Ternary-valued finite-difference time-domain method: asymptotic agreement with the Yee scheme through noise-shaped quantisation

Abstract

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. Numerical convergence studies show asymptotic agreement with 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 × 105 steps and the S required to achieve a prescribed accuracy for the driven mode. Extending the approach to acoustics, Virieux-type elastodynamics and Schrödinger-equation solvers is a prospective direction towards a broader class of quantised physics solvers for resource-constrained and specialised hardware.