transport equation by bridging the diffusion and transport scales [21]. The algorithm utilizes low-order (LO) continuum equations that are consistently derived from the high-order (HO) transport equation. Due to discrete consistency, the LO system can be used not only for accelerating the HO solution but also for coupling to other physics.Moment-based acceleration has been very successful in the field of nuclear reactor physics. One popular approach is the quasi-diffusion (QD) method [3,11], where the

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We review the state of the art in the formulation, implementation, and performance of so-called highorder/low-order (HOLO) algorithms for challenging multiscale problems. HOLO algorithms attempt to couple one or several high-complexity physical models (the high-order model, HO) with low-complexity ones (the low-order model, LO). The primary goal of HOLO algorithms is to achieve nonlinear convergence between HO and LO components while minimizing memory footprint and managing the computational complexity in a practical manner. Key to the HOLO approach is the use of the LO representations to address temporal stiffness, effectively accelerating the convergence of the HO/LO coupled system. The HOLO approach is broadly underpinned by the concept of nonlinear elimination, which enables segregation of the HO and LO components in ways that can effectively use heterogeneous architectures. The accuracy and efficiency benefits of HOLO algorithms are demonstrated with specific applications to radiation transport, gas dynamics, plasmas (both Eulerian and Lagrangian formulations), and ocean modeling. Across this broad application spectrum, HOLO algorithms achieve significant accuracy improvements at a fraction of the cost compared to conventional approaches. It follows that HOLO algorithms hold significant potential for high-fidelity system scale multiscale simulations leveraging exascale computing.

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