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Adaptive rational Krylov methods for exponential Runge--Kutta integrators

Kai BergermannMartin Stoll
Mar 2023
We consider the solution of large stiff systems of ordinary differentialequations with explicit exponential Runge--Kutta integrators. These problemsarise from semi-discretized semi-linear parabolic partial differentialequations on continuous domains or on inherently discrete graph domains. Aseries of results reduces the requirement of computing linear combinations of$\varphi$-functions in exponential integrators to the approximation of theaction of a smaller number of matrix exponentials on certain vectors.State-of-the-art computational methods use polynomial Krylov subspaces ofadaptive size for this task. They have the drawback that the required Krylovsubspace iteration numbers to obtain a desired tolerance increase drasticallywith the spectral radius of the discrete linear differential operator, e.g.,the problem size. We present an approach that leverages rational Krylovsubspace methods promising superior approximation qualities. We prove a novela-posteriori error estimate of rational Krylov approximations to the action ofthe matrix exponential on vectors for single time points, which allows for anadaptive approach similar to existing polynomial Krylov techniques. We discusspole selection and the efficient solution of the arising sequences of shiftedlinear systems by direct and preconditioned iterative solvers. Numericalexperiments show that our method outperforms the state of the art forsufficiently large spectral radii of the discrete linear differentialoperators. The key to this are approximately constant rational Krylov iterationnumbers, which enable a near-linear scaling of the runtime with respect to theproblem size.