A quadrature based method for evaluating exponential-type functions for exponential methods

We present a quadrature-based method to evaluate exponential-like operators required by different kinds of exponential integrators. The method approximates these operators by means of a quadrature formula that converges like O(e (-cK) ), with K the number of quadrature nodes, and it is useful when solving parabolic equations. The approach allows also the evaluation of the associated scalar mappings. The method is based on numerical inversion of sectorial Laplace transforms. Several numerical illustrations are provided to test the algorithm, including examples with a mass matrix and the application of the method inside the MATLAB package EXP4, an adaptive solver based on an exponential Runge-Kutta method.