A High-Accuracy Numerical Homogenization Framework for Quasiperiodic Hamilton--Jacobi Equations

Abstract

In this work, we develop an accurate numerical homogenization framework 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 approximation with the finite points recovery 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.