The Efficient High-Order Finite-Difference Scheme For 2-D Reaction-Diffusion Equations

This thesis studies the efficient high-order finite-difference schemes for 2-D reaction-diffusion equations. Three conventional high-order finite-difference schemes, two-layer implicit scheme, Crank-Nicolson scheme and Peaceman-Rachford scheme, for the special case of 2-D reaction-diffusion equation - 2-D constant coefficient reaction-diffusion equation are present at first, and the truncation error and stability condition for each scheme are analyzed by Fourier method and Von Neumann method. A high-order finite-difference scheme for 2-D constant coefficient reaction-diffusion equation is introduced by the combination of dimension-depress and order-depress methods. The truncation error expression for the finite-difference scheme is given. The algorithm of staggered-orientation finite-difference scheme is given by the introduction of transition layer. The a priori estimated expression of the solution to staggered-orientation finite-difference scheme, under discrete H~1 norm, is given using energy analysis method. The solvability, stability and convergence of the finite-difference scheme are demonstrated. Theconvergence order of the scheme is O(Ï„~2 +h~4). Then the method is generalized to the case of2-D variable coefficient reaction-diffusion equation and tested by numerical examples. The numerical results are coincident with the analytic ones. Conclusion and future study are given at the end.