Numerical Method

openEMS is a full-wave, time-domain electromagnetic field solver based on the Finite-Difference Time-Domain (FDTD) method. This page introduces FDTD at a conceptual level: what kind of method it is, why openEMS implements it as an Equivalent-Circuit (EC) formulation, and what FDTD is (and isn’t) good at compared to other computational electromagnetics (CEM) approaches. For the concrete mechanics of the grid itself — Yee cells, the staggered leapfrog update, and the CFL stability limit — see Mesh.

FDTD in a Nutshell

FDTD discretizes both space and time. Maxwell’s curl equations are approximated by finite differences on a spatial grid (the Yee mesh) and advanced one small timestep at a time (time-marching). Every timestep only updates each cell from its immediate neighbors, so the algorithm is entirely explicit: there’s no global system of equations to assemble or invert, and propagation delay across the model falls out naturally from the number of timesteps a wave takes to physically travel across it.

Because the simulation runs directly in the time domain, a single run can cover a wide frequency band at once: exciting the structure with a short, broadband pulse (e.g. a Gaussian) and Fourier-transforming the recorded time-domain response yields S-parameters, impedances, or far-fields across the entire excited spectrum, see Signal Waveforms. This is the main practical payoff of choosing a time-domain method in the first place, and it’s true of any time-domain solver, not something specific to FDTD or to openEMS.

The Equivalent-Circuit (EC) Formulation

openEMS implements FDTD using the Equivalent-Circuit (EC-FDTD) formulation described in the openEMS and Rennings/Liebig publications (see Mesh’s bibliography, and Publications). Instead of updating raw electric and magnetic field samples, EC-FDTD reformulates the same Yee-grid update in terms of the integrated voltages and currents along each cell’s edges and faces. Concretely, every Yee cell becomes a small lumped-element circuit: capacitances and conductances sit on the electric-field edges, inductances and resistances on the magnetic-field faces, and the familiar Yee leapfrog update becomes a set of discrete-time circuit (Kirchhoff) equations instead of discretized curl equations.

Important

EC-FDTD is a reformulation, not a different physical model. Standard Yee-FDTD (fields) and EC-FDTD (voltages/currents) are mathematically equivalent — anything that can be expressed and simulated in one can, in principle, also be expressed and simulated in the other. Choosing one over the other is an implementation and derivation decision, not a difference in what the solver can compute or how accurate it is. The name mainly describes how openEMS’s internals are derived and expressed, not a separate numerical method sitting alongside FDTD.

Where the circuit formulation does earn its keep is convenience, in a few recurring places:

Lumped elements. Because the solver’s state is already voltages and currents, discrete resistors, capacitors and inductors — see Lumped Elements — drop in as additional circuit branches without a separate field-to-circuit translation layer.
Dispersive materials. Representing a Debye/Drude/Lorentz pole as an extra RC/RLC-like branch makes deriving both the update equations and the stability criteria for these materials considerably more tractable, see Dispersive Materials.
Non-orthogonal, graded and cylindrical meshes. Circuit quantities are integrals over a cell’s edge or face rather than point samples of a field, which carries over more directly to non-uniform or curved cell geometry. This is part of why the same solver core handles both the graded Cartesian mesh and the cylindrical mesh (see Mesh and Cylindrical FDTD) without needing a distinct formulation per coordinate system. The same lineage of thinking — using integrated quantities on a grid rather than raw field samples, to generalize beyond simple Cartesian grids — traces back to Weiland’s Finite Integration Technique (FIT).

In short: EC-FDTD is best understood as an implementation choice that makes certain features easier to derive and prove stable, not a capability Cartesian-field FDTD inherently lacks. Whether it’s worth adopting for a given FDTD implementation is, in the end, largely a matter of the developer’s taste.

Why FDTD? Pros and Cons

Choosing FDTD (in either formulation) over another CEM method — such as the frequency-domain Method of Moments (MoM) or Finite Element Method (FEM) — involves trade-offs. None of these are specific to openEMS; they’re inherent to the explicit, time-domain, volumetric nature of FDTD itself.

Advantages

Broadband in a single run. As above, one simulation plus a Fourier transform covers a whole band, instead of one frequency-domain solve per frequency point.
Geometric flexibility. Arbitrary, inhomogeneous 3D geometry — including curved and layered dielectrics, metals, and lumped or dispersive materials — is handled directly by assigning per-cell material properties. There’s no need for surface meshing with a Green’s function (as in MoM), nor for generating basis functions on an unstructured volume mesh (as in FEM).
No global linear system. Because updates are local and explicit, there is no dense or sparse system to factor or iteratively solve for every excitation, unlike MoM or FEM.
Straightforward parallelization. Local, stencil-like updates scale well across threads, SIMD and GPUs; openEMS itself uses multi-threading and SSE.
Direct physical insight. Because the wave genuinely propagates timestep by timestep, field dumps in time make it easy to visually debug a setup (e.g. to catch a stray reflection or an incorrectly placed port).

Disadvantages

Volumetric meshing. FDTD must discretize the entire computational domain, including all surrounding free space out to the absorbing boundary — not just conductor or dielectric surfaces as in MoM. Memory and runtime therefore scale with the physical volume of the problem (in wavelengths cubed), which can be costly for electrically large, mostly empty problems.
Grid (staircasing) error. A Cartesian grid approximates curved or diagonal boundaries as steps; even where the boundary is well resolved, numerical dispersion accumulates with propagation distance. See Cylindrical FDTD for openEMS’s main mitigation for round structures, and Mesh for non-uniform (graded) Cartesian meshing.
Conditional stability. The timestep is tied to the smallest cell anywhere in the mesh (the CFL limit, see Mesh), so one small geometric feature can dominate the runtime of an otherwise coarse simulation.
Inefficient for narrowband, high-Q problems. A high-Q resonator’s energy decays slowly, so reaching the convergence criterion can require very many timesteps compared to a frequency-domain solve at a single frequency of interest.
Low-frequency / small-structure mismatch. As covered in Mesh’s CFL discussion, a physically small structure operated far below its natural resonance (e.g. a 10 cm board at 1 MHz) needs an impractically large number of timesteps; FDTD (of either formulation) is simply the wrong tool for that regime.

Note

Stability in practice. As long as the CFL/Rennings2 timestep limit is respected, the FDTD (and EC-FDTD) update itself is inherently numerically stable — it doesn’t drift or “blow up” on its own. The main known exception is the PML absorbing boundary: it’s a very effective absorber for transmission-line, waveguide-mode, and otherwise well-behaved radiating fields, but is itself an artificial material — this is a property of PML in any FDTD implementation, not something specific to openEMS. Fringe fields and evanescent waves reaching into it can destabilize it. Keep radiating structures — intentional or not — at least \(\lambda/4\) away from any PML boundary.