Iterative and non-iterative, full and approximate factorization methods for multidimensional reaction-diffusion equations

A variety of second-order accurate, in both space and time, full and approximate factorization methods for the numerical solution of two-dimensional reaction-diffusion equations is presented. These methods may use time linearization and yield linearly implicit techniques and one-dimensional operators in each direction. It is shown that, if the factorization errors are neglected, linearly implicit approximate factorization methods provide uncoupled equations, whereas, if these errors are considered, the equations are coupled and must be solved iteratively. It is also shown that the allocation of the reaction and diffusion terms to the one-dimensional operators plays a paramount role in determining the accuracy of approximate factorization methods and preserving the symmetry of the original differential problem. Iterative, full and approximate factorization methods that do require iterations are also presented, and, for the problem considered here, these methods are shown to converge in about two iterations and provide solutions in agreement with those obtained with linearly implicit full and approximate factorization techniques.