[Paper Review] A Dispersion Minimized Mimetic Method for Cold Plasma

Vrushali A. Bokil, Vitaliy Gyrya+1 more|arXiv (Cornell University)|Apr 5, 2016

Electromagnetic Simulation and Numerical Methods1 references3 citations

TL;DR

This paper proposes a dispersion-minimized mimetic finite difference (MFD) method for Maxwell's equations in cold plasma using exponential time differencing (ETD) and a novel m-adaptation optimization. By applying m-adaptation to a generalized mass-lumping MFD scheme in space and ETD in time, the method achieves fourth-order numerical dispersion error—reducing error by two orders of magnitude compared to standard second-order schemes—verified numerically and theoretically.

ABSTRACT

In this paper we consider the lowest edge-based mimetic finite difference (MFD) discretization in space for Maxwell's equations in cold plasma on rectangular meshes. The method uses a generalized form of mass lumping that, on one hand, eliminates a need for linear solves at every iteration while, on the other hand, retains a set of free parameters of the MFD discretization. We perform an optimization procedure, called m-adaptation, that identified a set of free parameters that lead to the smallest numerical dispersion. The choice of the time stepping proved to be critical for successful optimization. Using exponential time differencing we were able to reduce the numerical dispersion error from second to fourth order of accuracy in mesh size. It was not possible to achieve this order of magnitude reduction in numerical dispersion error using the standard leapfrog time stepping. Numerical simulations independently verify our theoretical findings.

Motivation & Objective

• To reduce numerical dispersion errors in electromagnetic wave simulations within cold plasma media, which are critical for long-time, electrically large domain simulations.
• To extend the m-adaptation technique—previously successful in vacuum—to linear dispersive media like cold plasma, where the problem is non-trivial due to added dynamics.
• To identify optimal free parameters in a family of mimetic finite difference schemes that minimize numerical dispersion while maintaining explicit time stepping.
• To demonstrate that exponential time differencing (ETD) enables higher-order dispersion reduction, unlike standard leapfrog time stepping.
• To develop a method with smaller stencil size than existing fourth-order methods, improving computational efficiency.

Proposed method

• A lowest-order edge-based mimetic finite difference (MFD) method is used for spatial discretization of Maxwell’s equations in cold plasma on rectangular meshes.
• A generalized form of mass lumping is applied to eliminate the need for linear solves at each time step, preserving free parameters for optimization.
• The m-adaptation procedure optimizes these free parameters to minimize numerical dispersion, using a dispersion relation analysis in the Fourier domain.
• Exponential time differencing (ETD) is employed for time integration, enabling exact integration of stiff linear terms and higher-order accuracy.
• The resulting Exponential Time Mimetic Finite Difference (ETMFD) scheme achieves fourth-order numerical dispersion error, unlike the base second-order error in standard ETMFD schemes.
• The method maintains explicit staggering of E and J fields from H, avoiding semi-implicit time averaging.

Experimental results

Research questions

• RQ1 Can m-adaptation be successfully extended from vacuum to cold plasma models, which include an auxiliary differential equation for polarization current?
• RQ2 Why does leapfrog time stepping fail to reduce numerical dispersion beyond second order in the MFD family for cold plasma, while ETD succeeds?
• RQ3 What is the optimal set of free parameters in the generalized mass-lumping MFD scheme that minimizes numerical dispersion in cold plasma?
• RQ4 Does the ETMFD method achieve fourth-order numerical dispersion error, and can this be verified independently through simulations?
• RQ5 How does the stencil size of the optimized ETMFD method compare to other fourth-order methods in the literature?

Key findings

• The m-adaptation procedure successfully reduces numerical dispersion error from second to fourth order in mesh size when using exponential time differencing (ETD).
• The ETMFD method achieves fourth-order convergence in numerical dispersion error, as confirmed by both theoretical analysis and numerical simulations.
• The optimal free parameters and local mass matrices in the ETMFD scheme are identical to those in the vacuum case, indicating structural consistency across models.
• The ETMFD method maintains a compact stencil size, offering computational advantages over other fourth-order methods that rely on higher-order base discretizations.
• Standard leapfrog time stepping fails to support higher-order dispersion reduction via m-adaptation in the cold plasma model, highlighting the critical role of time integration choice.
• Numerical simulations of special solutions show fourth-order convergence in L²-error for both electric field and current density, validating the theoretical findings.