On the application of the implicit “backward Euler” method for solving the diffusion equation

The present paper deals with numerical effects occurring in the application of the implicit (“backward Euler”) method to solve the diffusion equation in the case of a point source (i.e. singular initial data). The numerical over-estimation of the concentration at the source level as well as conditions for an over- or under-estimation of the ground-level concentration are investigated. To improve the results, a specific filtering of the initial concentration distribution is suggested. All theoretical results are illustrated by numerical examples; for this, an approach of constructing analytical ‘reference’ solutions, for special profiles of the diffusion coefficient, is presented.