AI 中文总结
该研究提出DF-ADI格式，用有理映射替代DF格式的矩阵指数，速度提升10-32倍，可精确守恒离散质量，适用于Fokker–Planck等方程，强交叉扩散区仅经验保正。
AI 中文摘要
一篇姊妹论文[ItkinDF2026]提出了适用于各向异性Fokker–Planck方程的对角蛙（Diagonal Frog, DF）保正格式，该格式在每个方向子步中使用Krylov方法计算矩阵指数，而这正是计算成本的主要来源。将该矩阵指数替换为有理映射$r(\gamma L)$可将子步简化为带状求解，但此时保正性论证不再适用。我们证明，对于最终指数正的生成元，在大步长下$r(\gamma L)$的逐元素符号由单个数值决定，即其在无穷刚性模态上的取值$r(\infty)$。当$0\le r(\infty)<1$时，非负性在一个可计算阈值以上成立；当$r(\infty)<0$时，非负性最多在一个有界区间上成立，在我们的所有测试中该区间要么为空要么极窄。该准则排除了$r(\infty)=-1$的Crank–Nicolson（梯形）方法，并选择了次对角Padé$(0,2)$方法，该方法为二阶、L稳定且在显式阈值以上可证正。由此得到的DF-ADI格式每步计算成本为$O(N)$，保持隐式分解的混合导数不变，在空间和时间上均为二阶，且能精确守恒离散质量。然而，在强交叉扩散区域，方向因子要求的步长大于混合导数允许的范围，因此该复合二阶格式仅在经验上是正的，此时该准则用于区分行为良好的乘性因子与稳定校正格式，而非保证保正性。在我们的测试中，与Krylov矩阵指数相比，该格式在匹配精度下运行速度快10至32倍，且加速比随网格规模增大而提升。我们将该构造扩展到了倒向Kolmogorov方程和跳扩散模型。
英文摘要
A companion paper \cite{ItkinDF2026} introduced the Diagonal Frog (DF) positivity-preserving schemes for anisotropic Fokker--Planck equations, advancing each directional substep by a Krylov-computed matrix exponential, which dominates the cost. Replacing that exponential by a rational map $r(γL)$ reduces the substep to a banded solve, but the positivity argument no longer applies. We prove that for eventually exponentially positive generators the entrywise sign of $r(γL)$ at large steps is decided by a single number, the value $r(\infty)$ taken on infinitely stiff modes. Nonnegativity holds above a computable threshold when $0\le r(\infty)<1$, and at most on a bounded interval, empty or vanishingly narrow in all our tests, when $r(\infty)<0$. The criterion rejects the Crank--Nicolson (trapezoidal) method, where $r(\infty)=-1$, and selects the subdiagonal Padé$(0,2)$ method, which is second order, L-stable and provably positive above an explicit threshold. The resulting DF-ADI scheme costs $O(N)$ per step, keeps the implicit factorized mixed derivative unchanged, is second order in space and time, and conserves discrete mass exactly. In the strong cross-diffusion regime, however, the directional factors demand a step larger than the mixed derivative permits, so the composite second-order scheme is only empirically positive there, and the criterion serves to discriminate the well-behaved multiplicative factors from the stabilizing-correction schemes rather than to guarantee positivity. Against the Krylov exponential it runs ten to thirty-two times faster at matched accuracy in our tests with the gain growing with the mesh. We extend the construction to the backward Kolmogorov equation and to jump-diffusion models.
Comments38 pages, 14 tables, 5 figures
