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This paper presents a new algorithm KIOPS for computing linear combinationsof $\varphi$-functions that appear in exponential integrators. This algorithmis suitable for large-scale problems in computational physics where little orno information about the spectrum or norm of the Jacobian matrix is known apriori. We first show that such problems can be solved efficiently by computinga single exponential of a modified matrix. Then our approach is to compute anappropriate basis for the Krylov subspace using the incompleteorthogonalization procedure and project the matrix exponential on thissubspace. We also present a novel adaptive procedure that significantly reducesthe computational complexity of exponential integrators. Our numericalexperiments demonstrate that KIOPS outperforms the current state-of-the-artadaptive Krylov algorithm phipm.

KIOPS: A fast adaptive Krylov subspace solver for exponential integrators