The linear third-order ordinary differential equation （ODE） can be transformed into a system of two second-order ODEs by introducing a variable replacement, which is different from the common order-reduced approach....The linear third-order ordinary differential equation （ODE） can be transformed into a system of two second-order ODEs by introducing a variable replacement, which is different from the common order-reduced approach. We choose the functions p（z） and q（x） in the variable replacement to get different cases of the special order-reduced system for the linear third-order ODE. We analyze the numerical behavior and algebraic properties of the systems of linear equations resulting from the sine diseretizations of these special second-order ODE systems. Then the block-diagonal preconditioner is used to accelerate the convergence of the Krylov subspace iteration methods for solving the discretized system of linear equation. Numerical results show that these order-reduced methods are effective for solving the linear third-order ODEs.展开更多
For nonsymmetric saddle point problems,Huang et al.in [Numer.Algor.75 (2017), pp.1161-1191]established a generalized variant of the deteriorated positive semi-definite and skew-Hermitian splitting (GVDPSS)precondition...For nonsymmetric saddle point problems,Huang et al.in [Numer.Algor.75 (2017), pp.1161-1191]established a generalized variant of the deteriorated positive semi-definite and skew-Hermitian splitting (GVDPSS)preconditioner to expedite the convergence speed of the Krylov subspace iteration methods like the GMRES method.In this paper,some new convergence properties as well as some new numerical results are presented to validate the theoretical results.展开更多
文摘The linear third-order ordinary differential equation （ODE） can be transformed into a system of two second-order ODEs by introducing a variable replacement, which is different from the common order-reduced approach. We choose the functions p（z） and q（x） in the variable replacement to get different cases of the special order-reduced system for the linear third-order ODE. We analyze the numerical behavior and algebraic properties of the systems of linear equations resulting from the sine diseretizations of these special second-order ODE systems. Then the block-diagonal preconditioner is used to accelerate the convergence of the Krylov subspace iteration methods for solving the discretized system of linear equation. Numerical results show that these order-reduced methods are effective for solving the linear third-order ODEs.
文摘For nonsymmetric saddle point problems,Huang et al.in [Numer.Algor.75 (2017), pp.1161-1191]established a generalized variant of the deteriorated positive semi-definite and skew-Hermitian splitting (GVDPSS)preconditioner to expedite the convergence speed of the Krylov subspace iteration methods like the GMRES method.In this paper,some new convergence properties as well as some new numerical results are presented to validate the theoretical results.
