We present an hp-version of the C⁰-continuous Petrov-Galerkin time stepping method for nonlinear second-order initial value problems. We derive a priori error bound in the H¹-norm that is fully explicit with respect to the local time steps and the local approximation orders. Moreover, we prove that the hp-version of the C⁰-continuous Petrov-Galerkin time stepping method based on geometrically refined time steps and on linearly increasing approximation orders yields exponential rates of convergence for solutions with start-up singularities. Numerical examples confirm the theoretical results.

中文翻译：

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非线性二阶初值问题的C 0连续Petrov-Galerkin时间步进方法的hp版本

我们提出了针对非线性二阶初值问题的C ⁰连续Petrov-Galerkin时间步进方法的hp版本。我们得出H ¹范数的先验误差界线，该误差域对于本地时间步长和本地近似阶数是完全明确的。此外，我们证明了基于几何精炼的时间步长和线性递增的近似阶次的C ⁰连续Petrov-Galerkin时间步长方法的hp版本会产生具有奇异启动项的解的指数收敛速度。数值例子证实了理论结果。