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 2×105 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.
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BibTeX
@misc{maksymov2026ternaryvalued,
title = {{Ternary-Valued Finite-Difference Time-Domain Method: Asymptotic Agreement with the Yee Scheme Through Noise-Shaped Quantisation}},
author = {Ivan S. Maksymov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.27069},
eprint = {2608.27069},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27069},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ternary-Valued Finite-Difference Time-Domain Method: Asymptotic Agreement with the Yee Scheme Through Noise-Shaped Quantisation
AU - Ivan S. Maksymov
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2608.27069
UR - https://arxiv.org/abs/2608.27069
AB - 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 2×10^5 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.
ER -