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Explicit Runge--Kutta (RK) methods are susceptible to a reduction in theobserved order of convergence when applied to initial-boundary value problemwith time-dependent boundary conditions. We study conditions on explicit RKmethods that guarantee high-order convergence for linear problems; we refer tothese conditions as weak stage order conditions. We prove a generalrelationship between the method's order, weak stage order, and number ofstages. We derive explicit RK methods with high weak stage order anddemonstrate, through numerical tests, that they avoid the order reductionphenomenon up to any order for linear problems and up to order three fornonlinear problems.

Explicit Runge Kutta Methods that Alleviate Order Reduction