Implicit summation by parts operators for finite difference approximations of first and second derivatives

Abstract

Implicit finite difference approximations are derived for both the first and second derivatives. The boundary closures are based on the banded-norm summation-by-parts framework and the boundary conditions are imposed using a weak (penalty) enforcement. Up to 8th order global convergence is achieved. The finite difference approximations lead to implicit ODE systems. Spectral resolution characteristics are achieved by proper tuning of the internal difference stencils. The accuracy and stability properties are demonstrated for linear hyperbolic problems in 1D and the 2D compressible Euler equations.