L-stable Methods For Nonlinear Dynamic Equation

Nonlinear dynamics is a complex science that studies force and motion,involving engineering,astronomy,electromagnetism,bioscience and so on.The construction principle of model equation is complex,involving initial value problem and boundary value problem,with high degree of nonlinearity and great difficulty in solving,which puts forward higher requirements for the stability of numerical methods.In this paper,the general nonlinear dynamic model is solved by L-stable numerical method.Through spatial discretization,the boundary value conditions are regarded as constraints,and Lagrange multipliers are introduced to transform the nonlinear dynamic model into differential-algebraic model,which is verified by numerical examples.L-stability method is a method with simple solution format and good numerical stability.The method is mainly based on Taylor expansion,constructs a numerical solution scheme in the time interval,defines the stability function of the scheme,makes a rational approximation to it according to Euler theorem and conjecture,and then obtains the solution scheme with L-stability.The L-stability method is applied to solve the nonlinear dynamic equations.Finally,the nonlinear dynamic partial differential equations with boundary value conditions are transformed into nonlinear algebraic equations,which are solved by Newton iteration.The numerical stability of L-stability method is explored by changing the number of nodes,step size and simulation time.Compared with differential quadrature method(DQ method),fourth-order Runge-Kutta method(RK4 method)and other stability methods,the L-stability method is applied to solve two typical nonlinear dynamic problems: beam vibration problem and fluid transportation pipeline vibration problem.The experimental results show that the L-stability method is suitable for solving nonlinear dynamic problems with high accuracy,can better meet the boundary conditions and maintain good stability for a long time.Although the L-stable method has lower solution efficiency,it can be solved in a large step size compared with the DQ method and the RK4 method,which makes up for the problem of computational efficiency,and has good numerical solution stability at the boundary.Applying the L-stability method to the processing of other nonlinear dynamics problems is helpful to the development of nonlinear dynamics.