  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 explicit RK
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 explicit RK 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.
