This paper proposes an event-driven Gaussian collocation strategy for computing periodic solutions of single-valued piecewise-smooth dynamical systems, by integrating a novel event-driven Gaussian sampling scheme into the Fourier collocation framework. The present formulation is restricted to periodic solutions with a finite number of isolated transversal switching events. This approach retains the advantages of the conventional collocation method in handling non-smooth functions, while significantly improving computational efficiency and accuracy through an optimized sampling scheme. It is motivated by the deduction that the accuracy of the Fourier collocation method is governed by the Fourier truncation error and the sampling-scheme-induced quadrature error, and for piecewise-smooth systems, the quadrature error is often dominant. Thus, aiming to reduce the quadrature error and achieve accelerated convergence of the solution, the event-driven Gaussian sampling scheme is developed, where the whole periodic time interval is partitioned into multiple smooth sub-intervals through transition conditions (or events) and then, the Gaussian quadrature scheme for sampling within each sub-interval is applied to gain exponential quadrature accuracy. Numerical examples on several typical single-valued non-smooth systems are conducted and the following findings are unveiled: the proposed approach achieves the optimal convergence rate of the non-smooth solution with H harmonics using approximately <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$4H+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>4</mn> <mi>H</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> Gaussian sampling points and significantly outperforms the conventional collocation method with uniform sampling in both accuracy and convergence rate, as it even surpasses the performance of quadratic-scale uniform sampling scheme using <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$2H^2+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <msup> <mi>H</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> sampling points.

An event-driven Gaussian collocation strategy for single-valued piecewise-smooth systems with periodic solutions

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发表日期：

2026年7月27日

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Zechang Zheng,
Loic Salles,
Yanmao Chen,
Zhong-Rong Lu,
Jike Liu,
Li Wang

摘要

This paper proposes an event-driven Gaussian collocation strategy for computing periodic solutions of single-valued piecewise-smooth dynamical systems, by integrating a novel event-driven Gaussian sampling scheme into the Fourier collocation framework. The present formulation is restricted to periodic solutions with a finite number of isolated transversal switching events. This approach retains the advantages of the conventional collocation method in handling non-smooth functions, while significantly improving computational efficiency and accuracy through an optimized sampling scheme. It is motivated by the deduction that the accuracy of the Fourier collocation method is governed by the Fourier truncation error and the sampling-scheme-induced quadrature error, and for piecewise-smooth systems, the quadrature error is often dominant. Thus, aiming to reduce the quadrature error and achieve accelerated convergence of the solution, the event-driven Gaussian sampling scheme is developed, where the whole periodic time interval is partitioned into multiple smooth sub-intervals through transition conditions (or events) and then, the Gaussian quadrature scheme for sampling within each sub-interval is applied to gain exponential quadrature accuracy. Numerical examples on several typical single-valued non-smooth systems are conducted and the following findings are unveiled: the proposed approach achieves the optimal convergence rate of the non-smooth solution with H harmonics using approximately $4H+1$ $4 H + 1$ Gaussian sampling points and significantly outperforms the conventional collocation method with uniform sampling in both accuracy and convergence rate, as it even surpasses the performance of quadratic-scale uniform sampling scheme using $2H^2+1$ $2 H^{2} + 1$ sampling points.