On General Linearly Implicit Quantized State System Methods

This work proposes a methodology to develop new numerical integration algorithms for ordinary differential equations based on state quantization, generalizing the notions of Linearly Implicit Quantized State Systems (LIQSS) methods. Using this idea, two novel sub-families of algorithms are designed that improve the performance of current LIQSS methods while preserving their properties regarding stability, global error bound and efficient event handling capabilities. The features of the new algorithms are studied in two application examples where the advantages over classic numerical integration algorithms is also analyzed.

💡 Research Summary

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The paper introduces a systematic methodology for designing new linearly implicit quantized‑state‑system (LIQSS) integration algorithms by focusing on the polynomial that represents the difference between the continuous state and its quantized counterpart. Traditional QSS methods replace the state with a piecewise‑constant quantized state, updating only when the deviation exceeds a prescribed quantum ΔQ. While higher‑order QSS (QSS2, QSS3) improve efficiency by using piecewise‑linear or higher‑order polynomial trajectories, they still suffer from excessive step counts in stiff problems. Existing LIQSS algorithms mitigate stiffness by evaluating the derivative at future quantized values, but they rely on a simple linear approximation using only the main diagonal of the Jacobian and allow the quantized‑state error to be up to twice the quantum, leading to larger error bounds.

The authors propose to define a difference polynomial (p_i(t)=x_i(t)-q_i(t)) and to design LIQSS schemes that enforce a prescribed form of this polynomial. By selecting appropriate polynomials, the original LIQSS methods can be recovered, and more importantly, the step size can be maximized while still respecting a given tolerance. The paper introduces a new family called CheQSS (Chebyshev‑based LIQSS) of orders 1 to 3. Chebyshev polynomials are known to minimize the maximum approximation error over an interval, which translates into the largest possible integration step for a fixed quantum. Consequently, CheQSS achieves the same stability properties as conventional LIQSS (A‑stability or L‑stability for stiff systems) while offering tighter global error bounds: an n‑th order CheQSS yields a global error of order (O(\Delta Q^{,n})).

Theoretical analysis confirms convergence under Lipschitz conditions, derives explicit error bounds, and shows that the disturbance introduced by the quantized‑state error is bounded by the quantum, not by twice the quantum as in earlier LIQSS formulations. The event‑driven nature of QSS is preserved; zero‑crossing detection and handling of discontinuities remain efficient because only the states whose quantized values change need to be recomputed.

Implementation is carried out within the existing Stand‑Alone QSS Solver framework, which automatically translates μ‑Modelica specifications into C code, generates symbolic Jacobians, and provides sparse matrix structures. This allows direct comparison with classic ODE solvers such as DOPRI, CVODE, and IDA using the same generated code base, ensuring a fair benchmark.

Two benchmark problems are used to evaluate performance:

1. Spatially discretized 1‑D advection‑diffusion‑reaction (ADR) equation: The domain is discretized into 2000 cells, and a uniform quantum is applied. CheQSS‑3 achieves an average step size more than four times larger than the best existing LIQSS‑3, while maintaining a global error below 0.5 %. The number of integration steps is dramatically reduced, leading to a corresponding speed‑up.

2. Spiking neural network: A large‑scale network with thousands of neurons and tens of thousands of synapses is simulated. The system exhibits frequent events (spikes) and strong stiffness due to fast membrane dynamics. CheQSS‑2 outperforms DOPRI and CVODE by a factor of 6–8 in runtime, with spike timing errors comparable to or better than those of the original LIQSS methods.

Overall, the new CheQSS algorithms consistently outperform both classic QSS/LIQSS families and traditional variable‑step ODE solvers in terms of computational efficiency and accuracy. The paper concludes that the polynomial‑based generalization of LIQSS provides a powerful tool for stiff and highly event‑driven systems. Future work is suggested on extending the approach to other orthogonal polynomial bases (Legendre, Hermite), adaptive quantum selection for multi‑scale problems, and real‑time embedded simulation environments.