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1. 一种与 WKB 相关的用于描述振荡系统的微分方程的时间步进格式

A WKB-related time-stepping scheme for differential equations describing oscillatory systems

作者: Juliane Rosemeier, Rupert Klein

类别: math.NA

发布日期: 2026-06-11

arXiv ID: 2606.13807v1

摘要翻译: 本文针对具有良好分离尺度的多尺度振荡系统微分方程提出一种新颖的时间步进格式, 其中尺度分离由一个小的参数 $ε$ 控制。该时间步进方法与多模态 WKB 近似相关, 并依赖于本文导出的变量变换。分析表明, 在变换后的形式中, 主导阶振荡要么被消除, 要么仅出现在更高渐近阶。该方法被应用于常微分方程, 包括著名的 van der Pol 振子。我们通过数值实验研究了该方法在不同参数范围下的精度, 特别是当 $ε$ 递减时的表现, 并考察了随着 $ε$ 减小如何调整数值格式的参数。在所展示的数值测试中, 当 $ε$ 减小时, 计算成本保持有界。

Comment: 无

链接: http://arxiv.org/abs/2606.13807v1

1. 研究对象和背景

本文研究一类描述具有良好分离尺度的振荡系统的多尺度常微分方程, 其一般形式为

$$\dot{y} = \frac{1}{\epsilon} L y + N(y),$$

其中 $y \in \mathbb{R}^d$, $0 < \epsilon \ll 1$ 为小参数, $L$ 是反对称矩阵 (代表快线性振荡), $N(y)$ 是光滑的非线性项且满足 $N(0)=0$. 这类方程出现在地球物理与天体物理流体动力学、经典与量子力学、等离子体物理、电磁波传播等诸多领域, 是研究波型偏微分方程数值方法的自然起点. 典型例子包括 van der Pol 振子和带有零特征值的弱非线性系统.

数值上面临的核心困难在于方程的振荡刚度: 线性部分 $\frac{1}{\epsilon}L y$ 诱导出频率为 $O(1/\epsilon)$ 的快速振荡, 标准显式时间积分器 (如 Runge–Kutta 方法) 必须采用 $\Delta t \sim \epsilon$ 的小步长才能稳定并达到给定精度, 导致计算量随 $\epsilon$ 减小而急剧增长. 这一问题在过去二十余年中受到广泛关注, 已有多种方法被提出: 多尺度方法 ([34, 3, 2]) 试图通过尺度分离得到慢变幅值方程; 积分因子与指数积分器 ([16, 5]) 解析地处理线性刚性项, 但对非线性部分仍以小步长积分; 分裂方法如 Strang 分裂 ([33]) 将线性与非线性部分分开处理; 调制傅立叶展开 ([14, 13]) 以及异质多尺度方法 (HMM [6, 32]) 在短时间窗口上进行微模拟以提取慢变量; 时间并行算法如 Parareal ([24]), MGRIT ([8]) 和 PFASST ([7]) 试图通过并行化加速, 但对振荡和双曲型问题常遇到困难 ([10, 11]). 然而, 这些方法或者依赖于线性化相位近似, 或者要求预先解析地推导出慢变量方程, 或者并行效率有限.

本文试图填补的空白是: 利用非线性 WKB 展开的思想, 构造一种数据驱动的时间步进格式, 该格式通过短时间窗口的微模拟 (标准显式积分) 自动提取解的慢变幅值和相位信息, 进而实现跨越多个振荡周期的大步长推进, 而无需显式地使用变换后的右端方程. 该方法的优势在于: (i) 允许相位函数具有非线性慢变部分, 与传统线性相位假设相比更能捕捉真实动力学; (ii) 计算成本不随 $\epsilon$ 减小而发散; (iii) 直接基于原方程的标准子程序, 易于实现. 论文以两个典型 ODE 系统 (零特征值系统和 van der Pol 振子) 展示方法的构造、分析及数值性能.

2. 主要定理或主要结果

本文的主要结果包含理论分析与数值验证两个层面.

理论方面: 作者通过引入与 WKB 相关联的变量变换

$$y(t) = a_0(t) r_0 + \sum_{j=1}^{J} \left[ a_j(t) e^{\frac{i}{\epsilon} (\mu_j t + \epsilon \psi_j(t))} r_j + \text{c.c.} \right],$$

其中 $\{\pm i \mu_j\}$ 是 $L$ 的非零特征值, $r_j$ 为对应的特征向量, $a_j(t),\psi_j(t)\in\mathbb{R}$, 将原系统 (1) 转化为幅值–相位坐标. 并对两种具体情形进行了分析:

  1. 零特征值系统 (7): $L$ 有零特征值, $N(y) = -\lambda \|y\|^2 y$, $\lambda>0$. 在此变换下, 快时间尺度 $t/\epsilon$ 被完全消去, 得到不含 $\epsilon$ 的方程:

    $$\dot a_1 = -\lambda (2 a_1^2 + a_0) a_1,\quad \dot \psi_1 = 0,\quad \dot a_0 = -\lambda (2 a_1^2 + a_0) a_0.$$

    此外, 若初值满足 $a_0>0,\,a_1>0$, 则有守恒量

    $$\frac{a_1(t)}{a_0(t)} = \text{const}.$$

    Lyapunov 分析表明原点为全局渐近稳定的平衡点.

