KIOPS: A fast adaptive Krylov subspace solver for exponential integrators

•KIOPS, a new Krylov subspace algorithm for computing linear combinations of φ-functions is presented.•KIOPS uses the incomplete orthogonalization procedure and an adaptive strategy to determine the optimal parameters.•Numerical experiments demonstrate that KIOPS outperforms the current state-of-the-art adaptive Krylov algorithm. This paper presents a new algorithm KIOPS for computing linear combinations of φ-functions that appear in exponential integrators. This algorithm is suitable for large-scale problems in computational physics where little or no information about the spectrum or norm of the Jacobian matrix is known a priori. We first show that such problems can be solved efficiently by computing a single exponential of a modified matrix. Then our approach is to compute an appropriate basis for the Krylov subspace using the incomplete orthogonalization procedure and project the matrix exponential on this subspace. We also present a novel adaptive procedure that significantly reduces the computational complexity of exponential integrators. Our numerical experiments demonstrate that KIOPS outperforms the current state-of-the-art adaptive Krylov algorithm phipm.