Numerical Scaling Invariance Applied to the van der Pol Model

In this study we use the van der Pol model to explain a novel numerical application of scaling invariance. The model in point is not invariant to a scaling group of transformations, but by introducing an embedding parameter we are able to recover it from an extended model which is invariant to an extended scaling group. As well known, within a similarity analysis we can define a family of solutions from a computed one, so that the solution of a target problem can be obtained by rescaling the solution of a reference problem. The main idea is to use scaling invariance and numerical analysis to find a reference problem easier to solve, from a numerical viewpoint, than the target problem. This allows us to save human efforts and computational resources every time we have to solve a challenging problem. We test our approach using three stiff solvers available within the most recent releases of MATLAB. Independently from the solver used, by employing the described scaling invariance we are able to significantly reduce the computational cost of the numerical solution of the van der Pol model.