Linear Equation Systems are most concerned in modern scientific and engineering computation, because most problems can be described by (partial) differential equations which could be discretized to a system of algebraic equations which can be solved on a computer. There are many methods to solve these algebraic equations, but it can be divided into two groups: one is direct methods, such as Gauss elimination, LU decomposition, tridiagonal systems, another is iterative methods.
The following solution for equation systems are mainly used in OpenFOAM which is an Open Source CFD Toolkit.
PBiCG: Preconditioned Bi-Conjugate Gradient solver for asymmetric lduMatrices using a run-time selectable preconditiioner.
BICCG: Diagonal incomplete LU preconditioned BiCG solver derived from the general preconditioned BiCG solver PBiCG but with the choice of preconditioner pre-selected. This solver is present for backward-compatibility and the PBiCG solver should be used for preference.
ICCG: Incomplete Cholesky preconditioned CG solver derived from the general preconditioned CG solver PCG but with the choice of preconditioner pre-selected. This solver is present for backward-compatibility and the PCG solver should be used for preference.
PCG: Preconditioned Conjugate Gradient solver for symmetric lduMatrices using a run-time selectable preconditiioner.
GAMG: Geometric Agglomerated algebraic MultiGrid solver.
For more detailed information about above solutions, please the references or other books.
References
1. Ferziger, J. H., Peric, M., Computational Methods for Fluid Dynamics. 3rd, rev. ed, Springer, 2002.
2. OpenFOAM sourcefiles in the folder of OpenFOAM-\src\OpenFOAM\matrices\lduMatrix\solvers.