A Galerkin Alternating Projection Method for Kinetic Equations in the Diffusive Limit

TL;DR

The paper tackles the computational burden of high-dimensional kinetic equations by introducing the Galerkin Alternating Projection (GAP) scheme within Dynamical Low-Rank Approximation (DLRA). GAP uses a two-step process that alternates projections to update angular and spatial low-rank factors, with rigorous local and global error bounds and proven asymptotic-preserving behavior for the Radiative Transfer Equation, ensuring correct diffusion limits as $\varepsilon \to 0$. For the RTE, the method yields a diffusion equation $\partial_t \rho = \tfrac{1}{3} \partial_{xx} \rho$ in the limit, while maintaining a low-rank representation that corresponds to moments, and it achieves CFL-free time stepping via exponential integrators. Numerical results confirm AP across regimes, demonstrate robustness and efficiency, and show that GAP can outperform traditional micro-macro or fully implicit DLRA schemes by avoiding stiffness constraints and well-prepared data requirements. The approach provides a PDE-oriented DLRA framework with reduced computational cost and clear links between low-rank factors and physical moments, making it attractive for large-scale kinetic simulations.

Abstract

The numerical approximation of high-dimensional evolution equations poses significant computational challenges, particularly in kinetic theory and radiative transfer. In this work, we introduce the Galerkin Alternating Projection (GAP) scheme, a novel integrator derived within the Dynamical Low-Rank Approximation (DLRA) framework. We perform a rigorous error analysis, establishing local and global accuracy using standard ODE techniques. Furthermore, we prove that GAP possesses the Asymptotic-Preserving (AP) property when applied to the Radiative Transfer Equation (RTE), ensuring consistent behavior across both kinetic and diffusive regimes. In the diffusive regime, the K-step of the GAP integrator directly becomes the limit equation. In particular, this means that we can easily obtain schemes that even in the diffusive regime are free of a CFL condition, do not require well prepared initial data, and can have arbitrary order in the diffusive limit (in contrast to the semi-implicit and implicit schemes available in the literature). Numerical experiments support the theoretical findings and demonstrate the robustness and efficiency of the proposed method.

Figures (2)

• Figure 1: Left: GAP numerical approximation for representative values of $\varepsilon$, including the AP-limit. Right: Relative $L^2$ error between the GAP approximation and the discrete AP-limit as a function of $\varepsilon$. The parameters used are $N_x = 1000$, $N_\mu = 100$, rank $r = 5$, and time step $\Delta t = 0.1$. The exponential of the linear operators $\mathcal{A}_L$ and $\mathcal{A}_K$ is computed using the expmv function.
• Figure 2: Left: Pointwise absolute error between the reference solution and the low-rank approximation at final time $T = 1$ for the scaled radiative transfer equation. The low-rank solution is computed with rank $r = 10$ and time step $\Delta t \approx 10^{-4}$, using expmv for time integration. Right: Convergence of the relative $L^2$ error as a function of the time step size $\Delta t$, illustrating first-order accuracy of the scheme in the kinetic regime. The dashed line shows the reference slope $\mathcal{O}(\Delta t)$, while the horizontal line indicates the saturation level associated with the $(r+1)$th singular value $\sigma_{11}(F(T))$ of the reference solution.

Theorems & Definitions (13)

• Lemma 2.1
• Proof 1
• Lemma 3.1: PSI Local error kieri2016discretized
• Lemma 3.2
• Proof 2
• Theorem 3.3: GAP Local error
• Proof 3
• Corollary 3.4: GAP Global Error
• Theorem 4.1: AP of RTE
• Theorem 4.2: AP for GAP