The thermal radiative transfer (TRT) equations form an integrodifferential system that describes the propagation and collisional interactions of photons. Computing accurate and efficient numerical solutions TRT are challenging for several reasons, the first of which is that TRT is defined on a highdimensional phase space that includes the independent variables of time, space, and velocity. In order to reduce the dimensionality of the phase space, classical approaches such as the P$_N$ (spherical harmonics) or the S$_N$ (discrete ordinates) ansatz are often used in the literature. In this work, we introduce a novel approach: the hybrid discrete (H$^T_N$) approximation to the radiative thermal transfer equations. This approach acquires desirable properties of both P$_N$ and S$_N$, and indeed reduces to each of these approximations in various limits: H$^1_N$ $\equiv$ P$_N$ and H$^T_0$ $\equiv$ S$_T$. We prove that H$^T_N$ results in a system of hyperbolic partial differential equations for all $T\ge 1$ and $N\ge 0$. Another challenge in solving the TRT system is the inherent stiffness due to the large timescale separation between propagation and collisions, especially in the diffusive (i.e., highly collisional) regime. This stiffness challenge can be partially overcome via implicit time integration, although fully implicit methods may become computationally expensivemore »
New SAVpressure correction methods for the NavierStokes equations: stability and error analysis
We construct new first and secondorder pressure correctionschemes using the scalar auxiliary variable approach for the NavierStokes equations. These schemes are linear, decoupled and only require solving a sequence of Poisson type equations at each time step. Furthermore, they are unconditionally energy stable. We also establish rigorous error estimates in the two dimensional case for the velocity and pressure approximation of the firstorder scheme without any condition on the time step.
 Award ID(s):
 2012585
 Publication Date:
 NSFPAR ID:
 10329244
 Journal Name:
 Mathematics of Computation
 Volume:
 91
 Issue:
 333
 Page Range or eLocationID:
 141 to 167
 ISSN:
 00255718
 Sponsoring Org:
 National Science Foundation
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