All Equalities Are Equal, but Some Are More Equal Than Others: The Effect of Implementation Aliasing on the Numerical Solution to Conservation Equations

TL;DR

The paper analyzes implementation aliasing in high-order conservation-law solvers, focusing on whether primitive or conserved-variable storage minimizes aliasing-driven errors. Using Flux Reconstruction and polynomial aliasing theory, it shows that constructing conserved variables from primitives introduces factorially increasing aliasing with order, while storing conserved variables generally reduces dissipation. It further finds that gradient calculations for viscous fluxes benefit from a product-rule approach due to lower computational cost, with negligible accuracy loss in many regimes. Precision studies indicate single precision suffices for pseudo-free-stream-turbulence flows, though low Mach numbers can amplify sensitivity. Overall, the work guides storage and gradient strategies to improve fidelity and efficiency in high-order CFD methods.

Abstract

We investigate the effect of aliasing when applied to the storage of variables, and their reconstruction for the solution of conservation equations. In particular, we investigate the effect on the error of storing primitives versus conserved variables for the Navier-Stokes equations. It was found that storing the conserved variables introduces less dissipation and that the dissipation caused by constructing the conversed variable from the primitives grows factorially with the order. Hence, this problem becomes increasingly important with the continuing move towards higher orders. Furthermore, the method of gradient calculation is investigated, as applied to the viscous fluxes in the Navier-Stokes equations. It was found that in most cases the difference was small, and that the product rule applied to the gradients of the conserved variables should be used due to a lower operation count. Finally, working precision is investigated and found to have a minimal impact on free-stream-turbulence-like flows when the compressible equations are solved, except at low Mach numbers.

Figures (8)

• Figure 1: Variation of error in ICV density with time for FR, $p=4$, using methods A, B and D on $16\times16\times2$ and $20\times20\times2$ element grids.
• Figure 1: Computation time comparison for one full RK44 explicit time step on a $8^3$, p=4, mesh. Time saving shown relative to scheme A.
• Figure 2: Variation in total kinetic energy of the ICV, FR $p=4$, for two grid resolutions. using methods A, B, and D.
• Figure 3: Enstrophy of the Taylor-Green Vortex with $R_e=1600$, $p=4$ and $80^3$ degrees of freedom for storage methods A-D.
• Figure 4: Enstrophy of the Taylor-Green Vortex with $R_e=1600$, $p=3$ and $80^3$ degrees of freedom.