Simple test cases for validating a finite element unstructured grid fecal bacteria transport model

This paper presents analytical test cases for tracer advection–diffusion-decay problems. The test cases are used to validate a finite element, unstructured grid fecal bacteria transport model. The test cases include the following domains: one-dimensional infinitely long river, two-dimensional half plane and two-dimensional infinitely long channel. In this work the bacteria are considered to enter the domain only through point sources. Analytical solutions are derived using either a Dirac delta function or a variable-width Gaussian function as a point source. Both analytical derivations and numerical simulations suggest that the error is maximised at the source. We present formulae for estimating the error caused by replacing a Dirac source with a Gaussian function in the numerical model. Furthermore, numerical simulations suggest that the best approximation for a Dirac source is a Gaussian whose width parameter is one third of the local mesh size.