Research On Solving Algorithm Of Large Sparse Linear Equations With Multiple Right-hand Sides And Its Application In CFD

In computational fluid dynamics,iterative methods,such as Krylov subspace method,are usually used to solve the large sparse linear equations after discretization.However,in the linear system obtained by discrete method in CFD,the coefficient matrices are the same but the right-hand sides are different.For example,the matrices of the linear system which is generated by discreting the governing equation of different velocity components are the same while the right-hand sides are different.Both the species equation in the specie transport model and the volume fraction equation in the multiphase problem have the same characteristics.For this kind of problems,solving the same coefficient matrix one by one using the single right-hand method requires repeated operations,such as the pretreatment process,which will introduce redundant calculation consumption.The multiple right-hand sides method can solve all the right-hand sides at the same time,which reasonably avoids this problem.To explore the applicability of multiple right-hand sides algorithm for solving multiple right-hand sides problems in CFD can improve the efficiency of solving linear equations.This paper developed and implemented the Krylov subspace methods of linear system with single right-hand side,include BiCGSTAB and other algorithms,and solved the eigenvalues of the sparse matrix generated by momentum equation,species equation and volumn fraction equation.Combined with the distribution characteristics of eigenvalues of the sparse matrices,some tests of the commonly used Krylov subspace methods have been conducted.According to the results,the efficiency of BiCGSTAB is higher than GMRES and CGS.On this basis,combined with the current situation of solving equations in CFD at domestic and abroad,the solution of the linear system with multi-right-hand sides has been further studied.The BlBiCGSTAB algorithm and Seed BiCGSTAB algorithm are developed and implemented.The calculation process of residual for Seed BiCGSTAB algorithm was improved.The two methods are tested numerically.And the results are compared with those of the BiCGSTAB algorithm.The effect of the linear correlation of each right-hand side on solving efficiency is analyzed.The results show that the improved Seed BiCGSTAB algorithm is more efficient than the original one.When some of the basis vectors which form the solutions overlap,the LU decomposition of the block vector dot product matrix in the construction process.For this reason,when the linear correlation of the right-hand sides is strong,the BlBiCGSTAB algorithm will have large residual oscillation and ever collapse in the iterative process,while the Seed BiCGSTAB algorithm is more stable.The Seed BiCGSTAB algorithm is proved to be more efficient than BiCGSTAB algorithm in solving problems with multiple right-hand sides generated by species models in CFD.And its efficiency becomes more obvious with the increase of grid magnitude.This indicates that Seed BiCGSTAB algorithm has a certain application value in CFD calculation.