发表机构
Wuhan Institute for Math & AI, Wuhan University; Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University; School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSC, Shanghai Jiao Tong University（武汉大学数学与人工智能研究院; 湘潭大学数学与计算科学学院; 上海交通大学数学科学学院自然科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对准周期哈密顿-雅可比方程数值计算的挑战，提出带准周期边界条件的SL-FPR格式，扩展均匀化结果至一般$k>1$情况，经数值实验验证了方法的收敛性与适用性。
AI 中文摘要
本研究中，我们开发了一种精确的数值均匀化框架，用于计算具有形如$H(x,p)=|p|^k/k-f(x)$（$k>1$，其中$f$为准周期函数）的凸哈密顿量的准周期哈密顿-雅可比方程（QHJEs）的有效哈密顿量。在准周期场景下计算有效哈密顿量需要求解定义在整个空间上的QHJEs，其解通常既不具备平移对称性也不具备衰减性，且可能呈现低正则性，这些特征给数值计算带来了重大挑战。为应对这些困难，我们引入了一种准周期边界条件，该条件可在有界域上处理原全空间问题的同时保留边界处的准周期性。随后，我们提出了一种SL-FPR格式，该格式结合了半拉格朗日近似与有限点恢复方法，并为所得格式建立了稳定性和误差估计。我们还将准周期均匀化结果从二次情况扩展到一般$k>1$的情况，并应用所提方法来精确近似对应的有效哈密顿量。数值实验验证了该方法的收敛性和适用性，也证实了扩展后的均匀化结果。
英文摘要
In this work, we develop an accurate numerical homogenization method for computing effective Hamiltonians of quasiperiodic Hamilton--Jacobi equations (QHJEs) with convex Hamiltonians of the form $H(x,p) = |p|^k/{k}-f(x), ~k>1$, where $f$ is quasiperiodic. Computing effective Hamiltonians in the quasiperiodic setting requires solving QHJEs posed on the whole space. Their solutions generally possess neither translational symmetry nor decay and may exhibit low regularity. These features pose substantial challenges for numerical computation. To address these difficulties, we introduce a quasiperiodic boundary condition, which allows the original whole-space problem to be treated on a bounded domain while preserving quasiperiodicity at the boundary. We then propose an SL--FPR scheme that combines a semi-Lagrangian (SL) approximation with the finite points recovery (FPR) method and establish stability and error estimates for the resulting scheme. We also extend the quasiperiodic homogenization result from the quadratic case to general $k>1$ and apply the proposed method to accurately approximate the corresponding effective Hamiltonians. Numerical experiments illustrate the convergence and applicability of the method and validate the extended homogenization results.
