A Theory for Modified Conservation Principles Optimization of CFD Algorithm Fidelity

A theory is developed in one and two space dimensions that successfully predicts optimal algorithm constructions for the convection operator intrinsic to unsteady Navier- Stokes (NS) problem statements. The analysis statement is parameterized via a Taylor series (TS) modification to the parent NS conservation principles statements. Phase velocity and amplification factor error analyses are enabled via weak form discretized implementations assembled at the generic node. The parameterized error statement is then resolved into a Taylor series expansion in non-dimensional wave number space, admitting identifications that progressively annihilate lowest order error terms. The theory computational implementation is via a Galerkin weak statement on the TS modified formulation, discretely implemented using linear and bi-linear finite element basis functions for one and two dimensions respectively.

The theory is extended to one dimension FE quadratic basis. A general formulation for TWS class of algorithms enabling analysis for phase accuracy is derived. Matrix stability analysis approach pertinent to TWS algorithms is presented. Theory suggested results are ported to other verification and validation problems and analyzed for solution fidelity. One dimensional space test cases include advection-diffusion and non-linear Burgers equation. Two-dimensional space test cases include a pure advection verification problem, an advection-diffusion-source verification problem and 8x1 full Navier-Stokes validation-class thermal cavity problem. Algorithm predictability is also compared for the selected algorithms on non-uniform Cartesian and regular but non- Cartesian triangular mesh.

A computational approach to obtain progressively higher order phase accurate solutions using a Matlab enabled optimization theory has also been examined. The unusual behavior algorithms thus generated are analyzed under the anomalous behavior topic generated by this approach.