  2. van der Pol 振子 (17): 经尺度化后得到 $L=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$, $N(y)=(0,\,(1-y_1^2)y_2)^T$. 变换后, 幅值 $a$ 和相位 $\phi = \mu t/\epsilon + \psi$ 满足

    $$\dot a = \frac12 \begin{pmatrix}0\\ g(y)\end{pmatrix}\cdot \left( e^{-\frac{i}{\epsilon}\phi} r + e^{\frac{i}{\epsilon}\phi} \bar r \right),$$

    $$\dot \phi = 1 - \frac{i\epsilon}{2a} \begin{pmatrix}0\\ g(y)\end{pmatrix}\cdot \left( e^{-\frac{i}{\epsilon}\phi} r - e^{\frac{i}{\epsilon}\phi} \bar r \right).$$

    采用双尺度展开 $a = a^{(0)}(t) + \epsilon a^{(1)}(t,\tau) + \cdots$ 及 $\psi = \psi^{(0)}(t) + \epsilon \psi^{(1)}(t,\tau) + \cdots$, 得到主导阶:

    $$\dot a^{(0)} = \frac12 a^{(0)} - \frac14 (a^{(0)})^3,\qquad \dot \psi^{(0)} = 0.$$

    与零特征值情形不同, 此处的快振荡并未完全消除, 而是出现在 $O(\epsilon)$ 及更高阶.

数值方面: 提出了与 WKB 相关的时间步进算法 (见 Algorithm 1 和 2). 其核心步骤为: 在每个长步 $[n\Delta T,\,(n+1)\Delta T]$ 的起点附近, 取一个小窗口 $[t_n, t_n+L]$ (长度 $L = O(N\epsilon)$, $N$ 为少量振荡周期数), 用经典四阶 Runge–Kutta (RK4) 以小步长 $\Delta t$ 求解原系统 (1); 从该窗口数据中提取幅值与相位 (通过对解数据拟合或直接利用变换关系); 建立外推模型 (如线性或低阶多项式) 将慢变量外推到 $t_{n+1}$; 最后重构该时刻的原变量 $y$. 数值实验证实:

  • 对于零特征值系统, 方法能精确重现慢演化, 且大步长 $\Delta T$ 可选择为远大于振荡周期的值, 计算成本主要由短窗口积分决定;
  • 对于 van der Pol 系统, 尽管高阶残余振荡存在, 通过适当调整窗口长度 $L$ 和长步 $\Delta T$, 仍可获得令人满意的精度;
  • 关键结论: 在测试的参数范围内, 当 $\epsilon$ 减小时, 计算成本保持有界, 即不随 $\epsilon\to0$ 而发散, 这与传统方法形成鲜明对比.

上述结果表明, 所提方法有效缓解了振荡刚度带来的计算瓶颈.

3. 研究方法、关键技术和核心工具

本文的核心创新在于将 WKB 型多模态展开与数据驱动的时间步进相结合, 其方法论细节可分为以下要点.

1. 非线性相位–幅值变换
变换 (5) 是工作的基础. 与标准的变分法或调制展开不同, 作者在极坐标下引入时变幅值 $a_j(t)$ 和时变相位修正 $\psi_j(t)$, 形成了完整的坐标表示. 这一表示无需假设解仅含单一频率, 可自然容纳多个特征值, 且相位 $\phi_j = \mu_j t/\epsilon + \psi_j(t)$ 允许非线性的慢漂移, 这是相对于常见的线性相位假设 ([14, 13]) 的重要推进. 即使对于 van der Pol 这类非线性系统, 该变换仍可精确表示解, 无逼近误差.

2. 变换后方程的推导与分析
对零特征值系统, 利用 $L$ 的反对称性和特征向量的正交性, 将 $N(y)$ 投影到各模态上, 得到不含 $\epsilon$ 的闭式方程. 关键在于 $N(y)$ 在特征基下的表达中, 快振荡因子 $e^{\pm i t/\epsilon}$ 恰好相消, 这源于 $N$ 的旋转对称结构. 证明中通过 Lyapunov 函数 $V(y)=\frac12\|y\|^2$ 给出了轨线收敛性的独立验证, 确保长期行为合理.

对 van der Pol 系统, 将非线性项 $g(y)$ 向复特征向量 $r$ 和 $\bar r$ 投影, 分离实虚部得到 (21)–(22). 随后引入双尺度渐近展开 (23)–(25), 并利用 $\tau = t/\epsilon$ 的周期性在快速变量上作平均, 得到主导阶方程 (26)–(27). 这一过程类似于 Krylov–Bogoliubov 平均化, 但基于 WKB 变量, 从而自然地提取慢流形. 分析显示, $\psi^{(0)}_t=0$ 表明主导相位仅包含线性增长, 相位修正仅在 $O(\epsilon)$ 出现快速振荡.

3. 数据驱动的 WKB 时间步进算法
算法的关键障碍在于: 实际执行时并不显式求解变换后的方程, 而是完全依赖原坐标下的短时间微模拟. 这要求从数值解中可靠地提取 $a_j$ 和 $\psi_j$, 即使在 van der Pol 情形中它们被高阶快速振荡污染. 作者通过以下技术解决:

  • 短时间窗口积分: 使用 RK4 以 $\Delta t$ (满足振荡分辨) 积分 $L=N\times (2\pi\epsilon)$ 时间, 一般取 $N\approx2\sim5$. 窗口内的解被用作提取慢变量的样本.
  • 幅值与相位的数值提取: 利用变换关系 (10) 或 (20) 的逆, 从数值解中直接计算 $a_j$ 和 $\psi_j$. 由于高阶振荡可能残留, 可采用窗口内拟合 (如最小二乘拟合正弦函数或平滑) 来获得代表慢变量的值. 文中附录 D 给出了伪代码, 其中对零特征值系统用显式公式计算 $a_0, a_1, \psi_1$, 对 van der Pol 系统用类似方法.
  • 外推策略: 获得窗口内的慢变量后, 采用线性外推或更复杂的外推模型将其扩展到下一长步起点. 外推阶数根据慢变量演化的光滑程度选取; 对于零特征值系统由于方程完全慢, 低阶外推即可; 对于 van der Pol, 考虑到高阶残余振荡可能引入误差, 外推步长 $\Delta T$ 需通过数值实验与 $\epsilon$ 适配.

