Recently, several publications have been published, where continuous and piecewise-affine Lyapunov functions are constructed for nonlinear systems by numerically computing their values on a grid and then interpolating these values over the simplices of a simplicial complex. The value of such a Lyapunov function is computed at a grid point by numerically solving an initial-value problem and then integrating a positive definite function of the solution on a given time-interval. In this paper we systematically investigate how different initial-value solution methods compare in this application. Further, we propose a method to compute the integrals that is superior to former approaches.