A Galerkin Alternating Projection Method for Kinetic Equations in the Diffusive Limit
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| author | Ceruti, Gianluca Crouseilles, Nicolas Einkemmer, Lukas |
| author_facet | Ceruti, Gianluca Crouseilles, Nicolas Einkemmer, Lukas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19929 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Galerkin Alternating Projection Method for Kinetic Equations in the Diffusive Limit Ceruti, Gianluca Crouseilles, Nicolas Einkemmer, Lukas Numerical Analysis 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. |
| title | A Galerkin Alternating Projection Method for Kinetic Equations in the Diffusive Limit |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2505.19929 |