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

Abstract

Motivated by the rapid development of quantised large language models, which substantially reduce computational cost and enable efficient artificial intelligence on resource-constrained and consumer hardware, I demonstrate that finite-difference time-domain (FDTD) dynamics can be reproduced with field variables restricted to the ternary alphabet ${-1,0,+1}$. The resulting update requires no run-time multiplication or floating-point arithmetic and represents each field component with just $\log_2 3\simeq1.58$ bits. A true state accumulator performs the integration, while a second-order noise-shaped encoder with an independent error register performs the quantisation. The Courant number S serves simultaneously as an exact fixed-point ratio and as the encoder's oversampling ratio. Ternary FDTD converges to the standard Yee scheme in the small-S limit while substantially reducing state storage and arithmetic complexity. Its extension to acoustics, Virieux-type elastodynamics and Schrodinger-equation solvers points to a broader class of quantised physics solvers for resource-constrained and specialised hardware.