Numerical Perturbation Algorithm and Its CFD Schemes

The numerical perturbation algorithm presented by the author is to couple fluid dynamics effects with mathematical basic schemes (MBS), especially with the most simplest MBS, i.e. first order upwind and second order central schemes for the Navier-Stokes(NS) equations and convective diffusion equation. As a result, many new schemes are obtained and called fluid mechanics basic scheme (FMBS), for convenience and emphasizing MBS coupling with fluid dynamics effects. The main steps of constructing FMBS are that flux and coefficient of convective derivative in MBS are reconstructed as power-series of grid interval and that by splitting resultant scheme above and operating the splitted scheme, the high-order fluid mechanics relation is obtained and that the variables at upstream and downstream nodes are expanded in Taylor series and that by eliminating truncated error terms in the modified differential equation of the reconstructed scheme the undetermined coefficients in the power-series are determined and then the FMBS is obtained. Formulations of FMBS is product of MBS and the numerical perturbation reconstruction functions, that are simple polynomial of (or ), where and are grid Reynolds number and grid CFL number, respectively.FMBS and original MBS utlize the same nodes and are nearly identical simple. But FMBS have higher accurate and larger stable-range than MBS. For example, the most simplest and the most important six FMBS for the convection diffusion (CD) equation are : sixth-order upwind finite-difference (FD) FMBS, dual perturbation (DP) fourth-and eighth-order central FD-FMBS, dual perturbation third- and fifth-order finite volume (FV) central FMBS and sixth upwind FV-FMBS. This six schemes are absolute stability or absolute positive and are non-oscillatory schemes for any values of grid Reynolds number. In the case of one dimension, this six schemes are TVD scheme for any values of grid Reynolds number. However, the same order MBS must use multi-nodes and oscillate on coarse grids, Besides FMBS being directly used to calculate flow, FMBS is also acted as a basic or starting scheme for reconstructing high resolution scheme by self-adjust numerical dissipation. The above six FMBS and others have been used to calculate incompressible flows, compressible flows, mass transfer and Marangoni convection in a falling drop, two phase flows etc., some excellent numerical results are given. For example, FMBS solve lib-driven and buoyancy-driven cavity flows and result in several new Benchmark solutions. The numerical perturbation algorithm and corresponding schemes are also called Gao's algorithm and Gao's schemes. However, the name of FMBS is more reasonable than the numerical perturbation scheme, this is because that FMBS may depict exactly physical connotation of MBS coupling with fluid dynamics effects. Several subjects being worthy of study further are discussed. The present method is also suitable for reconstruction of MBS of other mathematical physics equation (such as the simplified Boltzmann equation, magnetic fluid mechanic equations, KdV-Burgers equation etc.) coupling with physics dynamics effects.