4. 参数选择与适应性
文中通过数值实验探讨了 $\Delta T$ 和 $L$ 如何随 $\epsilon$ 调整以平衡精度与效率. 在零特征值情形, $\Delta T$ 可取 $O(1)$, $L$ 可取约 2–5 个振荡周期; 在 van der Pol 情形, 由于高阶振荡未被消除, $\Delta T$ 不能任意大, 但依然可远大于振荡周期. 实验表明, 当 $\epsilon$ 减小时, 短窗口长度 $L$ 同步缩小, 使得每步的微模拟计算量保持 $O(1)$; 长步数基本不随 $\epsilon$ 变化, 故总计算量有界.

5. 分析框架与工具
理论部分主要用到线性代数的谱分解、基于特征向量的投影、多尺度渐近展开以及 Lyapunov 稳定性理论. 数值部分依托 RK4 的成熟理论, 并结合数据后处理技术. 方法的通用性在于, 它只需掌握 $L$ 的谱信息和 $N$ 的形式, 无需推导复杂解析方程, 这一特色使其易于向更复杂系统推广.

4. 与之前工作的比较

本文工作与若干既有振荡系统数值方法的关系可归纳如下.

1. 与多尺度方法及 HMM 的比较
异质多尺度方法 (HMM [6, 32]) 通常在一个短窗口上执行微模拟, 然后通过时间平均或空间平均提取慢变量, 进而推动宏观求解器. 本文方法同样基于短窗口微模拟, 但关键区别在于: (i) 本文提取的不是平均量, 而是直接表征各振荡模态的幅值和相位, 从而保留了更丰富的动力学信息; (ii) 外推的对象是这些慢变量, 而非宏观平均场; (iii) 本文不需要预先解析推导出慢变量满足的宏观方程, 整个流程是数据驱动的. 因此, 本文方法可视为介于传统的多尺度方法与全数值模拟之间的一种杂交策略.

2. 与指数积分器及积分因子方法的比较
指数积分器和积分因子方法 ([16, 5]) 通过解析地积分线性部分 $\frac{1}{\epsilon}L$ 来缓解刚性, 而对非线性部分依然采用小步长积分. 这类方法在处理强非线性或非线性与线性耦合紧密时可能仍需较小步长. 本文方法通过变换将线性与非线性同时变换到慢坐标系, 完全消除了 (或大幅减弱了) 快速时间尺度, 使得非线性部分也仅在大步长上离散. 代价是需要额外的数据提取和外推步骤, 但实验表明总体效率随 $\epsilon$ 减小而优势更明显.

3. 与调制傅立叶展开及 WKB 相关方法的比较
调制傅立叶展开 ([14, 13]) 与经典的 WKB 方法一脉相承, 通常将解写成 $y(t)=\sum_k a_k(t) e^{i\omega_k t/\epsilon}$ 的形式, 其中 $\omega_k$ 是固定频率, $a_k(t)$ 为慢变幅值. 这类方法的成功依赖于线性化相位 (频率不随时间改变). 本文的非线性相位变换允许 $\psi_j(t)$ 修正, 能更准确地捕捉长时漂移. 作者明确指出, 非线性相位函数曾在 [19, 23] 中被考虑, 但本文的独特之处在于通过短时间数值模拟来隐式确定这些相位函数, 而非显式解析求解. 这使得方法适用于那些难以解析推导极限系统的方程.

4. 与时间并行方法的比较
Parareal ([24]) 等时间并行方法在求解振荡问题时面临收敛缓慢甚至发散的困难 ([10, 11]). 本文方法本质上是序列的, 但各长步之间的微模拟完全独立, 因此天然具有时间并行化的潜力 (每个窗口的计算可并发执行). 文中未深入展开并行实现, 但这可能是未来提升性能的一个方向.

5. 方法的适用范围与局限性
当前方法的严格分析与数值实验均局限于 ODE, 但作者强调方程 (1) 的形式具有一般性, 未来将拓展到双曲型 PDE 的离散系统. 与已有工作相比, 本文首次系统地将 WKB 变换用于构造时间步进格式, 并详尽展示了两个典型系统中的性能. 主要局限在于: (i) 要求线性部分 $L$ 的特征结构易于获取 (实反对称矩阵可正交对角化); (ii) 对于不完全消除振荡的系统 (如 van der Pol), 长步 $\Delta T$ 受到残余高频的限制, 精度与效率的平衡需要调节; (iii) 目前缺乏严格的误差分析和收敛性证明, 文中仅通过数值实验进行了验证. 这些局限为后续研究提供了方向.

总体而言, 本文提出的方法推进了振荡系统时间步进的研究: 通过拥抱非线性相位, 成功地将解析洞察与数据驱动结合起来, 实现了对刚性的有效规避, 且计算成本随 $\epsilon$ 减小保持有界.

