面向刚性输运-弛豫系统的、采用ADER轨迹导数的L稳定序列两阶段四阶方法
An L-Stable Sequential Two-Stage Fourth-Order Method with ADER Trajectory Derivatives for Stiff Transport--Relaxation Systems
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中文总结 AI 辅助
本文提出一种面向刚性输运-弛豫系统的L稳定序列两阶段四阶方法，结合ADER轨迹导数构建守恒近似，通过依次求解两个系统实现四阶L稳定，经多组实验验证了其精度、稳定性与奇异极限特性。
中文摘要 AI 辅助
一种全隐式两阶段四阶双导数时间离散格式此前已作为时间方法被提出。本文通过将守恒型有限体积残差$\u27e8\uc2a4\uc2a2_h$与其离散轨迹导数$\u27e8\uc2a2_h^{\rm tr}=D\u27e8\uc2a4\uc2a2_h\u27e8\uc2a4\uc2a2_h$配对，为刚性输运-弛豫方程完善了该序列积分器。ADER/Cauchy–Kowalevski预测子提供界面状态和物理时间导数；对同一数值通量求导并取共享面差分，可得到守恒近似$\u27e8\uc2a6\uc2a7\uc2a4\uc2a2_h$。对于线性常系数平衡律，尽管导数算子是独立组装而非通过残差矩阵平方得到，$\u27e8\uc2a6\uc2a7\uc2a4\uc2a2_h=\u27e8\uc2a2_h^{\rm tr}=\u27e8\uc2a4\uc2a2_h^2$严格成立。对于非线性离散格式，四阶时间理论适用于$\u27e8\uc2a2_h^{\rm tr}$，而轨迹闭合一致性估计控制ADER近似。两个未知阶段向量通过两个含N个未知量的系统依次求解。完整的时间步为四阶且具有L稳定性；参数$C_q=5/183$抵消了主导逆幂项，将深刚性放大效应从$O(|z|^{-1})$变为$O(|z|^{-2})$。对于固定的相容空间函数空间，快慢分解证明了全步渐近保持算子极限具有$O(\u27e8\uc2a6)$估计，并给出了依赖于初值准备的一致精度分类。线性有限体积、非线性弛豫、一维和二维阻尼、扩散极限以及模态实验，在所述范围内验证了相应的闭合性、精度、稳定性和奇异极限结论。
英文摘要
A fully implicit two-stage fourth-order two-derivative time discretization was introduced previously as a temporal method. This paper closes that sequential integrator for stiff transport--relaxation equations by pairing a conservative finite-volume residual $\mathcal L_h$ with its discrete trajectory derivative $\mathcal G_h^{\rm tr}=D\mathcal L_h\,\mathcal L_h$. An ADER/Cauchy--Kowalevski predictor provides interface states and physical time derivatives; differentiating the same numerical flux and taking shared face differences yields a conservative approximation $\widetilde{\mathcal G}_h$. For linear constant-coefficient balance laws, $\widetilde{\mathcal G}_h=\mathcal G_h^{\rm tr}=\mathcal L_h^2$ exactly, although the derivative operator is assembled independently rather than by squaring the residual matrix. For nonlinear discretizations, the fourth-order temporal theory applies to $\mathcal G_h^{\rm tr}$, while a trajectory-closure consistency estimate controls the ADER approximation. The two unknown stage vectors are solved successively through two $N$-unknown systems. The completed step is fourth order and L-stable; the parameter $C_q=5/183$ cancels the leading inverse-power term and changes the deep-stiff amplification from $O(|z|^{-1})$ to $O(|z|^{-2})$. For fixed compatible spatial spaces, a slow--fast decomposition proves a full-step asymptotic-preserving operator limit with an $O(δ)$ estimate and gives a preparation-dependent uniform-accuracy classification. Linear finite-volume, nonlinear relaxation, one- and two-dimensional damping, diffusion-limit, and modal experiments verify the corresponding closure, accuracy, stability, and singular-limit claims within their stated scopes.
