Local radial basis function collocation method preserving maximum and monotonicity principles for nonlinear differential equations

In this paper, a hybrid numerical scheme based on combining exponential time differencing (ETD) and local radial basis function collocation method was constructed. Model problems with different boundary conditions were considered, and the resulting linear system was carefully analyzed. The relation between the number of points employed in the local radial basis function collocation method and the condition number of the coefficient matrix was given. For application, three typical differential equations were investigated, that is, the Allen–Cahn equation for checking the maximum‐preserving property, the combustion equation for checking the monotonicity‐preserving property, and the Gray–Scott system for checking the robustness of the proposed scheme. Numerical examples show the effectiveness of the proposed method.