A Study On Solving Nonlinear Differential Equations Using Accelerated Search-Extension Method And New Extrapolation Cascadic Multi-grid Method

With the development of natural science,more and more nonlinear models are established in numerous fields,such as astrophysics,quantum mechanics. A great number of results on numerical methods are obtained.In this paper, we shall study the accelerated search-extension method and new extrapolation cascadic multi-grid method for solving the nonlinear differential equations.Firstly,we shall propose an accelerated search-extension method(ASEM) for solving the multiple solutions of nonlinear differential equations.The main algorithm of this approach is as follows:(1).On the coarse grid,we solve the discrete nonlinear system by the search-extension method(SEM)proposed by C.M.Chen and Z.Q.Xie,and the solution is denoted by u_h;(2).On the fine grid,we construct a linear system based on the solution u_h of the coarse grid and solve it to obtain e_H.Then u_h+e_H is the first correction of u_h;(3).On the coarse grid,we construct another linear system based on u_h and e_H,and we denote its solution as e_h.AS a consequence u_h+e_H+e_h is a further correction of u_h.In order to demonstrate the efficiency of ASEM,we compute a solution of the one-dimensional semilinear elliptic equations using ASEM and SEM, respectively.When these two approaches reach the same error,we compare the elapsed time and find that ASEM reduces the computation cost dramatically. Furthermore,we compute the multiple solutions of two-dimensional semilinear elliptic equations corresponding to the 1-4pies eigenvalues by ASEM.Secondly,we shall investigate a new extrapolated cascadic multigrid method (NECMM)for solving the nonlinear differential equations with an unique solution. The main algorithm of this method is as follows:(1).We solve the semilinear system on the uniform grid with the step-length h by the finite element method;(2).We refine the grid with the step-length h/2 and solve it with the same method;(3).We refine the grid with the step-length h/4 once again and use the aforementioned solution to transform the nonlinear system to linear system,then solve it by the conjugate gradient method with a good initial guess obtained by a new extrapolation approach based on the aforemen- tioned two solutions;(4).Repeat the procedures above.In this paper,we solve the semilinear elliptic equations by the cascadic multigrid method with the initial guess using the linear interpolated method,extrapolated method and new extrapolated method,respectively.The numerical results will demonstrate the efficiency of NECMM.