An ADI extrapolated Crank-Nicolson orthogonal spline collocation method for nonlinear reaction-diffusion systems: a computational study

An alternating direction implicit (ADI) orthogonal spline collocation (OSC) method is described for the approximate solution of a class of nonlinear reaction-diffusion systems. Its efficacy is demonstrated on the solution of well-known examples of such systems, specifically the Brusselator, Gray-Scott, Gierer-Meinhardt and Schnakenberg models, and comparisons are made with other numerical techniques considered in the literature. The new ADI method is based on an extrapolated Crank-Nicolson OSC method and is algebraically linear. It is efficient, requiring at each time level only $O({\cal N})$ operations where ${\cal N}$ is the number of unknowns. Moreover,it is shown to produce approximations which are of optimal global accuracy in various norms, and to possess superconvergence properties

Higher order compact finite difference method for the solution of 2-D time fractional diffusion equation

The main purpose of this study is to work on the solution of two-dimensional time fractional diffusion equation In this research work we apply the HOC scheme to approximate the second order space derivative. To obtain a discrete implicit scheme, Grunwald-Letnikov descritization is used in sense to approximate the Riemann-Liouville time fractional derivative. The scheme thus obtained is based on block pentadiagonal matrix and each matrix has five-point stencil in order to reduce the computational cost we use AOS method. In AOS method, before taking the average of two solutions first we split the n-dimensional problems into a sum of n-one dimensional problem