One-Minute Derivation of the Conjugate Gradient Algorithm

One of the great triumphs in the history of numerical methods was the discovery of the Conjugate Gradient (CG) algorithm. It could solve a symmetric positive-definite system of linear equations of dimension N in exactly N steps. As many practical problems at that time belonged to this category, CG algorithm became rapidly popular. It remains popular even today due to its immense computational power. But despite its amazing computational ability, mathematics of this algorithm is not easy to learn. Lengthy derivations, redundant notations, and over-emphasis on formal presentation make it much difficult for a beginner to master this algorithm. This paper aims to serve as a starting point for such readers. It provides a curt, easy-to-follow but minimalist derivation of the algorithm by keeping the sufficient steps only, maintaining a uniform notation, and focusing entirely on the ease of reader

One-Minute Derivation of the Conjugate Gradient Algorithm

One of the great triumphs in the history of numerical methods was the discovery of the Conjugate Gradient (CG) algorithm. It could solve a symmetric positive-definite system of linear equations of dimension N in exactly N steps. As many practical problems at that time belonged to this category, CG algorithm became rapidly popular. It remains popular even today due to its immense computational power. But despite its amazing computational ability, mathematics of this algorithm is not easy to learn. Lengthy derivations, redundant notations, and over-emphasis on formal presentation make it much difficult for a beginner to master this algorithm. This paper aims to serve as a starting point for such readers. It provides a curt, easy-to-follow but minimalist derivation of the algorithm by keeping the sufficient steps only, maintaining a uniform notation, and focusing entirely on the ease of reader

Symmetry-preserving discretization of the incompressible form of the Navier-Stokes equations under turbulent conditions. LES simulation of a turbulent channel flow

The incompressible form of the Navier-Stokes equations (conservation of mass, momentum and energy) is solved by applying a second-order symmetry-preserving spatial discretization which allows to preserve the symmetry of the operators. The physics behind turbulent flows and how those can be modelled is studied, considering both the RANS equations and the LES model. The Taylor-Green vortex problem is solved with no model and compared with the results of van Rees et al. [4], obtaining very good agreement regarding the time evolution of the volume-averaged kinetic energy, but higher discrepancies in the time evolution of the kinetic energy dissipation rate. Additionally, DNS results for a turbulent channel flow at Reτ “ 180 are obtained with coarse meshes. The same problem is also solved by applying the Smagorinsky, S3PR and Vreman’s LES models. DNS results obtained with a 323 mesh show relatively good agreement with the reference results of Moser et al. [5], while LES simulations employing the S3PR and Vreman’s model allow to improve the results in the buffer-layer region