Numerical Analysis and Scientific Computing Seminar | Helen C. Yee, Recent Advancement of Entropy Split Methods for Compressible Gas Dynamics and MHD

Tuesday, June 21, 2022 1:00 PM EDT

For Zoom Link please contact ddelreyfernandez@uwaterloo.ca

Speaker

Helen C. Yee, Senior Research Scientist, NASA AMES Research Center

Title

Recent Advancement of Entropy Split Methods for Compressible Gas Dynamics and MHD

Abstract

The entropy splitting of the compressible Euler flux derivatives based on Harten’s entropy function [6, 4, 24] in conjunction with classical spatial central and DRP (dispersion relation-preserving) finite discretizations with summation-by-parts (SBP) operators [22] for both periodic and non-periodic boundary conditions is proven to be entropy conservative and stable for a thermally-perfect gas by Sjogreen & Yee [18, 14, 19]. The various high order methods resulting from applying classical spatial central, DRP and Pade (compact) methods to the split form of the Euler flux derivative are referred to as entropy split methods as a function of the splitting parameter β. These entropy split methods are entropy conserving and stable but they are usually not conservative numerical methods without additional reformulation; e.g., those proposed in Sjogreen & Yee [19].

Here the same Harten entropy function is used for the ideal magnetohydrodynamic (MHD) governing equation set. This approach was not examined in great depth by Sjogreen & Yee and Yee & Sj¨ogreen in two conference proceedings papers [21, 27], which also included the high order Pade (compact) spatial dis-cretizations [7]. For the extension of the entropy split method to the ideal MHD, the Godunov symmetriz-able non-conservative form of the formulation is used. Due to the non-conservative portion (commonly referred to as a nonlinear source term vector) of the symmetrizable system [5], there are variants in formu-lating the entropy split methods. Two different numerical treatments for these particular source terms of the symmetrizable MHD governing equation set are presented.

The comparative studies concentrate on the Tadmor-type of discrete entropy conserving [23], momentum conserving [2], kinetic energy preserving [8, 10, 1], Yee et al., Sjogreen & Yee entropy split methods [24, 16, 13, 18, 14, 19], as well as the combination of these physical-preserving methods [11, 26]. All of these methods are not only preserve certain physical properties of the chosen governing equations but are also known to either improve numerical stability, and/or minimize aliasing errors in long time integration of turbulent flow computations without the aid of added numerical dissipation for selected compressible flow types. Extensive error norm comparison with grid refinement were performed to show how well nine of the eighth-order methods conserve the entropy, momentum and mass, and preserve the kinetic energy for long time integration of the various flows. In general the Tadmor-type entropy conserving methods are at least twice the CPU per time step than the rest of the considered nine methods

References

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