Chapter 4 Legacy CFD difference algebra derived stabilizations, O(h2) truncation error annihilation (TEA) theory, NS error freed PDE mathematics, monotonicity, stability4.1 CFD algorithm O(h2) TE instability, Fourier modal analysis4.2 Functional correlation of legacy CFD derived stabilizations4.3 Algebraic instability, O(h2) numerical diffusion infused legacy CFD4.4 Derivation, NS DP O(h2) truncation error continuum nonlinear functional, n = 14.5 Nonlinear fidelity, NS TEA altered O(h4) DmP GWSh FE algorithm, Newton Jacobian4.6 Validation, Burgers TEA O(h4) DmP GWSh FE DOF monotonicity ∀Re ≤ 1064.7 Validation, compressible NS TEA O(h4) PDE monotone shock interpolation, Re ≤ 1054.8 O(h4) error freed CFD mathematics diagnosed physics of fluids anomaliesChapter 5 Navier-Stokes error freed CFD mathematics, continuous weak formulation, FE theory, asymptotic convergence, error estimates, validations, monotone continuum shock interpolation5.1 Introduction, unstagnation options for current practice NS (and RaNS) codes5.2 CFD fundamentals, verification, validation, uncertainty quantification (VVUQ)5.3 Derivation, weak formulation continuous Galerkin NS CFD algorithm5.4 Compressible NS TEA altered O(h4) error freed CFD mathematics PDE system5.5 Weak formulation continuous Galerkin NS TEA O(h4) PDE FE basis algorithms5.6 Weak formulation finite volume NS TEA altered O(h4) PDE algorithm5.7 Theory synopsis, weak formulation error freed CFD, asymptotic convergence, optimal continuous GWSh FE p = 1,2,3 trial space basis algorithms5.8 Validation, shock continuum interpolation, monotonicity, stability, weak formulation compressible NS TEA O(h4) error freed PDE GWSh FE p = 1 basis algorithm5.9 Validation, weak formulation theory, asymptotic convergence, compressible NS TEA-altered O(h4) error freed PDE GWS FE p = 1,2,3 basis algorithms5.10 Validation, weak formulation theory, error quantification, compressible NS TEA altered O(h4) error freed PDE GWSh FE p = 1,2,3 basis algorithms5.11 Validation, weak formulation theory, optimal mesh identification, compressible NS TEA O(h4) error freed PDE GWSh FE p = 1,2,3 basis algorithms5.12 Incompressible NS (INS) error freed TEA altered O(h4) PDE system5.13 Validation, weak formulation theory, INS TEA altered O(h4) PDE GWSh FE p = 1 basis algorithm, driven cavity monotone DOF5.14 Validation, weak formulation theory, asymptotic convergence, INS TEA O(h4) error freed DmE GWSh FE algorithm, thermal stagnation point 5.15 Incompressible NS DM differential constraint, pressure Poisson PDE, DMh algorithm5.16 Validation, INS pressure, TEA O(h4) error freed DmP:O(h2)DMh + NBC GWSh FE p = 1 basis algorithm, n = 3 step wall diffuser 5.17 Validation, pressure, multiply connected domains, INS O(h4) error freed DmP: O(h2)DMh + NBC:DP + NBC GWSh + θTS FE p = 1 basis algorithm5.18 Conclusion, weak formulation Navier-Stokes continuous GWSh + θTS FE p = 1,2,3 basis algorithms cast on error freed CFD mathematicsChapter 6 Time-averaged (RaNS) Navier-Stokes error freed CFD weak formulation, theory, asymptotic convergence, validations6.1 Derivation, compressible RaNS:SA TEA O(h4) DmM:DmP:DmE:DmSA PDE system6.2 Validation, weak formulation continuous RaNS:SA TEA O(h4) error freed PDE GWSh FE p = 1 basis algorithm, shock monotone continuum interpolation, stability6.3 Validation, first-order effects predictions, same code/same mesh RaNS:SA O(h4) PDE GWSh and O(h2) PDE non-GWSh numerically stabilized CFD algorithms6.4 Validation/failure, GWSh/non-GWSh CFD first-, higher-order effects predictions6.5 Validation, shock-turbulent BL interaction CFD diagnostics, error freed PDE prediction of the dominant physics of fluids process6.6 Derivation, incompressible RaNS TEA O(h4) DmP:DmTKE:O(h2)DMh PDE system continuous GWSh FE basis algorithm, Newton Jacobian6.7 Validation, RaNS:TKE O(h4) PDE system GWSh + θTS FE p = 1 basis algorithm6.8 Spectral distribution of TEA annihilated O(h2) error, RaNS:TKE O(h4) PDE system6.9 Conclusion, RaNS error freed CFD mathematics O(h4) PDE systemsChapter 7 Annihilation of NS/RaNS PDE space-time discretization-induced O(m2, m3) phase dispersion and O(Δt3) truncation errors7.1 Characterization, NS scalar convective transport dispersion error mechanisms7.2 Derivation, NS DM scalar transport CFD algorithm amplification factor, n = 17.3 Derivation, NS mass transport DmY O(mp) TE annihilation, p = 2, 3, n = 17.4 Derivation, NS mass transport DmY PDE O(mp) TE annihilation, p = 2, 3, n-D7.5 Validation, space-time discretization aliasing error annihilation, NS O(mp) DmY, p = 2, 37.6 Derivation, trapezoidal rule O(Δt3) TE-annihilated NS/RaNS GWSh + θTS algebraic statementChapter 8 Hypersonics, aerothermodynamics, radiation CFD issues8.1 Issues uniquely challenging to hypersonic aerodynamics CFD8.2 Aerothermodynamics impact on hypersonic BL CFD prediction8.3 Radiation, weak formulation Stefan-Boltzmann RBC, Newton Jacobian8.4 Radiation, differential radiosity theory weak formulation NBC, Newton Jacobian8.5 Differential radiosity weak formulation, GWSh algorithm8.6 Validation, asymptotic convergence, differential radiosity weak formulation GWSh FE p = 1 basis algorithm8.7 Validation, long time stability, weak formulation gray body unsteady NS DE + NBC GWSh + θTS FE p = 1 basis trapezoidal rule algorithm8.8 Conclusion, NS/RaNS PDE system error-freed CFD mathematics
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