Lower-upper Symmetric Gauss-Seidel Algorithm Research Articles

An implicit free element method for simulation of compressible flow

A novel implicit scheme based on free element method (FECM) is proposed to solve Navier–Stokes equations. The proposed method has the mesh-free features but elements are essential instead of point clouds. The isoparametric transformations, which are usually used in finite element method, are employed to obtain the coefficients in the spatial derivative approximation. The method satisfies the geometric conservation law and first-order consistency. An approximate Riemann solver is employed to compute convective fluxes at the midpoints between collocated points and field points. Meanwhile, a reconstruction approach is used to reach higher order discretization. Viscous terms are approximated with a corrected average progress of gradients. Furthermore, a matrix-free Lower-Upper Symmetric Gauss–Seidel (LU-SGS) scheme is introduced to speed up the convergence. In order to validate the proposed method, three test cases varying from subsonic to supersonic flows are simulated and discussed in detail. The test cases validate that the proposed method is as accurate as second order finite volume solver and is robust with different node sets. What's more, LU-SGS algorithm can significantly improve convergence speed compared with the explicit scheme.

An investigation of natural convection in a three dimensional square wavy channel

Natural convection in a three dimensional wavy channel, which is composed of numerous two dimensional cross sections, is investigated numerically. The method of elliptic partial differential equations grid generation is adopted to construct a distribution of grids on each cross section. The distributions of the whole cross sections are piled up in sequence from the inlet to outlet to form a three dimensional channel. In order to solve a low speed compressible flow problem, the compressibility and viscosity of the working fluid are taken into consideration, and the non-reflecting boundaries at the inlet and outlet of the channel are held. The Bossinesq assumption is then no longer necessary. As well, the methods of the Roe scheme, preconditioning and dual time stepping matching the implicit LUSGS algorithm (Implicit Lower Upper symmetric Gauss–Seidel algorithm) are simultaneously used to solve governing equations. Effects of the Rayleigh number and the ratio of the amplitude to wave length on heat transfer rates of the channel are examined. Results of both this study and existing studies have good agreements.

Validation of Unstructured Grid Method for Incompressible Flows using Artificial Compressibility Method.

The artificial compressibility algorithm for incompressible flows is applied to an unstructed grid method. It is based on a finite volume cell-vertex upwind technique and uses a time-marching procedure to efficiently solve the continuity equation as well as the momentum equations. The Lower-Upper Symmetric Gauss-Seidel (LU-SGS) approximate factorization scheme is implemented with the Navier-Stokes solver on arbitrary three-dimensional unstructured grid. The LU-SGS algorithm provides a significant enhancement of the convergence rate. The accuracy of the method is validated by applying the method to incompressible flow problems of a 2-D cavity and 3-D duct.