Explicit Runge Kutta Methods that Alleviate Order Reduction

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.