A compact method for second-order boundary value problems on nonuniform grids (Q1921214)

The purpose of this paper is to present a new compact implicit method for the boundary value problem \[ a(x)y''+ b(x)y'+ c(x)y = r(x) \] \[ \alpha_0 y(0)+ \beta_0y'(0) = \gamma_0,\;\alpha_1y(1)+\beta_1y'(1)=\gamma_1. \] The proposed method is at least, formally, of order 4 on variable grids. Application of the method to linear and nonlinear ordinary differential equations and to a nonlinear partial differential equation is given. Numerical results are presented that support this order of convergence.