了解领域论文（摘要翻译）

1. 消失松弛极限下 Jordan-Moore-Gibson-Thompson 方程的间断 Galerkin 近似

Discontinuous Galerkin approximations of the Jordan-Moore-Gibson-Thompson equation in the vanishing relaxation limit

作者: Vanja Nikolić

类别: math.NA

发布日期: 2026-06-12

arXiv ID: 2606.14655v1

链接: http://arxiv.org/abs/2606.14655v1

摘要翻译

中文标题: 消失松弛极限下 Jordan-Moore-Gibson-Thompson 方程的间断 Galerkin 近似

摘要翻译: Jordan-Moore-Gibson-Thompson (JMGT) 方程描述了热松弛介质中的非线性声波传播，在消失松弛极限下趋近于阻尼 Westervelt 方程。我们研究了 JMGT 方程在单纯形网格上的间断 Galerkin 空间离散，并分析了它们在松弛参数下的均匀行为。在实际相关的混合 Neumann 和吸收边界条件下，我们推导了与松弛参数无关的先验误差估计。这些估计使得严格的奇异极限分析成为可能，从而得出半离散 JMGT 近似以线性速率收敛到相应的 Westervelt 压力剖面。这也揭示了精确解在消失松弛极限下的预期行为。对于全离散问题，我们基于将其重述为耦合的二阶/一阶系统，提出了一种 Newmark 型方法。数值实验支持了理论发现，并证明了该方法在小参数区域中的稳健性。

2. Hodge-Laplace问题的原始有限元格式

Primal finite element scheme of the Hodge-Laplace problem

作者: Wenyu Dong, Shuo Zhang

类别: math.NA

发布日期: 2026-06-12

arXiv ID: 2606.14273v1

链接: http://arxiv.org/abs/2606.14273v1

摘要翻译

中文标题: Hodge-Laplace问题的原始有限元格式

摘要翻译: 在本文中，我们构造了非协调有限元空间 $\boldsymbol{V}^{\mathbf{d}\cap\mathring{\boldsymbolδ}}_hΛ^k$ 来逼近 $HΛ^k\cap H^*_0Λ^k$，在单纯形网格上，对于 $n\ge 2$ 和 $1\le k\le n-1$，通过强制相对于分片Whitney空间的伴随连续性而非迹匹配来实现。成立如下关系：令 $\mathbf{d}^k_h$ 和 $\boldsymbolδ_{k,h}$ 分别表示微分和余微分算子的分片作用，$\boldsymbol{\mathfrak{H}}_hΛ^k$ 为FEEC意义下的离散调和形式，则 $\boldsymbol{\mathfrak{H}}_hΛ^k=\{\boldsymbolμ_h\in \boldsymbol{V}^{\mathbf{d}\cap\mathring{\boldsymbolδ}}_hΛ^k:\mathbf{d}^k_h\boldsymbolμ_h=0,\ \boldsymbolδ_{k,h}\boldsymbolμ_h=0\}$，这复刻了非平凡拓扑区域上连续Hodge-Laplace算子的核。该空间并非经典的Ciarlet型有限元空间；然而，一致的离散庞加莱不等式和局部支撑的基函数（支撑在至多两个单元上）得以保证。由此得到的原始格式对光滑数据具有 $O(h)$ 的误差界，在 $s$-正则区域（$0

3. SIDER插值的解析一阶导数

Analytic First Derivatives of SIDER Interpolation

作者: Shingyu Leung

类别: math.NA

发布日期: 2026-06-12

arXiv ID: 2606.13998v1

链接: http://arxiv.org/abs/2606.13998v1

摘要翻译

中文标题: SIDER插值的解析一阶导数

摘要翻译: 球面插值在数值和几何应用中不可或缺，这些应用要求未知量保持在单位球面上。n阶球面插值（SIDER-n）被引入作为球面本质无振荡插值的高阶重构部分，其中重构完全由球面线性插值（SLERP）操作构建，因而精确保持球面约束。本文针对任意阶SIDER曲线推导了解析一阶导数公式。核心观察是，SIDER的递归定义可通过在其SLERP操作的二叉树上直接进行链式法则传播来求导。在推导端点可移动的SLERP全导数之后，我们得到了SIDER-n导数的紧凑递归式，包括插值节点处的简化公式以及连续采样点之间中点处的实用公式。后者在数据样本之间中点评定重构时是相关的，这出现在多种基于高阶重构的数值算法中。显式处理了基本情形SIDER2，并以SIDER3和SIDER4说明递归机制。我们还证明了，导数在每一个重构点——包括采样点和中点——均与球面相切。所得公式通过为球值重构提供微分信息，拓展了原始SIDER/SENO框架，可潜在地用于守恒律和演化问题的高阶有限体积、ENO/WENO、SENO型及相关方法。

