• ADI;
  • compact;
  • finite difference scheme;
  • Neumann boundary condition;
  • reaction-diffusion equation


In this article, we extend the fourth-order compact boundary scheme in Liao et al. (Numer Methods Partial Differential Equations 18 (2002), 340–354) to a 3D problem and then combine it with the fourth-order compact alternating direction implicit (ADI) method in Gu et al. (J Comput Appl Math 155 (2003), 1–17) to solve the 3D reaction-diffusion equation with Neumann boundary condition. First, the reaction-diffusion equation is solved with a compact fourth-order finite difference method based on the Padé approximation, which is then combined with the ADI method and a fourth-order compact scheme to approximate the Neumann boundary condition, to obtain fourth order accuracy in space. The accuracy in the temporal dimension is improved to fourth order by applying the Richardson extrapolation technique, although the unconditional stability of the numerical method is proved, and several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed new algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013