Realizability-Preserving Discontinuous Galerkin Method for Spectral Two-Moment Radiation Transport in Special Relativity

TL;DR

The work addresses robustly solving a spectral two-moment model for relativistic neutral-particle transport with a moving background by developing a realizability-preserving DG-IMEX framework. It combines a phase-space discontinuous Galerkin discretization with strong stability-preserving IMEX time stepping, a realizability-conserving moment conversion, and a realizability-enforcing limiter, all under Minerbo maximum-entropy closure. The authors prove realizability of the scheme and validate it on streaming, diffusion, Doppler-shift, shock, shadow-casting, and vortex tests, demonstrating accurate SR transport and Doppler effects while maintaining conservation and physical bounds. This method lays groundwork for extending to more complex interactions, dynamic backgrounds, and general-relativistic contexts, with potential benefits for astrophysical radiation transport modeling.

Abstract

We present a realizability-preserving numerical method for solving a spectral two-moment model to simulate the transport of massless, neutral particles interacting with a steady background material moving with relativistic velocities. The model is obtained as the special relativistic limit of a four-momentum-conservative general relativistic two-moment model. Using a maximum-entropy closure, we solve for the Eulerian-frame energy and momentum. The proposed numerical method is designed to preserve moment realizability, which corresponds to moments defined by a nonnegative phase-space density. The realizability-preserving method is achieved with the following key components: (i) a discontinuous Galerkin (DG) phase-space discretization with specially constructed numerical fluxes in the spatial and energy dimensions; (ii) a strong stability-preserving implicit-explicit (IMEX) time-integration method; (iii) a realizability-preserving conserved to primitive moment solver; (iv) a realizability-preserving implicit collision solver; and (v) a realizability-enforcing limiter. Component (iii) is necessitated by the closure procedure, which closes higher order moments nonlinearly in terms of primitive moments. The nonlinear conserved to primitive and the implicit collision solves are formulated as fixed-point problems, which are solved with custom iterative solvers designed to preserve the realizability of each iterate. With a series of numerical tests, we demonstrate the accuracy and robustness of this DG-IMEX method.

Figures (19)

• Figure 1: Flowchart depicting how the Lemmas introduced in Section \ref{['sec:RealizabilityProof']} build upon each other to prove Theorem \ref{['thm:realizable_scheme']}. Realizability of the moment conversion process guarantees realizability-preserving properties that can be derived from our numerical fluxes introduced by the DG discretization. These properties, along with the set $\mathcal{R}$ being a convex cone, ensure the explicit update of the conserved moments maintains realizability. Then the realizability-preserving collision solver guarantees the implicit update maintains realizability. Finally, this allows us to conclude that updating $\mathbf{U}_{\mathbf{K}}^{n}$ to $\mathbf{U}_{\mathbf{K}}^{n+1}$ preserves the realizability of the conserved moments.
• Figure 2: Top left: Iteration counts for the proposed Picard iteration method for the moment conversion problem using $\lambda' = \lambda/W = [W(1+v)]^{-1}$. Top right: Iteration counts Newton's method when solving the moment conversion problem. Bottom left: Picard iteration using $\lambda' = (1+v)^{-1}$. Each square represents the average iteration count of $100$ randomly generated samples of $\mathbf{M}$ at each $(v,\mathsf{h})$. White squares indicate Picard iteration failed to converge within $10000$ iterations for at least one of the $100$ values of $\mathbf{M}$ tested for that specific pairing of $(v,\mathsf{h})$. Bottom right: Number of nonrealizable iterates produced by Picard iteration when using a nonrealizable $\lambda' = (1+v)^{-1}$. Each square represents the average number of nonrealizable iterates before the convergence criteria are satisfied for the random samples of $\mathbf{M}$. White squares have a value of zero, indicating realizability is maintained.
• Figure 3: Convergence of the method on the streaming sine wave problem for linear (black) and quadratic (red) elements. The dashed lines are order reference lines.
• Figure 4: The left panel shows the results of the first diffusion test, comparing the numerical (open circles) and approximate analytic (solid lines) solutions at times $t=0,10,20,30$. The solutions are shifted so the peaks are aligned at $x^1=1$. The right panel shows the results of the second diffusion test, comparing numerical and reference solutions at times $t=0,1,2$.
• Figure 5: Steady state solutions ($t=20$) for the streaming Doppler shift problem with various $v_{\mathrm{max}}\in\{0.0,0.1,0.3,0.6,0.9\}$. In the left panel, the spectra are plotted at $x^1=5$, with open circles representing cell centers in the $\varepsilon$-direction. In the right panel, the $\mathrm{RMS}$ energy defined in Eq. \ref{['eq:rmsEnergy']} is plotted versus $x^{1}$, with open circles representing every fourth cell center in the $x$-direction. In both panels, solid lines represent the analytical solution, while dotted lines with open circles represent the numerical solution.

Theorems & Definitions (50)

• Remark 1
• Lemma 1
• proof
• Corollary 1
• Remark 2
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3