This paper studies the spatial manifestations of order reduction that occur when time-stepping initial-boundary-value problems (IBVPs) with high-order Runge-Kutta methods. For such IBVPs, geometric structures arise that do not have an analog in ODE IVPs: boundary layers appear, induced by a mismatch between the approximation error in the interior and at the boundaries. To understand those boundary layers, an analysis of the modes of the numerical scheme is conducted, which explains under which circumstances boundary layers persist over many time steps. Based on this, two remedies to order reduction are studied: first, a new condition on the Butcher tableau, called weak stage order, that is compatible with diagonally implicit Runge-Kutta schemes; and second, the impact of modified boundary conditions on the boundary layer theory is analyzed.

翻译：本文研究了使用高阶龙格-库塔方法对初始边值问题进行时间步进时出现的降阶现象的空间表现。对于这类初始边值问题，会涌现出常微分方程初值问题中不具有类似物的几何结构：由内部近似误差与边界近似误差不匹配所诱导的边界层。为理解这些边界层，本文对数值格式的模态进行了分析，阐明了边界层在何种情况下会随多个时间步长持续存在。基于此，研究了两种降阶修正方案：首先，提出了一种与对角隐式龙格-库塔格式兼容的、名为弱阶条件的新Butcher表条件；其次，分析了修正边界条件对边界层理论的影响。