High Order Compact Central Spatial Discretization Under the Framework of Entropy Split Methods

Yee and Sjögreen (Comput Fluid 37:593–619, 2008) did a study on the performance between high order compact (Padé) spatial central finite discretizations and standard spatial central (explicit) finite discretizations. Their performance evaluation comparison used the Yee et al. (J Comput Phys 150:199–238, 1999) and Sjögreen and Yee (J Sci Comput 20:211–255, 2004) nonlinear filter approach for multiscale problems containing shock waves. For reference purposes, spatial compact and spatial explicit central discretizations are referred to as compact and explicit central schemes. High order compact schemes are methods of choice for many incompressible, nearly incompressible and low speed compressible turbulent/acoustic flows due to their advantage of requiring a low number of grid points per wavelength and low aliasing errors in a linear analysis. In the presence of multiscale shock interactions and under the Yee et al. and Sjögreen and Yee nonlinear filter approach (Yee et al., J Comput Phys 150, 199–238, 1999; Yee and Sjögreen, Comput Fluid 37:593–619, 2008; Sjögreen and Yee, J Sci Comput 20:211–255, 2004; Yee and Sjögreen, High order filter methods for wide range of compressible flow speeds; Proceedings of the ICOSAHOM09, June 22–26, 2009, Trondheim 2009), however, this desired property of high order compact schemes seems to have diminished in both the gas dynamic and MHD test cases that Yee and Sjögreen demonstrated in 2008 (Yee and Sjögreen, Comput Fluid 37:593–619, 2008). In addition, compact schemes are more CPU intensive, and less friendly in the parallel computation environment than their explicit central scheme cousins. Moreover, compact schemes are global methods. For non-periodic boundary condition case, any mishandling of the numerical boundary scheme treatment, it will contaminate the accuracy and stability of the interior domain, unlike explicit spatial schemes which are local schemes. The objective of this research is to examine the performance of the Yee et al. and Sjögreen and Yee high order entropy split methods (Yee et al., J Comput Phys 162:33–81, 2000; Sjögreen and Yee, J Comput Phys 364:153–185, 2018; Sjögreen and Yee, J Sci Comput 81:1359–1385, 2019; Sjögreen and Yee, Construction of conservative numerical fluxes for the entropy split method. Communication in Applied Mathematics and Computation (CAMC), 2021) by employing the high order compact central spatial schemes. Under the entropy split method framework, stability is greatly improved in the use of the compact schemes without the need of low pass high order compact linear filters to stabilize long time integration of flows, e.g., DNS (direction numerical simulation) and LES (large eddy simulation) computations. For the numerical test cases, an eighth-order compact scheme and the explicit scheme of the same order are compared. Both schemes have fourth-order accurate summation-by-parts (SBP) boundary closure at the domain boundaries. Furthermore, the compact scheme in an entropy split approximation of the equations of magnetohydrodynamics (MHD) is used, demonstrating good performance when used as a base discretization together with the dissipative portion of a seventh-order shock-capturing WENO method as nonlinear filter.