关键词: nonlinear equation, stability, Newton’s method, auto-adjustabledamping method, the vector of damping factors

Abstract: The general approach for solving the nonlinear equations is linearizing the equations and forming various iterative procedures,then executing the numerical simulation.For the strongly nonlinear problems,the solution obtained in the iterative prpcess is always difficult, even divergent due to the numerical instability.It can not fulfill the engineering requiements. Newton’s method and its variants can not settle this problem. As a result, the application of numerical sinulation for the strongly nonlinear problems is limited. An auto-adjustable damping method has been presentd in this paper. This is a further improvement of Newton’s method with damping factor A set of vector of damping factor is introduced. This set of vector can be adjsted continuously during the iterative process in accordance with the judgement and adjustment. An effective convergence coefficient and quichening coefficient are employed to relax the restricted requirements for the initial values and to shorten the iterative process. Then. the numerical stability will be ensured for the solution of complicated strongly nonlinear equations.Using this method.some complicated strongly nonlinear heat transfer problems in airplanes and aeroengines have been numerically simulated successfully. It can be used for the numerical simulation ofstrongly nonlinear problems in engineering such as nonlinear hydrodynamic and aerodynamics, heat transfer and struactural dynamic response etc.

Key words: nonlinear equation, stability, Newton’s method, auto-adjustabledamping method, the vector of damping factors