Simulation of a pollutant motion and decay in polytropic atmospheres with spherical symmetry

The motion and decay of a pollutant immersed in an atmosphere are described by using a hyperbolic model in which the atmosphere is assumed an ideal gas undergoing polytropic processes and the pollutant is transported radially. The simplified mathematical model is based on mass and linear momentum conservation for the air-pollutant mixture as well as the mass balance for the pollutant, giving rise to a nonlinear system of hyperbolic equations that admits discontinuities in addition to smooth or classical solutions. Numerical approximations for this nonlinear system are obtained by combining Glimmʹs method – which, in turn, requires the solution of a Riemann problem for each two consecutive steps – with an operator splitting technique to account for the non-homogeneous part of the operator. This technique consists in decomposing the operator in a merely hyperbolic part (the homogeneous associated system) and a purely time evolutionary one. The numerical methodology is illustrated by some representative results considering a spherical shell configuration.