Construction of Conservative Numerical Fluxes for the Entropy Split Method

The entropy split method is based on the physical entropies of the thermally perfect gas Euler equations. The Euler flux derivatives are approximated as a sum of a conservative portion and a non-conservative portion in conjunction with summation-by-parts (SBP) difference boundary closure of (Gerritsen and Olsson in J Comput Phys 129: 245–262, 1996; Olsson and Oliger in RIACS Tech Rep 94.01, 1994; Yee et al. in J Comp Phys 162: 33–81, 2000). Sjögreen and Yee (J Sci Comput https://doi.org/10.1007/s10915-019-01013-1) recently proved that the entropy split method is entropy conservative and stable. Standard high-order spatial central differencing as well as high order central spatial dispersion relation preserving (DRP) spatial differencing is part of the entropy stable split methodology framework. The current work is our first attempt to derive a high order conservative numerical flux for the non-conservative portion of the entropy splitting of the Euler flux derivatives. Due to the construction, this conservative numerical flux requires higher operations count and is less stable than the original semi-conservative split method. However, the Tadmor entropy conservative (EC) method (Tadmor in Acta Numerica 12: 451–512, 2003) of the same order requires more operations count than the new construction. Since the entropy split method is a semi-conservative skew-symmetric splitting of the Euler flux derivative, a modified nonlinear filter approach of (Yee et al. in J Comput Phys 150: 199–238, 1999, J Comp Phys 162: 3381, 2000; Yee and Sjögreen in J Comput Phys 225: 910934, 2007, High Order Filter Methods for Wide Range of Compressible flow Speeds. Proceedings of the ICOSAHOM09, June 22–26, Trondheim, Norway, 2009) is proposed in conjunction with the entropy split method as the base method for problems containing shock waves. Long-time integration of 2D and 3D test cases is included to show the comparison of these new approaches.