{'fraction_non_alphanumeric': 0.029159519725557463, 'fraction_numerical': 0.0017152658662092624, 'mean_word_length': 5.864705882352941, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Runge–Kutta Methods for Systems of Differential Equation with Piecewise Continuous Arguments: Convergence and Stability

ABSTRACT This work deals with the convergence and stability of Runge–Kutta methods for systems of differential equation with piecewise continuous arguments x′(t) = Px(t)+Qx([t+1∕2]) under two cases for coeﬃcient matrix. First, when P and Q are complex matrices, the suﬃcient condition under which the analytic solution is asymptotically stable is given. It is proven that the Runge–Kutta methods are convergent with order p. Moreover, the suﬃcient condition under which the analytical stability region is contained in the numerical stability region is obtained. Second, when P and Q are commutable Hermitian matrices, using the theory of characteristic, the necessary and suﬃcient conditions under which the analytic solution and the numerical solution are asymptotically stable are presented, respectively. Furthermore, whether the Runge–Kutta methods preserve the stability of analytic solution are investigated by the theory of Padé approximation and order star. To demonstrate the theoretical results, some numerical experiments are adopted.