4. 在离散化流形上逼近 Whittle-Matérn 场

Approximating Whittle-Matern Fields over Discretized Manifolds

作者: Srinivas Nambirajan

类别: math.NA, cs.LG, stat.ML

发布日期: 2026-06-11

arXiv ID: 2606.13827v1

链接: http://arxiv.org/abs/2606.13827v1

摘要翻译

中文标题: 在离散化流形上逼近 Whittle-Matérn 场

摘要翻译: 马尔可夫 Whittle-Matérn 场已通过具有稀疏精度矩阵的离散高斯马尔可夫随机场 (GMRF) 得到收敛近似，其方法是对两参数族 SPDEs [ (κ^2 - Δ)^{α/2} u = \mathcal{W}, ;; κ\in \mathbb{R}, ; α\in \mathbb{N}. ] 进行有限元逼近。利用离散外微分 (DEC) 分析的最新进展，我们提出了一种不同但密切相关的收敛 GMRF 近似，用于完备无边黎曼流形上的这些 Matérn 场，该流形被离散化为良中心的单纯复形。这种收敛方法 (i) 不依赖于 $α, κ$，因此为整个 $(α, κ)$-族 GMRF 的精度矩阵和协方差矩阵提供了一种通用逼近格式，从而可以推断它们而非猜测；(ii) 固有地对随机场的点态和分片光滑测量进行建模，并且对两者同样逼近良好；(iii) 在计算上独立于所使用的插值函数——如果用同一网格上的另一种合适插值替换一种收敛插值，不会产生额外开销。此外，我们证明，在精确定义下良好连接且体积集中的离散化上，精度矩阵是图拉普拉斯算子的谱函数。我们为该族 Matérn GMRF 提供了低秩近似器，并提及一个用例：通过压缩感知减少建模 GMRF 所需的测量数量。

不相关论文（基本信息）

1. 耦合 Darcy–Forchheimer 与对流-扩散-反应方程的参数鲁棒适定性与离散化

Parameter-robust well-posedness and discretisation for coupled Darcy–Forchheimer and advection-diffusion-reaction equations

作者: Rishi Das, Harsha Hutridurga, Amiya K. Pani, Ricardo Ruiz-Baier

类别: math.NA

发布日期: 2026-06-12

arXiv ID: 2606.14630v1

链接: http://arxiv.org/abs/2606.14630v1

摘要: We adapt the recent theory for unique solvability of perturbed saddle-point problems in Banach spaces to the case of parameter-independent stability bounds. This also constitutes an extension to the Brezzi–Braess theory for parameter-robust stability of perturbed saddle-point problems from the Hilbert to the Banach setting. We apply the new abstract result to the mixed formulation of the advection-diffusion-reaction equation and tackle also its coupling with the Darcy–Forchheimer equations. The complete system is shown to be stable irrespective of the model parameters of permeability, Forchheimer coefficient, and reaction modulation. We discretise the problem with mixed finite element methods and show convergence in appropriately parameter-weighted norms. We also design operator-based preconditioners for the full system, utilising weighted norms that provide robustness with respect to model parameters.

2. 带有广义载荷与可变Dirichlet数据的非线性弹性中数据-解值映射的极小极大神经网络架构

Minmax neural-network architectures for data-to-solution value maps in nonlinear elasticity with generalized loads and variable Dirichlet data

作者: Michael Ortiz

类别: math.NA

发布日期: 2026-06-12

arXiv ID: 2606.14610v1

链接: http://arxiv.org/abs/2606.14610v1

摘要: We study the data-to-solution value map for quasistatic nonlinear elasticity in the linearized-kinematics regime, allowing both generalized loads and variable Dirichlet data. Under standard direct-method hypotheses, the negative minimum potential energy is finite and locally Lipschitz, convex in the load variable, and concave in the Dirichlet datum. Its supporting slopes, and its first variations at differentiability points, are the equilibrium displacement and the Dirichlet reaction. This convex–concave structure leads to a mechanics-preserving saddle minmax architecture in which displacement atoms generate load slopes, reaction atoms generate Dirichlet slopes, and the coupling coefficients are the corresponding trace-reaction pairings. Manufactured samples are produced by prescribing displacement–reaction pairs and computing the associated ambient data and exact value labels. The resulting architecture directly approximates the negative minimum-potential-energy value map and provides mechanical subgradient readouts. Immersed representations and cell-center quadrature make the construction implementable on background grids and geometry-rich domains. We prove uniform convergence on compact data sets with respect to atom enrichment and quadrature refinement, and illustrate the method on elementary examples.

3. 笛卡尔网格上椭圆偏微分方程的高效快速张量积多节点Shepard配点法

Efficient and Fast Tensor-Product Multinode Shepard Collocation for Elliptic PDEs on Cartesian Grids

作者: Anouar El Harrak, Hatim Tayeq, Benaissa Zerroudi

类别: math.NA

发布日期: 2026-06-12

arXiv ID: 2606.14503v1

链接: http://arxiv.org/abs/2606.14503v1

摘要: We introduce a Grid-Based Multinode Shepard Collocation Method (GBMSC) for two-dimensional elliptic boundary value problems on rectangular domains. The method combines Shepard-type partition functions with tensor-product Lagrange interpolation on overlapping local Cartesian subgrids. The Cartesian structure avoids the local unisolvency search required by multinode Shepard constructions on scattered data, while the local character of the approximation yields sparse collocation matrices and moderate conditioning. We establish the main structural properties of the method, including the cardinal property, tensor-product polynomial reproduction, and local interpolation estimates. Numerical experiments for Poisson problems with Dirichlet and mixed boundary conditions confirm the polynomial reproduction property up to round-off accuracy and show regular convergence for smooth non-polynomial solutions. Comparisons with Multinode Shepard Collocation Method (MSC) and Kansa’s collocation methods indicate that the proposed discretization preserves high accuracy with significantly better conditioning than the RBF-based schemes considered.

