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When applied to stiff, linear differential equations with time-dependentforcing, Runge-Kutta methods can exhibit convergence rates lower than predictedby the classical order condition theory. Commonly, this order reductionphenomenon is addressed by using an expensive, fully implicit Runge-Kuttamethod with high stage order or a specialized scheme satisfying additionalorder conditions. This work develops a flexible approach of augmenting anarbitrary Runge-Kutta method with a fully implicit method used to treat theforcing such as to maintain the classical order of the base scheme. Our methodsand analyses are based on the general-structure additive Runge-Kutta framework.Numerical experiments using diagonally implicit, fully implicit, and evenexplicit Runge-Kutta methods confirm that the new approach eliminates orderreduction for the class of problems under consideration, and the base methodsachieve their theoretical orders of convergence.

Eliminating Order Reduction on Linear, Time-Dependent ODEs with GARK Methods