Improving the Spatial Resolution and Stability by Optimizing Compact Finite Differencing Templates

Parameter optimization is an excellent path for easily raising the resolution efficiency of compact finite differencing schemes. Their low-resolution errors are attractive for resolving the fine-scale turbulent physics even in complex flow domains with difficult boundary conditions. Most schemes require optimizing closure stencils at and adjacent to the domain boundaries. But these constituents can potentially degrade the local resolution errors and destabilize the final solution scheme. Current practices optimize and analyze each participating stencil separately, which incorrectly quantifies their local resolution errors. The proposed process optimizes each participant simultaneously. The result is a composite template that owns consistent spatial resolution properties throughout the entire computational domain. Additionally, the optimization technique leads to templates that are numerically stable as understood by an eigenvalue analysis. Finally, the predictive accuracy of the optimized schemes are evaluated using four canonical test problems that involve resolving linear convection, nonlinear Burger wave, turbulence along a flat plate, and circular cylinder wall pressure.