_{1}

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The paper first introduces two-dimensional convection-diffusion equation with boundary value condition, later uses the finite difference method to discretize the equation and analyzes positive definite, diagonally dominant and symmetric properties of the discretization matrix. Finally, the paper uses fixed point methods and Krylov subspace methods to solve the linear system and compare the convergence speed of these two methods.

In the case of a linear system

The goal of this paper is to find an efficient iterative method combined with preconditioning for the solution of the linear system

For parameter

top. Now, we take

equals to 0, using central difference method to the diffusion term, we get the discretization matrix of the Equation (1)

For

we obtain the discretization matrix of the Equation (1)

where

For

we obtain the discretization matrix of the Equation (1)

where

In this section, we would first compute the eigenvalues of the discretization matrices

Using MATLAB, the eigenvalues of the discretization matrices

Matrix A is positive definite if and only if the symmetric part of A i.e.

we have all the eigenvalues of

Therefore

For all the discretization matrices

Therefore, the discretization matrices

It is easy to see that only

The goal of this section is to find an efficient iterative method for the solution of the linear system

Without loss of generality, take

In order to achieve

Computational results using fixed point methods such as Jacobi, Gauss-Seidel, SOR etc. and projection methods such as PCG, BICG, BICGSTAB, CGS, GMRES and QMR are listed out in figures (Figures 1-21), for all three different

The tables (

From the figures (Figures 1-21) and tables (

• The convergence speeds of SOR, Backward SOR and SSOR are faster than that of GS and Backward GS; while GS and Backward GS are faster than Jacobi;

• If matrix A is symmetric

• The convergence speeds of SOR and Backward SOR are the same, also for GS and Backward GS;

• From

• The upwind difference method is more suitable to be applied to convection dominant problem than the cen-

Gamma | Function | Preconditioning | flag | No. of iterations | Relres | Delta (tol) |
---|---|---|---|---|---|---|

0 | PCG | jacobi | 0 | 52 | 1.31E−06 | 1.55E−06 |

luinc | 0 | 23 | 1.38E−06 | |||

cholinc | 0 | 25 | 9.05E−07 | |||

BICG | jacobi | 0 | 52 | 1.31E−06 | ||

luinc | 0 | 23 | 1.38E−06 | |||

cholinc | 0 | 25 | 9.05E−07 | |||

BICGSTAB | jacobi | 0 | 40.5 | 1.42E−06 | ||

luinc | 0 | 15.5 | 9.36E−07 | |||

cholinc | 0 | 16 | 1.54E−06 | |||

CGS | jacobi | 0 | 43 | 1.20E−06 | ||

luinc | 0 | 16 | 8.27E−07 | |||

cholinc | 0 | 16 | 1.18E−06 | |||

GMRES | jacobi | 0 | 49 | 1.32E−06 | ||

luinc | 0 | 23 | 9.01E−07 | |||

cholinc | 0 | 24 | 1.21E−06 | |||

QMR | jacobi | 0 | 52 | 1.05E−06 | ||

luinc | 0 | 23 | 1.12E−06 | |||

cholinc | 0 | 24 | 1.37E−06 | |||

16 | PCG | jacobi | 1 | 4 | 0.483607757 | 1.14E−05 |

luinc | 1 | 2 | 0.312145391 | |||

cholinc | 1 | 2 | 0.38563672 | |||

bicg | jacobi | 0 | 89 | 7.33E−06 | ||

luinc | 0 | 25 | 1.11E−05 | |||

cholinc | 0 | 35 | 4.37E−06 | |||

BICGSTAB | jacobi | 0 | 56.5 | 1.03E−05 | ||

luinc | 0 | 16.5 | 6.30E−06 | |||

cholinc | 0 | 23.5 | 3.98E−06 | |||

CGS | jacobi | 0 | 61 | 6.61E−06 | ||

luinc | 0 | 20 | 1.01E−08 | |||

cholinc | 0 | 27 | 5.29E−06 | |||

GMRES | jacobi | 0 | 68 | 9.70E−06 | ||

luinc | 0 | 22 | 7.08E−06 | |||

cholinc | 0 | 30 | 1.10E−05 | |||

QMR | jacobi | 0 | 89 | 6.43E−06 | ||

luinc | 0 | 25 | 2.64E−06 | |||

cholinc | 0 | 33 | 9.57E−06 |

64 | PCG | jacobi | 1 | 1 | 0.710978037 | 4.13E−05 |
---|---|---|---|---|---|---|

luinc | 1 | 1 | 0.547110057 | |||

cholinc | 1 | 1 | 0.660611555 | |||

BICG | jacobi | 0 | 70 | 3.15E−05 | ||

luinc | 0 | 21 | 1.21E−05 | |||

cholinc | 0 | 44 | 3.47E−05 | |||

BICGSTAB | jacobi | 0 | 59.5 | 2.12E−05 | ||

luinc | 0 | 16.5 | 5.27E−06 | |||

cholinc | 0 | 27.5 | 1.60E−05 | |||

CGS | jacobi | 1 | 87 | 0.000123357 | ||

luinc | 0 | 19 | 6.36E−07 | |||

cholinc | 0 | 35 | 3.14E−06 | |||

GMRES | jacobi | 0 | 63 | 2.66E−05 | ||

luinc | 0 | 19 | 2.50E−05 | |||

cholinc | 0 | 37 | 3.11E−05 | |||

QMR | jacobi | 0 | 69 | 3.56E−05 | ||

luinc | 0 | 21 | 1.25E−05 | |||

cholinc | 0 | 45 | 3.31E−05 |

The fixed point method | No. of iterations | |
---|---|---|

Gamma = 0 | Jacobi | 2246 |

Gauss-Seidel | 1124 | |

SOR | 370 | |

Backward Gauss-Seidel | 1124 | |

Backward SOR | 370 | |

SSOR | 370 | |

Gamma = 16 | Jacobi | 459 |

Gauss-Seidel | 231 | |

SOR | 68 | |

Backward Gauss-Seidel | 231 | |

Backward SOR | 68 | |

SSOR | 42 | |

Gamma = 64 | Jacobi | 191 |

Gauss-Seidel | 97 | |

SOR | 54 | |

Backward Gauss-Seidel | 97 | |

Backward SOR | 54 | |

SSOR | 20 |

tral difference method;

• The convergence speed of the six projection methods including PCG, BICG, BICGSTAB, CGS, GMRES and QMR under luinc preconditioning are faster than under cholinc preconditioning, while under cholinc preconditioning are faster than Jacobi preconditioning;

• The six projection methods under Jacobi, luinc and cholinc are convergent when

I thank the editor and the referee for their comments. I would like to express deep gratitude to my supervisor Prof. Dr. Mark A. Peletier whose guidance and support were crucial for the successful completion of this paper. This work was completed with the financial support of Foundation of Guangdong Educational Committee (2014KQNCX161, 2014KQNCX162).