A tailored integration scheme is developed to treat stiff reaction-diffusion prob- lems. The construction adapts a stiff solver, namely VODE, to treat reaction im- plicitly together with explicit treatment of diffusion. The second-order Runge-Kutta- Chebyshev (RKC) scheme is adjusted to integrate diffusion. Spatial operator is de- scretised by second-order finite differences on a uniform grid. The overall solution is advanced over S fractional stiff integrations, where S corresponds to the number of RKC stages. The behavior of the scheme is analyzed by applying it to three simple problems. The results show that it achieves second-order accuracy, thus, preserving the formal accuracy of the original RKC. The presented development sets the stage for future extensions, particularly, to multidimensional reacting flows with detailed chemistry.

Air Quality Models are defined for prediction and simulation of air pollutant concentrations over a certain period of time. The predictions can be used in setting limits for the emission levels of industrial facilities. The input data for the air quality models are very large and encompass various environmental conditions like wind speed, turbulence, temperature and cloud density. Most air quality models are based on advection-diffusion equations. These differential equations are moderately stiff and require appropriate techniques for fast integration over large intervals of time. Implicit time stepping techniques for solving differential equations being unconditionally stable are considered suitable for the solution. However, implicit time stepping techniques impose certain data dependencies that can cause the parallelization of air quality models to be inefficient. The current approach uses Runge Kutta Chebyshev explicit method for solution of advection diffusion equations. It is found that even if the explicit method used is computationally more expensive in the serial execution, it takes lesser execution time when parallelized because of less complicated data dependencies presented by the explicit time-stepping. The implicit time-stepping on the other hand cannot be parallelized efficiently because of the inherent complicated data dependencies. / Master of Science