Explicit Runge--Kutta (\rk{}) methods are susceptible to a reduction in the observed order of convergence when applied to initial-boundary value problem with time-dependent boundary conditions. We study conditions on \erk{} methods that guarantee high-order convergence for linear problems; we refer to these conditions as weak stage order conditions. We prove a general relationship between the method's order, weak stage order, and number of stages. We derive \erk{} methods with high weak stage order and demonstrate, through numerical tests, that they avoid the order reduction phenomenon up to any order for linear problems and up to order three for nonlinear problems.

翻译：显式龙格-库塔（\rk{}）方法在应用于具有时变边界条件的初边值问题时，其观测到的收敛阶数易出现降低现象。我们研究保证线性问题高阶收敛的\erk{}方法条件，并将这些条件称为弱阶段阶条件。我们证明了方法的阶数、弱阶段阶数与阶段数之间的一般关系。通过推导具有高弱阶段阶数的\erk{}方法，并经数值试验验证，这些方法能避免线性问题中任意阶数的阶数降低现象，以及非线性问题中三阶以内的阶数降低现象。