In this paper, a class of high-order methods to numerically solve Functional Differential Equations with Piecewise Continuous Arguments (FDEPCAs) is discussed. The framework stems from the expansion of the vector field associated with the reference differential equation along the shifted and scaled Legendre polynomial orthonormal basis, working on a suitable extension of Hamiltonian Boundary Value Methods. Within the design of the methods, a proper generalization of the perturbation results coming from the field of ordinary differential equations is considered, with the aim of handling the case of FDEPCAs. The error analysis of the devised family of methods is performed, while a few numerical tests on Hamiltonian FDEPCAs are provided, to give evidence to the theoretical findings and show the effectiveness of the obtained resolution strategy.

翻译：本文讨论了一类用于数值求解分段连续参数泛函微分方程的高阶方法。该方法框架源于参考微分方程对应的向量场沿平移与缩放后的勒让德多项式标准正交基的展开，并基于哈密顿边值方法的适当推广构建。在方法设计过程中，考虑了来自常微分方程领域的摄动结果的正确推广，以处理分段连续参数泛函微分方程的情况。对所提出方法族进行了误差分析，并通过哈密顿型分段连续参数泛函微分方程的数值实验，验证理论发现并展示所得求解策略的有效性。