Summation by parts operators for finite difference approximations of second derivatives
The authors design central difference operators to approximate the derivatives \(d/dx\), \(d^2/dx^2\) at evenly spaced points in an interval \([a,b]\) which satisfy numerical formulas analogous to the integration by parts formulas from calculus (summation by parts (SBP) formulas). Using discrete diagonal norms the authors construct third, fourth, and fifth degree accurate SBP operators approximating \(d/dx\), \(d^2/dx^2\) and using discrete matrix norms fourth, sixth, and eighth accurate approximations. These approximations lead to easily obtained energy estimates for mixed hyperbolic parabolic problems in \((x,t)\) variables provided that the initial-boundary conditions are imposed such that the SBP property is preserved. For this purpose the authors employ a method referred to as the SAT method for boundary conditions [\textit{M. Carpenter}, \textit{D. Gottlieb} and \textit{S. Abarbanel}, J. Comput. Phys. 111, No. 2, 220--236 (1994; Zbl 0832.65098)]. Energy estimates and an error analysis are made for the convection diffusion equation and a model for the compressible Navier-Stokes equation. To obtain \(2p\)-order of accuracy it is proved that the approximating second derivatives should have order \(2p\) in the interior region and \(2p-2\) near the boundary.
scientific article; zbMATH DE number 2107334
scientific article