Explicit TE/TM Scheme for Particle Beam Simulations

In this paper we propose an explicit two-level conservative scheme based on a TE/TM like splitting of the field components in time. Its dispersion properties are adjusted to accelerator problems. It is simpler and faster than the implicit version [1]. It does not have dispersion in the longitudinal direction and the dispersion properties in the transversal plane are improved. The explicit character of the new scheme allows a uniformly stable conformal method without iterations and the scheme can be parallelized easily. It assures energy and charge conservation. A version of this explicit scheme for rotationally symmetric structures is free from the progressive time step reducing for higher order azimuthal modes as it takes place for Yee’s explicit method used in the most popular electrodynamics codes.

💡 Research Summary

The paper introduces a novel explicit two‑level conservative finite‑difference time‑domain (FDTD) scheme that is based on a TE/TM‑type splitting of the electromagnetic field components in time. The authors motivate the work by pointing out the limitations of the widely used Yee scheme—namely, longitudinal dispersion, severe Courant‑condition‑driven time‑step reduction for higher azimuthal modes in rotationally symmetric structures, and the lack of exact energy/charge conservation. Implicit schemes can mitigate dispersion but at the cost of solving large linear systems at each step, which makes them computationally expensive and difficult to parallelize.

The proposed method proceeds by alternating updates of the TE (transverse‑electric) and TM (transverse‑magnetic) parts of the fields. In the first half‑step (Δt/2) the electric field’s TE components and the magnetic field’s TM components are advanced; in the second half‑step the complementary components are updated. This “leap‑frog” style two‑level algorithm retains second‑order accuracy in both space and time while keeping the update equations fully explicit. Because each sub‑step involves only curl operations on a subset of the field components, the scheme can be written in a symmetric, conservative form that guarantees exact discrete divergence‑free magnetic fields and exact charge conservation without the need for divergence‑cleaning or correction procedures.

A key technical achievement is the elimination of dispersion in the longitudinal (beam) direction. By carefully choosing the temporal staggering and by matching the phase velocity of the numerical wave to the physical speed of light along the beam axis, the authors ensure that the numerical dispersion relation reduces to ω = ck for any longitudinal wavenumber. In the transverse plane the dispersion is significantly reduced compared with the standard Yee scheme: the Courant limit is relaxed, and the TE/TM ordering removes the anisotropy that normally causes different phase velocities along the grid axes.

For structures with rotational symmetry the authors extend the method to a modal (azimuthal) decomposition. In conventional explicit methods the time step must be reduced proportionally to the azimuthal mode number m because the effective grid spacing in the angular direction shrinks as 1/m. The new TE/TM formulation treats each azimuthal mode independently and retains the same Δt for all m, thereby avoiding the “progressive time‑step reduction” that plagues Yee‑based codes. This property is especially valuable for accelerator cavities and waveguides where higher‑order modes are important for beam dynamics and wakefield calculations.

The paper also discusses implementation aspects. Because each half‑step updates only local stencil operations, the algorithm is naturally amenable to domain decomposition and MPI‑based distributed memory parallelism. No global matrix solves are required, which dramatically reduces communication overhead and makes the scheme well‑suited for GPU acceleration. The authors present benchmark results on three‑dimensional accelerator structures, showing that the explicit TE/TM scheme achieves a 2–3× speed‑up over comparable implicit solvers while delivering longitudinal dispersion errors essentially at machine precision and transverse dispersion errors an order of magnitude lower than Yee. Energy and charge conservation errors remain at the level of round‑off, even after thousands of time steps.

In summary, the explicit TE/TM scheme offers a compelling combination of (i) exact longitudinal dispersion‑free propagation, (ii) reduced transverse dispersion, (iii) unconditional energy and charge conservation, (iv) uniform time‑step size for all azimuthal modes in rotationally symmetric geometries, and (v) excellent scalability on modern parallel architectures. These attributes make it a strong candidate for next‑generation particle‑beam and accelerator‑cavity simulation tools, bridging the gap between the simplicity and speed of explicit methods and the accuracy and stability traditionally associated with implicit approaches.