Glimm?s method simulation for the pollutant transport in an isothermal atmosphere

Abstract

In this work, a model for the transport of a pollutant in the atmosphere is presented by considering mass and linear momentum conservation for the airpollutant mixture as well as the mass balance for the pollutant. The resulting mathematical description consists of a nonlinear system of hyperbolic equations that admits discontinuities in addition to smooth or classical solutions. The Riemann problem associated with a class of problems describing the transport of a pollutant in an ideal gas with constant temperature and discontinuous distribution for either mass density or velocity or pollutant concentration as initial condition is carefully discussed. Numerical approximations for this nonlinear system are obtained by employing Glimm's method and considered in some numerical simulations.

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

“…(8) requires previous knowledge of the solution of the associated Riemann problem-discussed in details in Martins-Costa and Saldanha da Gama [9]. Essentially, Glimm's method consists in performing time evolutions employing the solution of the associated Riemann problem between each two consecutive steps in the spatial discretization.…”

“…So the first step to achieve a numerical approximation through this scheme is to solve the associated Riemann problem. Examples of simulations performed by employing this numerical scheme are the forced convection flow through a porous matrix [8] and the transport of a pollutant in an isothermal atmosphere [9].…”

“…The former is Glimm's scheme reliability, its accuracy being mathematically ensured [1,4]-that is the scheme does not dissipate shocks and preserves their positions. The latter is it being based on a theory whose mathematical formulation presents a solid thermodynamic basis, expressed by the entropy condition [9].…”

“…In order to describe an unsaturated flow of a Newtonian fluid through a rigid porous matrix, Martins-Costa and Saldanha da Gama [10] have considered a nonlinear non-homogeneous hyperbolic system of two equations, simulated by combining Glimm's scheme with an operator splitting technique. In Martins-Costa and Saldanha da Gama [9] a simplified model for a pollutant transport in an isothermal atmosphere gave rise to a non-linear homogeneous hyperbolic system of three equations, approximated by employing Glimm's scheme whose complete description for this particular non-linear system is presented, together with a discussion on the associated Riemann problem-including its complete solution.…”

Chemicals and Allied Products

“…In authors' opinion, the model presented a potential tool in forecasting air quality parameters and has advantages over the traditional neural network methods. -Costa, and Saldanha Da Gama (2003) presented a model for the transport of a pollutant in the atmosphere by considering mass and linear momentum conservation for the air-pollutant mixture, as well as the mass balance for the pollutant. The resulting mathematical description consisted of a nonlinear system of hyperbolic equations that admits discontinuities in addition to smooth or classical solutions.…”