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Two Frameworks and their Fourth Order Implicit Schemes for Time Discretization of Maxwell's Equations

Archana Arya, Kaushik Kalyanaraman

TL;DR

The paper addresses structure-preserving, high-order time integration for Maxwell's equations by formulating a three-field system within a FEEC framework. It introduces two complementary fourth-order implicit schemes: a spatial LF$_{4}$ strategy and a temporal TS$_{4}$ strategy, and proves their stability and convergence for both time semi- and fully discretized problems. The authors establish discrete energy conservation, derive optimal $O((\Delta t)^4)$ temporal accuracy, and demonstrate FEEC-compatible spatial discretization with de Rham spaces, validated by 2D numerical experiments. The work provides a solid foundation for energy-preserving, high-accuracy simulations of Maxwell's equations and offers a pathway to generalize to higher-order schemes and broader linear/quasi-linear systems.

Abstract

Our work is about energy conserving fourth-order time discretizations of a three-field formulation of Maxwell's equations in conjunction with a spatial discretization using higher-order and compatible de Rham finite element spaces. Toward this end, we delineate two broad classes of strategies for general higher-order time discretizations which we term spatial and temporal strategies. We provide a description of these two strategies and develop fourth-order time accurate schemes in the context of our Maxwell's system. However, our description can be used to prescribe similar fourth- or even higher-order time-integration methods for any linear (or quasi-linear) system of time-dependent partial differential equations. Our organizing principle in our proposed two strategies is to Taylor expand the unknown solution in time by assuming sufficient regularity. Then, in the spatial strategy, we use Maxwell's equations themselves to replace the fourth-order time derivatives in an appropriately truncated Taylor expansion with corresponding higher-order spatial derivatives. On the other hand, in the temporal strategy, we simply use higher-order finite difference schemes for the various higher-order time derivative terms in the truncated Taylor approximation. In both cases, we then defer to a standard finite element exterior calculus manner of compatible discretization for the spatial component of the Maxwell's solution. For our proposed schemes corresponding to the two strategies, we show that they are both stable and convergent and provide some validating numerical examples in $\mathbb{R}^2$. Our main contributions are in the development of the fourth-order time discretization methods that are energy conserving using our two outlined strategies and proofs of their convergence for semi- and full-discretizations of our three-field system of Maxwell's equations.

Two Frameworks and their Fourth Order Implicit Schemes for Time Discretization of Maxwell's Equations

TL;DR

The paper addresses structure-preserving, high-order time integration for Maxwell's equations by formulating a three-field system within a FEEC framework. It introduces two complementary fourth-order implicit schemes: a spatial LF strategy and a temporal TS strategy, and proves their stability and convergence for both time semi- and fully discretized problems. The authors establish discrete energy conservation, derive optimal temporal accuracy, and demonstrate FEEC-compatible spatial discretization with de Rham spaces, validated by 2D numerical experiments. The work provides a solid foundation for energy-preserving, high-accuracy simulations of Maxwell's equations and offers a pathway to generalize to higher-order schemes and broader linear/quasi-linear systems.

Abstract

Our work is about energy conserving fourth-order time discretizations of a three-field formulation of Maxwell's equations in conjunction with a spatial discretization using higher-order and compatible de Rham finite element spaces. Toward this end, we delineate two broad classes of strategies for general higher-order time discretizations which we term spatial and temporal strategies. We provide a description of these two strategies and develop fourth-order time accurate schemes in the context of our Maxwell's system. However, our description can be used to prescribe similar fourth- or even higher-order time-integration methods for any linear (or quasi-linear) system of time-dependent partial differential equations. Our organizing principle in our proposed two strategies is to Taylor expand the unknown solution in time by assuming sufficient regularity. Then, in the spatial strategy, we use Maxwell's equations themselves to replace the fourth-order time derivatives in an appropriately truncated Taylor expansion with corresponding higher-order spatial derivatives. On the other hand, in the temporal strategy, we simply use higher-order finite difference schemes for the various higher-order time derivative terms in the truncated Taylor approximation. In both cases, we then defer to a standard finite element exterior calculus manner of compatible discretization for the spatial component of the Maxwell's solution. For our proposed schemes corresponding to the two strategies, we show that they are both stable and convergent and provide some validating numerical examples in . Our main contributions are in the development of the fourth-order time discretization methods that are energy conserving using our two outlined strategies and proofs of their convergence for semi- and full-discretizations of our three-field system of Maxwell's equations.
Paper Structure (11 sections, 8 theorems, 165 equations, 8 figures)

This paper contains 11 sections, 8 theorems, 165 equations, 8 figures.

Table of Contents

  1. Introduction and Preliminaries
  2. Three-field Formulation of Maxwell's Equations
  3. A Few Notes on Finite Element Spaces and Other Preliminaries
  4. Two Strategies for Higher-Order Time Derivatives
  5. Characterization of LF$_4$ Scheme
  6. Time Discretization Stability
  7. Error Estimate for Full Discretization
  8. Characterization of TS$_4$ Scheme
  9. Time Discretization Stability
  10. Error Estimate for Full Discretization
  11. Numerical Results and Future Outlook

Key Result

Theorem 1

ArFaWi2006 Denote by $\Pi_h$ the canonical projection of $\Lambda^k(\Omega)$ onto either $\mathcal{P}_r \Lambda^k(\mathcal{T}_h)$ or $\mathcal{P}_{r+1}^- \Lambda^k(\mathcal{T}_h)$. Let $1 \leq p \leq \infty$ and $(n-k)/p < s \leq r+1$. Then $\Pi_h$ extends boundedly to $W_p^s \Lambda^k(\Omega)$, and

Figures (8)

  • Figure 1: Linear finite elements with LF$_4$: Plots of solutions at different time steps for the problem described in Example 1 of Section \ref{['sec:numerics']} using the LF$_4$ and linear Whitney forms as basis for the FEEC spaces. The solutions for $p$ are not shown due to them being identically equal to $0$. The computed solutions for $E$ and $H$ visually match with the analytical solutions near identically.
  • Figure 2: Quadratic finite elements with LF$_4$: Plots of solutions at different time steps for Example 1 of Section \ref{['sec:numerics']} using LF$_4$ and quadratic Whitney forms as basis for the FEEC spaces. The solutions for $p$ are not shown due to them being identically equal to $0$. The computed solutions for $E$ and $H$ visually match with the analytical solutions near identically.
  • Figure 3: Linear finite elements with TS$_4$: Plots of solutions at different time steps for Example 1 of Section \ref{['sec:numerics']} using TS$_4$ and linear Whitney forms for the FEEC spaces. The solutions for $p$ are again not shown due to them being identically $0$.
  • Figure 4: Quadratic finite elements with TS$_4$: Plots of solutions at different time steps for Example 1 of Section \ref{['sec:numerics']} using TS$_4$ and quadratic Whitney forms for the FEEC spaces.
  • Figure 5: Linear finite elements with LF$_4$: Plots of solutions at different time steps for Example 2 of Section \ref{['sec:numerics']} using the LF$_4$ and linear Whitney forms in FEEC. The computed solutions for $p$, $E$ and $H$ visually match with the analytical solutions near identically.
  • ...and 3 more figures

Theorems & Definitions (14)

  • Theorem 1
  • Lemma 2: Discrete Gronwall, AnAn2019
  • Theorem 3: Discrete Energy Conservation
  • proof
  • Theorem 4: Discrete Error Estimate
  • proof
  • Theorem 5: Full Error Estimate
  • proof
  • Theorem 6: Discrete Energy Conservation
  • proof
  • ...and 4 more