Ren, Jincheng; Sun, Zhi-Zhong; Zhao, Xuan Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions. (English) Zbl 1291.35428 J. Comput. Phys. 232, 456-467 (2013). Summary: An effective finite difference scheme is considered for solving the time fractional sub-diffusion equation with Neumann boundary conditions. A difference scheme combining the compact difference approach the spatial discretization and \(L_{1}\) approximation for the Caputo fractional derivative is proposed and analyzed. Although the spatial approximation order at the Neumann boundary is one order lower than that for interior mesh points, the unconditional stability and the global convergence order \(O(\tau^{2-\alpha}+h^{4})\) in discrete \(L_{2}\) norm of the compact difference scheme are proved rigorously, where \(\tau\) is the temporal grid size and \(h\) is the spatial grid size. Numerical experiments are included to support the theoretical results, and comparison with the related works are presented to show the effectiveness of our method. Cited in 79 Documents MSC: 35R11 Fractional partial differential equations 35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs 35K05 Heat equation 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs Keywords:fractional sub-diffusion equation; compact difference scheme; discrete energy method; stability; convergence; Neumann boundary conditions Software:FODE PDFBibTeX XMLCite \textit{J. Ren} et al., J. Comput. Phys. 232, 456--467 (2013; Zbl 1291.35428) Full Text: DOI