On the optimally convergent splines difference scheme for one-dimensional reaction-diffusion problem
The author proposes a second-order optimally accurate difference scheme for selfadjoint perturbation problems of the following form: \(\varepsilon u'' + p(x)u = f(x)\), \(x \in [0,1]\), \(u(0) = \alpha_ 0\), \(u(1) = \alpha_ 1\), where \(p,f \in C^ 2[0,1]\), \(0 < \varepsilon \ll 1\), \(p(x) > \beta > 0\), \(x\in [0,1]\). The second section describes the difference discretization for the selfadjoint perturbation problem based on the exponential splines. The main idea is to use the continuous condition of the first derivative of exponential splines at midpoints. Section 3 proves that the scheme possesses second-order optimal convergence. The results are supported with the numerical experiments from Section 4.
scientific article; zbMATH DE number 679971
scientific article