On numerical simulations of the 1d wave equation with a distributional coefficient and source term

Abstract

In this note, we illustrate numerical experiments for the one-dimensional wave equation with �-like (delta like) terms. Our research is connecting the theory with the numerical realisations. By using results on very weak solutions introduced by Michael Ruzhansky with his co-authors, we investigate a corresponding regularized problem. In contrast to our expectations, the experiments show that the solution of the regularized problem has a "good" behaviour. Indeed, numerical experiments show that approximation methods work well in situations where a rigorous mathematical formulation of the problem is difficult in the framework of the classical theory of distributions. The concept of very weak solutions eliminates this difficulty, giving results of correctness for equations with singular coefficients. In the framework of this approach (very weak solutions), the expected physical properties of the equation can be reconstructed, for example, the distribution profile and the decay of the solutions for large times. Finally, we give a number of illustrations.