Adaptive Numerical Simulation of Reaction - Diffusion Systems

The aim of this work is to find efficient and reliable numerical solutions of two complex problems under consideration. In the first application problem, an improved continuum model has been derived to describe the temperature and concentration distributions in gas-solid-fluidized beds with spray injection. The model equations for the nozzle spray are also reformulated to achieve reliable numerical solutions. The model equations are strongly coupled and semilinear partial differential equations with boundary conditions. The model equations are more flexible to compute the numerical solution on unstructured meshes than previous models. Solutions to these equations are approximated using a finite element method for the spatial discretization and an implicit Euler method in time. A study has been conducted to see the behavior of process parameters for heat and mass transfer in fluidized beds. The numerical results demonstrate that the method has a convergence order that agrees with theoretical considerations. The numerical results are validated with experimental results for two cases in three space dimensions. From parallelized numerical results, using domain decomposition methods, we show that good parallel efficiency is achieved with different numbers of processors. The second application problem concerns the adaptive numerical simulation of intracellular calcium dynamics. The modeling of diffusion, binding and membrane transport of calcium ions in cells leads to a system of reaction-diffusion equations. The strongly localized temporal behavior of calcium concentration due to opening and closing of channels as well as their spatial localization are effectively treated by an adaptive finite element method. The discrete approximation of deterministic equations produces a system of stiff ordinary differential equations with multiple time scales. The time scales are handled using linearly implicit time stepping methods with an adaptive step size control. The opening and closing of channels is typically a stochastic process. A hybrid method is adopted to couple the deterministic and stochastic equations. The adaptive numerical convergence of solutions is studied with different cluster arrangements. The deterministic equations are solved with parallel numerical methods to reduce the computational time using domain decomposition methods. A good parallel efficiency is achieved with different numbers of processors.