Ternary-Valued Finite-Difference Time-Domain Method: Asymptotic Agreement with the Yee Scheme Through Noise-Shaped Quantisation

Abstract

I show that finite-difference time-domain (FDTD) dynamics can be reproduced using field variables restricted to the ternary alphabet $-1,0,+1$. The integration is carried out by a state accumulator, while quantisation is done by a second-order noise-shaped encoder with a separate error register. The Courant number $S$ plays two roles: it is the exact fixed-point ratio and also sets the encoder's oversampling ratio. When $S$ is chosen as a power of two, each cell update requires neither run-time multiplication nor floating-point arithmetic, and each emitted field symbol is represented by just two bits. The ternary scheme converges asymptotically to the standard Yee scheme as $S$ decreases towards zero. I report wall-clock and instruction counts relative to a floating-point reference, long-time energy integrations over $2\times 10^5$ steps and the $S$ required to achieve a prescribed accuracy for the driven mode. Extending the approach to acoustics, Virieux-type elastodynamics and Schrodinger-equation solvers is a prospective direction towards a broader class of quantised physics solvers for resource-constrained and specialised hardware.

arXiv comment5 figures, 2 tables