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.

翻译：我们研究使用显式指数龙格-库塔积分器求解大型刚性常微分方程组。这些问题源自连续域或固有不连续图域上半离散化的半线性抛物型偏微分方程。一系列研究结果将指数积分器中计算φ函数线性组合的需求简化为在特定向量上近似计算更少数量的矩阵指数作用。当前最先进的计算方法采用自适应大小的多项式Krylov子空间完成此任务。其缺点在于，为达到所需容差所需的Krylov子空间迭代次数会随离散线性微分算子的谱半径（例如问题规模）急剧增加。我们提出一种采用有理Krylov子空间方法的方法，该方法具有更优的逼近性能。我们针对单时间点上矩阵指数在向量上作用的有理Krylov逼近，证明了新颖的后验误差估计，从而可实现与现有多项式Krylov技术类似的自适应方法。我们讨论了极点选择，以及通过直接法和预处理迭代法高效求解所生成的序列化移位线性系统。数值实验表明，当离散线性微分算子的谱半径足够大时，我们的方法优于现有技术。其关键在于有理Krylov迭代次数近似恒定，这使得运行时间相对于问题规模具有近线性缩放特性。