Alternatively triangular screw-symmetric method for solving linear systems with non-symmetric definite matrix

Alternatively triangular screw-symmetric method (ATSM) – variant of implicit alternatively triangular iterative scheme – is proposed for solving large linear systems with strongly non-symmetric definite sparse matrix. Convergence criterion was proved in a relevant energy space, so constructive sufficient condition of convergence was established which permit us to propose practical way to choose parameter. In addition implicity matrix of the ATSM may be independently used as preconditioning matrix while solving a system by other methods. This preconditioning technique can be treated as simplified ILU-factorization, and we name it MSSILU (modified screw-symmetric ILU) preconditioning. Numerical tests have included solving of the steady convection-diffusion problem by five methods: ATSM, triangular screwsymmetric method TSM, alternately triangular symmetric method, GMRES(2) and GMRES(10). The last two methods have been used independently as well as with MSSILU-preconditioning. Tests reveal advantage of ATSM which appears to be noticeably better even than GMRES(10) with preconditioning.