用于中微子输运的多维广义相对论玻尔兹曼求解器：实现、离散化与优化

中文摘要 AI 辅助

我们介绍了使用有限体积法实现的多维广义相对论玻尔兹曼求解器，用于中微子输运。我们将广义相对论磁流体动力学求解器Gmunu扩展以离散化完整的6维相空间。我们在随动系中以球坐标离散动量空间，并在实验室系中以笛卡尔、柱坐标或球坐标离散位置空间。与M1格式类似，我们将相互作用核展开至勒让德级数的一阶。我们提出了一种离散化方案，确保在空平坦时空、多维度及多种坐标系下，数守恒与非守恒公式之间的一致性，并在1D测试中，使用20个能量箱将能量守恒控制在约1%以内。我们讨论了针对刚性源项的隐式求解器的优化，以最小化其计算成本。特别地，我们引入了一种利用核的勒让德展开来降低问题维度的方法。在1D核心坍缩超新星快照中，当包含能量和物种耦合相互作用时，与全矩阵LU方法相比，我们的方法在14个角度箱下实现了300倍的加速比。最后，我们在标准测试案例上验证了我们的实现，并报告了在空间、能量和传播角度上的二次收敛。我们将我们的求解器与M1格式在简化的1D配置中进行了比较，发现在弛豫测试案例中，自由流光度差异约为10%，我们主要将其归因于M1闭合关系。另一方面，两种方法的平均能量一致。我们在核心坍缩超新星测试案例上比较了两种方法，发现在核心反弹时刻，使用20个能量箱时，流体变量剖面总体吻合良好。

英文摘要

We present the implementation of a multidimensional general-relativistic Boltzmann solver for neutrino transport using the finite volume method. We extend the general-relativistic magnetohydrodynamics solver \texttt{Gmunu} to discretize the full $6$D phase space. We discretize the momentum space in spherical coordinates in the comoving frame, and the position space in the lab frame in Cartesian, cylindrical or spherical coordinates. As in the M1 scheme, we expand the interaction kernels up to first order in a Legendre series. We present a discretization scheme that ensures consistency between the number-conservative and non-conservative formulations in empty flat spacetime, multiple dimensions and coordinate systems, conserves energy to $\sim 1\%$ with $20$ energy bins in 1D tests. We discuss optimizations of the implicit solver for stiff source terms to minimize its computational cost. In particular, we introduce a method that leverages the Legendre expansion of the kernels to reduce the dimensionality of the problem. In a 1D core-collapse supernova snapshot, our method yields a speed-up factor of $300$ compared to a full-matrix LU method for $14$ angular bins when including energy- and species-coupling interactions. Finally, we validate our implementation on standard test cases and report quadratic convergence in space, energy and propagation angles. We compare our solver to the M1 scheme in simplified 1D configurations and find differences of about $10\%$ in the free-streaming luminosities in relaxation test cases, which we attribute primarily to the M1 closure relation. The average energies, on the other hand, agree between the two methods. We compare both methods on a core-collapse supernova test case and find an overall good agreement in the fluid variables' profiles at the time of core bounce for $20$ energy bins.

发表机构

• KU Leuven（荷语鲁汶大学）
• California Institute of Technology（加州理工学院）
• University of California, Berkeley（加州大学伯克利分校）
• University of Tennessee, Knoxville（田纳西大学诺克斯维尔分校）

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