Analytical O(h2) CFD error annihilation theory: FREE O(h4) upgrade for second-order numerics codes

• Published in: Numerical heat transfer. Part B, Fundamentals Vol. 71; no. 5; pp. 397 - 424
• Main Authors: Baker, A. J., Orzechowski, Joe
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 04.05.2017; Taylor & Francis Ltd
• Subjects: Algorithms; Computational fluid dynamics; Destabilization; Differential calculus; Diffusion; Discretization; Error analysis; Fluid dynamics; Fluid flow; Galerkin method; Grid refinement (mathematics); Heat transfer; Large eddy simulation; Mathematical analysis; Navier-Stokes equations; Nonlinear systems; Operators (mathematics); Partial differential equations; Reynolds averaged Navier-Stokes method; Reynolds number; Simulation; Taylor series; Truncation errors
• ISSN: 1040-7790; 1521-0626
• DOI: 10.1080/10407782.2017.1309156

Worldwide, computational fluid dynamics (CFD) codes for Navier-Stokes (NS), Reynolds-averaged Navier-Stokes (RaNS), and/or large eddy simulation NS (LES) partial differential equation (PDE) systems are invariably based on second-order discrete numerics. Resulting nonlinear convection term discretizations inject an O(h 2 ) dispersive error mechanism, h the mesh measure, inducing code algebraic destabilization for practical Reynolds numbers (Re). Code universal resolution is PDE discretization augmentation with a (usually) difference algebra derived numerical diffusion scheme to render O(h 2 ) dispersion error destabilization nonpathological. The penalty of such schemes is artificial diffusion compromising of sharp fronts and/or discontinuities and generation of nonmonotone CFD approximations. Such legacy practices are now rendered obsolete by a totally analytical theory that rigorously identifies, in the continuum (!), the O(h 2 ) truncation error terms resident but unspecified in NS/RaNS/LES PDE system second-order CFD spatial discretizations. The theory removes identified O(h 2 ) error terms by alteration of the continuum appearance of NS/RaNS/LES PDE systems with nonlinear vector differential calculus operators. Theory is amenable to any second-order "tri-diagonal stencil" equivalent CFD discretization and, upon implementation, elevates the original second-order numerics code to O(h 4 ) with no further action. This Taylor series error estimate is weak form theory formalized to a regular solution adapted nonuniform mesh refinement O(h 4 ) asymptotic error estimate. Theory implementation in a linear basis optimal Galerkin criterion weak form algorithm CFD code enables a posteriori data generation validating annihilation of O(h 2 ) dispersive error mechanisms for reduced NS, full NS, and RaNS PDE systems. In every instance, theory implementation leads to CFD monotone solution distributions free from artificial diffusion influence on sufficiently refined meshes. Differential definition Galerkin weak forms, code post-processed, quantify theory annihilated O(h 2 ) dispersion error spectra, RaNS state variable member specific.