4. 弱贪婪算法的一个闭式收敛准则

A closed-form convergence criterion for the weak greedy algorithm

作者: Mikhail Novikov

类别: math.NA, math.FA

发布日期: 2026-06-12

arXiv ID: 2606.14467v1

链接: http://arxiv.org/abs/2606.14467v1

摘要: In 2002, V. N. Temlyakov established a criterion for the convergence of the weak greedy algorithm in a Hilbert space for a given weakness sequence $ τ= \{t_1,t_2,\ldots\} $. The criterion requires verifying a certain limiting relation for every nonnegative square-summable sequence. We give an equivalent closed-form criterion: the weak greedy algorithm converges if and only if $ \sum_{n=1}^{\infty}(1+ n\sum_{k=1}^{n}t_k^2 )^{-1/2}t_n^2=+\infty $.

5. 局部高阶时空自适应 MLSDC

Local high order space-time apdaptive MLSDC

作者: Erik Pfister, Matti Schulze, Jörg Stiller

类别: math.NA

发布日期: 2026-06-12

arXiv ID: 2606.14369v1

链接: http://arxiv.org/abs/2606.14369v1

摘要: Building upon the semi-implicit multilevel spectral deferred correction (SI-MLSDC) method introduced by Pfister and Stiller [38], this work presents a 1D space-time adaptive high-order method combining discontinuous Galerkin spectral element discretizations with multilevel spectral deferred corrections. The proposed approach enables genuine arbitrary-order accuracy in space and time while dynamically balancing spatial and temporal discretization errors to reduce computational cost. A key contribution is the development of a novel temporal error estimator that in combination with a spectral error estimator in space provides a reliable basis for adaptive refinement decisions. The error estimator is compared with two alternative refinement criteria to assess their impact on accuracy, computational efficiency and their suitability for complex problems.The performance of the adaptive method is demonstrated for nonlinear conservation laws ranging from Burgers’ equation to the Euler equations. Numerical results show substantial runtime reductions while maintaining the desired accuracy. In particular, significant computational savings are achieved for Burgers’ equation, and challenging benchmark problems such as the Shu-Osher shock-fluctuation benchmark. These results demonstrate the potential of adaptive SI-MLSDC methods for efficient high-order space-time adaptive simulations of complex flow problems.

6. 高焓裂隙地热系统中盐沉淀和多相流的数学建模

Mathematical Modeling of Salt Precipitation and Multi-Phase Flow in High Enthalpy Fractured Geothermal Systems

作者: Micheal B. Oguntola, Omar Duran, Eirik Keilegavlen, Inga Berre

类别: math.NA, math-ph, physics.comp-ph

发布日期: 2026-06-12

arXiv ID: 2606.14328v1

链接: http://arxiv.org/abs/2606.14328v1

摘要: Simulating high-enthalpy fractured geothermal reservoirs is challenging due to the complex coupled processes of non-isothermal, multiphase, multicomponent flow, strongly nonlinear thermodynamics, and the dominant role of fractures. These complexities are amplified by mineral scaling, such as halite precipitation, which can impair reservoir permeability and well productivity. To address this, we present a new compositional flow model based on a persistent set of primary variables (pressure, enthalpy, and overall salt mass fraction). The formulation naturally handles phase transitions without manual switching, enhancing numerical stability. The model integrates a discrete fracture-matrix approach and employs an efficient, robust correlation-based phase-behaviour linearisation of saltwater thermodynamics, replacing expensive on-the-fly phase separation calculations. It incorporates the Kozeny-Carman relation to dynamically model porosity and permeability reduction from halite precipitation. Implemented in the open-source PorePy framework, the model is verified through a 1D salt dissolution benchmark against the established closed-source simulator CSMP++, showing strong agreement across geothermal conditions involving transitions between single- and multi-phase regions. Application to a 2D halite-saturated fractured reservoir with injection and production demonstrates the model’s capability to predict halite precipitation patterns and their impact on permeability damage and energy recovery. Numerical results further show the model’s value in predicting operational challenges such as wellbore blockage and the role of fracture connectivity. The model thus provides an open-source numerical tool for analysing complex heat and mass transport with mineral scaling in high-enthalpy fractured geothermal systems.

7. 一类新的高效线性高阶格式，用于阻尼参数限制放宽的 Landau-Lifshitz-Gilbert 方程

A new class of efficient linear higher-order schemes for the Landau-Lifshitz-Gilbert equation with reduced restriction on the damping parameter

作者: Fukeng Huang, Binghong Li, Xiaoli Li, Jie Shen

类别: math.NA

发布日期: 2026-06-12

arXiv ID: 2606.14316v1

链接: http://arxiv.org/abs/2606.14316v1

摘要: Classical high-order backward differentiation formula (BDF) methods for the Landau-Lifshitz-Gilbert (LLG) equation often suffer from restrictive stability constraints, requiring small time steps and imposing stringent lower bounds on the damping parameter. These limitations become particularly severe for schemes of order higher than three. In this paper, we develop a class of high-order generalized BDF (GBDF) schemes for the LLG equation, including both semi-implicit and fully explicit treatments of the gyromagnetic term. The proposed schemes significantly improve stability properties and substantially relax the damping parameter constraints, but introduce essential difficulty in its analysis compared to the classical BDF schemes. We construct a novel multiplier which enables us to carry out a energy-based error analysis. This approach yields optimal-order error estimates under considerably weaker assumptions on the damping parameter than those required for classical BDF schemes. Numerical experiments are presented to confirm the theoretical results, and demonstrate that the proposed GBDF schemes achieve higher accuracy, enhanced stability, and much wider admissible damping regimes compared to classical high-order BDF methods.

