A modified conjugate gradient (MCG) method, recently derived for symmetric positive definite matrices, has been extended to solve sparse sets of arbitrary linear non-symmetric equations. The original system Ax = b is ideally transformed into the symmetric positive definite system AT Ax = AT and an approximate inverse K-1 of A is obtained with the incomplete LU (Crout) factorization of A. A straightforward derivation of the symmetric MCG scheme is applied to the transformed equations. If K-1 is a satisfactorily good approximation to A- in the sense required by MCG, the eigenvalues of the iteration matrix are favourably distributed in the vicinity of 1 and MCG turns out to be very rapidly convergent. This condition occurs when the vast majority of the rows of A are diagonally dominant. A further acceleration is obtained by taking the initial guess equal to the initial solution of the Newton iteration. The new scheme has been experimented on sparse matrices arising from the finite element integration of the diffusive-convective equation in various physical environments (e.g. surface water, subsurface water and atmosphere). The results show that when the time integration step Δt is relatively large and the convection is not negligible, most of the equations of A are not diagonally dominant and therefore MCG is slowly convergent. If Δt is kept sufficiently small to allow A to be diagonally or nearly diagonally dominant, K-1 is close to A-1 and convergence is very fast. © 1979.
Solution to unsymmetric finite element diffusive convective equations by a modified conjugate gradient method
GAMBOLATI, GIUSEPPE
1979
Abstract
A modified conjugate gradient (MCG) method, recently derived for symmetric positive definite matrices, has been extended to solve sparse sets of arbitrary linear non-symmetric equations. The original system Ax = b is ideally transformed into the symmetric positive definite system AT Ax = AT and an approximate inverse K-1 of A is obtained with the incomplete LU (Crout) factorization of A. A straightforward derivation of the symmetric MCG scheme is applied to the transformed equations. If K-1 is a satisfactorily good approximation to A- in the sense required by MCG, the eigenvalues of the iteration matrix are favourably distributed in the vicinity of 1 and MCG turns out to be very rapidly convergent. This condition occurs when the vast majority of the rows of A are diagonally dominant. A further acceleration is obtained by taking the initial guess equal to the initial solution of the Newton iteration. The new scheme has been experimented on sparse matrices arising from the finite element integration of the diffusive-convective equation in various physical environments (e.g. surface water, subsurface water and atmosphere). The results show that when the time integration step Δt is relatively large and the convection is not negligible, most of the equations of A are not diagonally dominant and therefore MCG is slowly convergent. If Δt is kept sufficiently small to allow A to be diagonally or nearly diagonally dominant, K-1 is close to A-1 and convergence is very fast. © 1979.
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