In this seminar the problem of numerical approximation of evolution PDEs is considered. A peculiarity of the considered equations is their dependence on a few parameters, which is associated with the need to calculate multiple solutions as the parameters vary, possibly real-time. The proposed method approximates a complex contour integral in order to invert numerically the Laplace transform of the solution \cite{GLM}. The considered contours are constructed in such a way as to approximate certain curves, so-called pseudospectral, which characterize the space operator of the equation. The proposed approach is particularly well-suited to parabolic equations, where the contour can be essentially bounded. When the operator is hyperbolic the application of the method becomes critical, as the inversion of the Laplace transform on the continuous problem cannot be limited to a bounded contour. However, many numerical approaches introduce an artificial diffusion, which makes the proposed approach feasible. For the purpose of real-time computations for several instances of the parameters, various methodologies based on reduced bases or on model reduction methods have been proposed in the literature, which allow to solve small dimensional problems, maintaining a control of the accuracy. However, when the operator is hyperbolic, reduced basis methods and model order reduction based on classical time stepping schemes fail to provide the desired performances due to a critical decay of the Kolmogorov n-width. The use of reduction methods based on the Laplace transform on pseudospectral contours is new and seems to have several advantages (see \cite{GM}) with respect to those considered in the literature. The communication is inspired by collaborations with M. Lopez Fernandez (Malaga) and C. Lubich (Tuebingen).