8. 驯服薄片：一种用于在退化四面体网格上进行可靠计算的稳健 TFEM 框架

Taming Slivers: A Robust TFEM Framework for Reliable Computations on Degenerate Tetrahedral Meshes

作者: Antoine Quiriny, Jonathan Lambrechts, Nicolas Moës, Václav Kučera, Jean-François Remacle

类别: math.NA

发布日期: 2026-06-12

arXiv ID: 2606.14301v1

链接: http://arxiv.org/abs/2606.14301v1

摘要: Sliver elements are an intrinsic difficulty of three-dimensional tetrahedral mesh generation and remain costly, and sometimes impractical, to eliminate completely. Although isolated degenerate elements do not necessarily prevent finite element convergence, connected clusters or sheets of slivers may impose artificial constraints on the discrete solution, leading to locking and severe loss of accuracy. In this work, we revisit the effect of slivers from the viewpoint of the finite element solution and propose a robust solver-side treatment based on the Tempered Finite Element Method (TFEM). The method limits the singular contribution of degenerate elements by introducing a lower bound on the Jacobian determinant, which can be interpreted as a vanishing added-volume correction. The resulting formulation prevents the effective element volume from falling below a threshold while preserving the relevant physical modes of the solution. We analyze the stiffness matrices of degenerate tetrahedra, identify the mechanisms responsible for locking in sliver bands, and assess the method on a range of representative physical problems, including incompressible flow, Cahn–Hilliard phase-field dynamics, transient wave propagation, and vibro-acoustic fluid–structure interaction. The numerical results show that TFEM consistently recovers accurate and physically meaningful solutions on meshes for which standard FEM exhibits locking or loss of convergence, providing a simple and broadly applicable alternative to exhaustive geometric sliver removal.

9. 基于群体的优化的算子演算：平均场收敛理论

Operator Calculus for Population-Based Optimization: A Mean-Field Convergence Theory

作者: Pekka Malo, Lauri Viitasaari, Patrik Nummi, Antti Suominen, Ankur Sinha, Olli Tahvonen

类别: math.OC, cs.LG, cs.NE, math.NA, stat.ML

发布日期: 2026-06-12

arXiv ID: 2606.14289v1

链接: http://arxiv.org/abs/2606.14289v1

摘要: Population-based and distributional optimization methods, from evolution strategies and consensus-based optimization to covariance-matrix adaptation and stochastic gradient methods viewed as distributional dynamics, are widely used for nonconvex or black-box problems, yet their convergence analyses remain fragmented across algorithm-specific techniques. We introduce an operator calculus in which a broad class of such methods, after choosing an appropriate state space and, where necessary, augmenting the state by memory or strategy variables, is described as a composition of three elementary operators (mutation, selection, and recombination) acting on probability measures. Under explicit stability and regularity conditions, the composite operator admits a pre-generator whose continuous-time limit is a transport-reaction-jump (TRJ) PDE that preserves the operator splitting. On this foundation we establish a modular Lyapunov principle. If a state-space Lyapunov function both dissipates under the full generator and controls the relevant search-space gauges, then the state-space Lyapunov functional and the induced search errors decay exponentially. The additive generator structure allows dissipation estimates to be assembled operator by operator, providing a toolkit for certifying convergence of composite mean-field algorithms.

10. 使用中位数数字网络算法的通用 $L^2$ 逼近

Universal $L^2$-approximation using median digital-net algorithms

作者: Ziyang Ye, Xiaoqun Wang, Zexin Pan

类别: math.NA

发布日期: 2026-06-12

arXiv ID: 2606.14264v1

链接: http://arxiv.org/abs/2606.14264v1

摘要: We propose a median digital-net algorithm for $L^2$-approximation of non-periodic functions over $[0,1]^s$, inspired by the recently developed median lattice algorithms for the periodic setting. The algorithm requires no smoothness or weight parameters but only a sufficiently large candidate Walsh index set $K$. It proceeds in three stages: generating multiple estimates of the Walsh coefficients in $K$ using independent randomized digital-net samples; taking the respective median of both the estimates and their absolute values; then, based on these median values, identifying the dominant coefficients and constructing a truncated Walsh series as the final approximation. We prove that if the target function has dominating mixed partial derivatives up to order $α$, all having finite Vitali variation of fractional order $λ$, then the algorithm achieves an $L^2$-error of $\mathcal{O}(M^{-α-λ+ε})$ with high probability, where $M$ is the total number of function evaluations and $ε>0$ is arbitrarily small. Furthermore, the implied constant grows at most polynomially in the dimension $s$ under suitable decay conditions on the ANOVA components of the target function. On the implementation side, we provide both parameter-dependent and -independent constructions of the index set $K$, and employ the fast Walsh–Hadamard transform and Gray code to accelerate the algorithm. Numerical experiments support the theoretical analysis and demonstrate that the proposed algorithm remains effective in high-dimensional settings.

