  We consider the solution of large stiff systems of ordinary differential
equations with explicit exponential Runge--Kutta integrators. These problems
arise from semi-discretized semi-linear parabolic partial differential
equations on continuous domains or on inherently discrete graph domains. A
series of results reduces the requirement of computing linear combinations of
$\varphi$-functions in exponential integrators to the approximation of the
action of a smaller number of matrix exponentials on certain vectors.
State-of-the-art computational methods use polynomial Krylov subspaces of
adaptive size for this task. They have the drawback that the required number of
Krylov subspace iterations to obtain a desired tolerance increase drastically
with the spectral radius of the discrete linear differential operator, e.g.,
the problem size. We present an approach that leverages rational Krylov
subspace methods promising superior approximation qualities. We prove a novel
a-posteriori error estimate of rational Krylov approximations to the action of
the matrix exponential on vectors for single time points, which allows for an
adaptive approach similar to existing polynomial Krylov techniques. We discuss
pole selection and the efficient solution of the arising sequences of shifted
linear systems by direct and preconditioned iterative solvers. Numerical
experiments show that our method outperforms the state of the art for
sufficiently large spectral radii of the discrete linear differential
operators. The key to this are approximately constant numbers of rational
Krylov iterations, which enable a near-linear scaling of the runtime with
respect to the problem size.
