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We present a method for computing actions of the exponential-like φ -functions for a Kronecker sum K of d arbitrary matrices A μ . It is based on the approximation of the integral representation of the φ -functions by Gaussian quadrature formulas combined with a scaling and squaring technique. The resulting algorithm, which we call phiks, evaluates the required actions by means of μ -mode products involving exponentials of the small sized matrices A μ , without forming the large sized matrix K itself. phiks, which profits from the highly efficient level 3 BLAS, is designed to compute different φ -functions applied on the same vector or a linear combination of actions of φ -functions applied on different vectors. In addition, thanks to the underlying scaling and squaring techniques, the desired quantities are available simultaneously at suitable time scales. All these features allow the effective usage of phiks in the exponential integration context. In fact, our newly designed method has been tested in popular exponential Runge–Kutta integrators of stiff order from one to four, in comparison with state-of-the-art algorithms for computing actions of φ -functions. The numerical experiments with discretized semilinear evolutionary 2D or 3D advection–diffusion–reaction, Allen–Cahn, and Brusselator equations show the superiority of the proposed μ -mode approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider an approach to improve the performance of exponential Runge–Kutta integrators and Lawson schemes in cases where the solution of a related, but usually much simpler, problem can be computed efficiently. While for implicit methods such an approach is common (e.g., by using preconditioners), for exponential integrators this has proven more challenging. Here we propose to extract a constant coefficient differential operator from the semilinear advection-diffusion-reaction equation for which, in many situations, efficient methods are known to compute the required matrix functions. Both a linear stability analysis and extensive numerical experiments show that the resulting schemes can be unconditionally stable. In fact, we find that exponential integrators of Runge–Kutta type and Lawson schemes can have better stability properties than similarly constructed implicit-explicit schemes. We also derive two new Lawson-type integrators that further improve on these stability properties. The overall effectiveness of the approach is highlighted by a number of performance comparisons on examples in two and three space dimensions. [ABSTRACT FROM AUTHOR]