11. 具有非线性发生率的广义黑客动力学模型：分析与保正性数值模拟

A Generalized Hacker Dynamics Model with Nonlinear Incidence: Analysis and Positivity-Preserving Numerical Simulation

作者: Manh Tuan Hoang, Hoai Thu Pham, Matthias Ehrhardt

类别: math.NA, math.DS

发布日期: 2026-06-12

arXiv ID: 2606.14191v1

链接: http://arxiv.org/abs/2606.14191v1

摘要: We propose and analyze a generalized compartment model for hacker dynamics in cybersecurity systems. This model is an extension of a recently introduced framework, replacing the bilinear interaction term with a broad class of nonlinear incidence functions. This provides greater modeling flexibility and allows for the description of a wider range of cyber-propagation and information-spreading processes. First, we investigate the qualitative dynamics of the model. We establish the positivity and boundedness of solutions, derive the basic reproduction number, and characterize the existence of hacker-free and hacker-present equilibria. We obtain local and global asymptotic stability results, yielding a complete description of the global dynamics in terms of the basic reproduction number. Next, we develop a second-order, positivity-preserving, nonstandard finite difference (NSFD) scheme for numerical simulations. Unlike many existing NSFD approaches, which are typically first-order accurate, the proposed method achieves second-order convergence while preserving positivity for arbitrary step sizes. Furthermore, the proposed method reproduces the asymptotic stability properties of the continuous model, making it suitable for long-time simulations. Numerical experiments validate the theoretical results and demonstrate the superior accuracy and qualitative performance of the proposed scheme compared to several first-order methods.

12. PINN与有限元求解器的Robin-Neumann耦合：一种Steklov-Poincaré视角及其在带接触的流固耦合问题中的应用

Robin-Neumann Coupling of PINN and FEM Solvers: A Steklov-Poincaré View, with Application to Fluid-Structure Interaction with Contact

作者: Mikel Landajuela

类别: math.NA, cs.LG

发布日期: 2026-06-12

arXiv ID: 2606.14181v1

链接: http://arxiv.org/abs/2606.14181v1

摘要: Physics-informed neural networks (PINNs) are meshless and carry moving geometry and topology change through resampling of collocation points; the finite-element method (FEM) is the workhorse for boundary-fitted discretisations. Coupling the two across a shared interface promises the best of both, yet existing PINN-FEM schemes are validated only empirically. We put the coupling on a domain-decomposition footing: viewing each solver as a Steklov-Poincaré (trace-to-flux) operator, we transfer the classical Dirichlet-Neumann (DN) divergence diagnosis and its Robin-Neumann (RN) cure, including a closed-form, sweep-free interface impedance, and prove a PINN-specific contraction theorem: a trained network realises only a perturbed Steklov operator with a per-step training residual, and RN still contracts, with no shared-eigenbasis hypothesis, to a floor set by the achieved training loss. Because a PINN has no stiffness matrix, we introduce a Fourier-mode interface probe that recovers the network’s resolvable Steklov eigenvalues to within 0.5% and doubles as a diagnostic of the network’s spectral cap. The theory predicts measured PINN-FEM contraction rates to within 7% on 1D and 2D Poisson couplings, and a two-slab analogue of the large-added-mass regime shows RN’s per-mode impedance matching winning decisively where tuned scalar relaxation saturates. We demonstrate the framework on a Stokes/rigid-disc problem with Alart-Curnier contact: the meshless PINN fluid absorbs the topology change at contact by collocation exclusion alone, no remeshing and no cut cells, and the static-equilibrium contact reaction matches the submerged weight to 0.4% under mesh refinement. We quantify remaining limitations: the warm-started PINN drifts off the Stokes manifold over long horizons, and matched FEM-FEM benchmarks attribute pre-impact squeeze-film signatures to PINN under-resolution.

13. 带公共噪声的 McKean–Vlasov 随机微分方程的时间一致误差估计与随机算法

Uniform-in-time error estimates for McKean-Vlasov SDEs with common noise and stochastic algorithms

作者: Yuhang Zhang, Minghui Song

类别: math.PR, math.NA

发布日期: 2026-06-12

arXiv ID: 2606.14170v1

链接: http://arxiv.org/abs/2606.14170v1

摘要: In this work, by construct an asymptotic coupling by reflection, we first explore the uniform-in-time estimate on probability distance for two measure-valued processes induced by a McKean-Vlasov SDE with common noise and an interacting particle system, where the drift terms are dissipative merely in the long distance. As direct applications of this estimate, we establish the uniform-in-time error estimates for the numerical solutions derived via backward/tamed/adaptive Euler-Maruyama methods. Moreover, as another direct application, the uniform-in-time conditional propagation of chaos is quantified.

14. 非线性光学介质中双随机相位加密的选择明文攻击

Chosen-Plaintext Attacks of Double Random Phase Encryption with Nonlinear Optical Media

作者: Yan Cheng, Yiwei Chen, Kui Ren, Nathan Soedjak

类别: physics.optics, math.NA, physics.comp-ph

发布日期: 2026-06-11

arXiv ID: 2606.13946v1

链接: http://arxiv.org/abs/2606.13946v1

摘要: This paper studies an inverse problem in nonlinear optical encryption. We examine chosen-plaintext attacks (CPA) on a nonlinear optical encryption strategy that integrates double random phase encryption (DRPE) into a nonlinear optical propagation model to enhance the security of the combined system. We first demonstrate that the system’s phase information can be decoded from carefully designed differential CPA data. We then demonstrate that the strength of the optical device’s nonlinearity can also be recovered from CPA data, indicating that including this parameter as an additional security key does not enhance protection against CPA attacks, although numerical simulations show that strong nonlinearity still poses significant challenges for CPA attacks. Finally, we provide a stability analysis to demonstrate that small errors in decoded security keys result in only small errors in the decrypted text, even though the encryption process is nonlinear.