5.23 Summary of algorithms and solvers
- The ﬁnite volume method produces sparse matrices, Sec. 5.1 .
- Symmetric matrices have the same coeﬃcients across the diagonal and advection generally produces an asymmetric matrix.
- The matrix equations are solved using iterative methods that reduce a residual to a speciﬁed tolerance, Sec. 5.4 .
- Diagonal dominance is a suﬃcient condition for convergence, see Eq. (5.9 ) and Sec. 5.5 .
- Under-relaxation may be required to maintain diagonal dominance, Sec. 5.6 , in particular for steady-state solutions.
- Simple iterative method for solutions with cost , Sec. 5.2 .
- Eﬀective when the range of inﬂuence is short, i.e. in each cell is inﬂuenced by changes in close neighbouring cells.
- The symmetric variant and mesh renumbering may improve convergence, Sec. 5.8 .
Conjugate gradient method
- A “descent” method which solves a minimisation problem for cases with a longer range of inﬂuence, Sec. 5.14 .
- Solved iteratively, which requires preconditioning to make it eﬃcient, Sec. 5.16 .
- Symmetric matrix: PCG with DIC preconditioning.
- Asymmetric matrix: PBiCGStab with DILU preconditioning.
Geometric-algebraic multi-grid (GAMG)
- Well suited to equations with a long range of inﬂuence, in particular the pressure equation, Sec. 5.17 .
- Simple algebraic agglomeration is corrected by a scaling based on minimisation for Laplacian-dominated equations, Sec. 5.18 .
Systems of equations
- Systems of equations are solved iteratively with solutions from one equation substituted into the next, Sec. 5.9 .
- Mass and momentum conservation are coupled by a pressure equation and momentum and ﬂux correctors, Sec. 5.10 .
- Steady ﬂow solutions use the SIMPLE algorithm, Sec. 5.12 .
- Equations are solved in an iterative sequence with under-relaxation required to ensure convergence, Sec. 5.13.
- Transient solutions use a PISO loop, Sec. 5.19 .
- Correction for non-orthogonality in the pressure equation may require a distinct corrector loop, Sec. 5.20 .
- In the PISO loop, typical conﬁgurations solve: 2 pressure equations, each including 1 non-orthogonal corrector; or 3 pressure equations without any non-orthogonal correctors.
- The PIMPLE algorithm provides additional controls for transient and pseudo-transient simulations, Sec. 5.21 .
Notes on CFD: General Principles - 5.23 Summary of algorithms and solvers