On improved methods for monotone CFD solution accuracy

The critical issues of improved solution accuracy, stability, monotonicity and efficiency for computational fluid dynamics (CFD) problems is the subject. The goal is to work towards a generalized algorithm that extracts the "best" solution out of an affordable "optimal” mesh. By optimal, we mean utilizing the least number of degrees of freedom possible for achieving ||u — u_h,p|| ≤ ε, where ε is a small positive real number. Computational fluid dynamics (CFD) solutions are especially prone to phase dis-persion error yielding moderate to high distortion and ultimately instability. Specif-ically, the universally intrinsic zero phase velocity for "2Δx" information induces an error cascading to smaller wave number modes that appears as mesh scale solution oscillations. Artificial viscosity and/or flux correction operators are designed to dif-fuse this error mode, but may produce unacceptable amounts of artificial dissipation especially in multidimensional embodiments. Alternatively, fluxvector splitting oper-ators are not sufficient to prevent oscillations near strict local extrema. Introduction of non-linear correction factors called limiters may attain monotone solution at the expense of a very complicated procedure in multi-dimensions, and even then for a moderately coarse mesh the solution remains nodally inexact. In this dissertation, three algorithms are derived to address the subject as follows: a) A totally new, non-linear sub-grid embedding (SGM) weak statement finite element algorithm is derived, containing a local extremization step for which the the-ory predicts attainment of nodally accurate monotone solutions on arbitrary meshes. Further, concomitant accurate boundary fluxes can be established, with a minimal degrees of freedom solution, for linear and nonlinear parabolic benchmark problems and a Navier-Stokes problem. b) The discrete approximate solution error, in general, can be expressed as a truncation of a Taylor series expansion. A Galerkin weak statement matrix pertur-bation (GMP) theory is derived yielding tridiagonal forms (in one-dimension) that can reduce, or annihilate in special cases, the Taylor series identified truncation error to high order. The theoretical analysis is via a von Neumann frequency analysis, and corresponding verification and benchmark solutions are documented in one and two dimensions. In concert, using an element-specific (local) Courant number, a con-tinuum (total) time integration procedure is derived that can directly produce an accurate temporal solution independent of mesh measure and time step. c) A discrete solution reconstruction (RON) procedure is derived and validated for accuracy enhancement of dispersed (poor quality) solutions produced by "any" CFD algorithm. This algorithm theory is based on the preservation of l_p norms during solution evolution, and can yield an absolutely dispersion-free reconstruction of any transient or steady-state CFD solution.