In this dissertation, we will focus on developing efficient higher order algorithms for solving systems of nonlinear reaction-diffusion equations with Dirichlet, Neumann and Robin boundary conditions. The equations are widely used in modeling and simulations of numerous important processes in science and engineering. We will start from Crank-Nicolson algorithm, which is second order accurate in both temporal and spatial dimensions and unconditional stable, to develop higher order efficient algorithms using approximate factorization and Richardson's extrapolation. The new algorithms are fourth order accurate in both temporal and spatial dimensions and only require a three-point stencil in each dimension. The main objectives of this dissertation include: (1) To develop efficient implicit higher-order algorithms for solving systems of two and three-dimensional nonlinear reaction-diffusion equations. (2) To develop efficient methods to approximate Dirichlet, Neumann and Robin boundary conditions. (3) To conduct stability analyses for the new algorithms. (4) To carry out numerical experiments to demonstrate the accuracy and stability of the new algorithms.
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