Stiffness resilient exponential integrators and $$\varphi $$-order conditions

Abstract In this paper we propose a new analytical framework for deriving exponential time integrators. Exponential methods are typically constructed using either classical or stiff order conditions. Either of these approaches lead to complex order conditions which are challenging to solve to build high order methods. Classically-derived integrators can also suffer from the order reduction phenomenon. In this work we propose novel $$\varphi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> </mml:math> -order conditions which enable construction of schemes with a particular error structure that prevents order reduction for many important stiff problems. Moreover, we develop a systematic procedure to parametrize the $$\varphi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> </mml:math> -order conditions and solve them exactly. Thus construction of the new high order exponential methods, we call stiffness resilient, is greatly simplified compared to the derivation of classical or stiffly accurate integrators. We develop stiffness resilient methods of various types such as exponential Runge–Kutta, multistep and multivalue schemes and explain how they are related to the interpolation integrators. Practical aspects of implementing these methods with variable time stepping and dense output are discussed and performance of specific schemes is demonstrated using several test problems.