Nonlinear stability of reaction–diffusion equations using wavelet-like incremental unknowns

Incremental unknowns of different types were proposed as a means to develop numerical schemes in the context of finite difference discretizations. In this article, we present a novel wavelet-like incremental unknowns (WIU) method for the two-dimensional reaction–diffusion equations with a polynomial nonlinearity, and verify that the WIU is small as expected and it has the property of L2 orthogonal decomposition. Euler explicit and semi-implicit schemes based on the WIU are presented. And sufficient stability conditions are derived to improve the stability constraints of the corresponding classical algorithms in the multilevel meshes. Numerical results of reaction–diffusion equations are given to exhibit the features of the WIU.