A solution to the stability issues with block norm summation by parts operators

Finite difference operators approximating first and second derivatives and satisfying a summation by parts (SBP) rule have been derived for the 4th, 6th, 8th and 10th order case by using the symbolic mathematics software Maple. The operators are based on block norms, to avoid the curse of losing accuracy at the boundaries that are present for corresponding operators based on diagonal norms. The reason why block norm operators have not been used in realistic applications before is related to the well-known stability issues on curvilinear grids. To avoid this problem an additional boundary stabilization is introduced, that removes unstable eigenvalues without interfering with accuracy and stiffness. The superior accuracy properties of the newly derived block norm SBP operators will be demonstrated for the second order wave equation in 1-D and for the compressible Euler equations in complex 2-D geometries.