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A $P$-Adaptive Hermite Method for Nonlinear Dispersive Maxwell's Equations

Yann-Meing Law, Zhichao Peng, Daniel Appelö, Thomas Hagstrom

TL;DR

This work addresses efficient, high-order simulation of Maxwell's equations in nonlinear dispersive media with Kerr-Lorentz-Raman effects by introducing a $p$-adaptive Hermite method. The approach discretizes space with Hermite interpolation and evolves polynomial coefficients in time via locally defined ODEs, using ADEs to model dispersion and Raman dynamics and avoiding nonlinear solvers. It achieves arbitrary odd-order spatial accuracy, demonstrated through 1D and 2D convergence tests and soliton-like scenarios, while the $p$-adaptive algorithm concentrates computational effort where the solution features are most complex, substantially reducing degrees of freedom. The method remains stable under appropriate sub-stepping, with linear small-system solves at each step and clear pathways for extension to 3D and complex interfaces, offering a scalable tool for nonlinear photonics simulations.

Abstract

In this work, we introduce a novel Hermite method to handle Maxwell's equations for nonlinear dispersive media. The proposed method achieves high-order accuracy and is free of any nonlinear algebraic solver, requiring solving instead small local linear systems for which the dimension is independent of the order. The implementation of order adaptive algorithms is straightforward in this setting, making the resulting p-adaptive Hermite method appealing for the simulations of soliton-like wave propagation.

A $P$-Adaptive Hermite Method for Nonlinear Dispersive Maxwell's Equations

TL;DR

This work addresses efficient, high-order simulation of Maxwell's equations in nonlinear dispersive media with Kerr-Lorentz-Raman effects by introducing a -adaptive Hermite method. The approach discretizes space with Hermite interpolation and evolves polynomial coefficients in time via locally defined ODEs, using ADEs to model dispersion and Raman dynamics and avoiding nonlinear solvers. It achieves arbitrary odd-order spatial accuracy, demonstrated through 1D and 2D convergence tests and soliton-like scenarios, while the -adaptive algorithm concentrates computational effort where the solution features are most complex, substantially reducing degrees of freedom. The method remains stable under appropriate sub-stepping, with linear small-system solves at each step and clear pathways for extension to 3D and complex interfaces, offering a scalable tool for nonlinear photonics simulations.

Abstract

In this work, we introduce a novel Hermite method to handle Maxwell's equations for nonlinear dispersive media. The proposed method achieves high-order accuracy and is free of any nonlinear algebraic solver, requiring solving instead small local linear systems for which the dimension is independent of the order. The implementation of order adaptive algorithms is straightforward in this setting, making the resulting p-adaptive Hermite method appealing for the simulations of soliton-like wave propagation.

Paper Structure

This paper contains 24 sections, 1 theorem, 82 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Hermite Method in One Space Dimension
  4. Hermite Interpolation
  5. System of Ordinary Differential Equations for the Polynomial Coefficients
  6. System of ODEs for the Magnetic Field Coefficients
  7. Systems of ODEs for the Auxiliary Variables Coefficients
  8. A Recursive System of ODEs for the Electric Field Coefficients
  9. The Complete Set of ODEs for the Polynomial Coefficients
  10. Time Evolution
  11. Extension to Multi-Dimensions
  12. Two Dimensional Case
  13. System of ODEs for the Polynomial Coefficients in Two Dimensions
  14. A Note on the Three Dimensional Case
  15. $P$-Adaptive Algorithm
  16. ...and 9 more sections

Key Result

proposition thmcounterproposition

If the matrix $M$ evaluated at $\mathbold{\phi}_{0,0}$, $\mathbold{Q}_{0,0}$ is positive definite then the system of equations for the time derivatives is uniquely solvable. This condition always holds when only the Kerr term is considered, $(\theta = 0)$ and $a\geq0$.

Figures (15)

  • Figure 1: Illustration of the Hermite method to evolve the data at the primal grid node $x_{i+1}$ from $t_{n}$ to $t_{n+1}$. Here the Hermite interpolation and time-evolution procedures are denoted respectively by $\mathcal{I}$ and $\mathcal{T}$.
  • Figure 2: Convergence plots for manufactured solutions problems for $m=1-4$ in one space dimension. Here $\hat{\mathbf{U}}$ is a vector containing all variables.
  • Figure 3: The error in maximum norm as a function of the tolerance $\epsilon_{p_{tol}}$ using the $p$-adaptive algorithm for the manufactured solution problem representing the propagation of a pulse. The left, middle and right plots represent respectively the mesh size $\Delta x = 1/12.5$, $\Delta x = 1/25$ and $\Delta x = 1/50$. Here $\hat{\mathbf{U}}$ is a vector containing all variables.
  • Figure 4: The values of $m$ as a function of the tolerance $\epsilon_{p_{tol}}$ using the $p$-adaptive algorithm for the manufactured solution problem representing the propagation of a pulse. The left, middle and right plots represent respectively the mesh size $\Delta x = 1/12.5$, $\Delta x = 1/25$ and $\Delta x = 1/50$.
  • Figure 5: Values of $m$ as a function of $x$ for different values of tolerance $\epsilon_{p_{tol}}$ using $\Delta x = 1/50$. The left, middle and right plots are for the tolerances $10^{-2}$ , $10^{-4}$ and $10^{-6}$.
  • ...and 10 more figures

Theorems & Definitions (4)

  • remark thmcounterremark
  • remark thmcounterremark
  • proposition thmcounterproposition
  • proof