Local Variational Iteration Method Research Articles

A quasi-linear local variational iteration method for orbit transfer problems

A novel Quasi-linear Local Variational Iteration Method (QLVIM) for solving two-point boundary value problems (TPBVPs) of strongly nonlinear systems is proposed. With quasi-linearization technique, the multidimensional nonlinear TPBVP is transformed into a series of iterative linear TPBVPs, and then into initial value problems (IVPs) which appear in pairs. Then these IVPs are solved by Local Variational Iteration Method (LVIM). Combining the rapid convergence of quasi-linearization with the high computational efficiency of LVIM, the proposed QLVIM can solve various orbit transfer problems in aerospace engineering efficiently and accurately. The high-performance of this method is verified in solving perturbed Lambert's problems and a circular restricted three-body orbit transfer problem. Comparisons with various methods show that the QLVIM has the advantages of high convergence speed, low computation cost and longer solvable time span. Rather than a refined initial guess of starting velocity, the QLVIM only needs a rough initial guess of the trajectory. The results also prove that the QLVIM is capable of solving long-time-span orbit transfer problem. A theoretical proof of its quadratic convergence is given at last.

Combination of the variational iteration method and numerical algorithms for nonlinear problems

A very simple and efficient local variational iteration method (LVIM), or variational iteration method with local property, for solving problems of nonlinear science is proposed in this paper. The analytical iteration formula of this method is derived first using a general form of first order nonlinear differential equations, followed by straightforward discretization using Chebyshev polynomials and collocation method. The resulting numerical algorithm is very concise and easy to use, only involving highly sparse matrix operations of addition and multiplication, and no inversion of the Jacobian in nonlinear problems. Apart from the simple yet efficient iteration formula, another extraordinary feature of LVIM is that in each local domain, all the collocation nodes participate in the calculation simultaneously, thus each local domain can be regarded as one “node” in calculation through GPU acceleration and parallel processing. For illustration, the proposed algorithm of LVIM is applied to various nonlinear problems including Blasius equations in fluid mechanics, buckled bar equations in solid mechanics, the Chandrasekhar equation in astrophysics, the low-Earth-orbit equation in orbital mechanics, etc. Using the built-in highly optimized ode45 function of MATLAB as a comparison, it is found that the LVIM is not only very accurate, but also much faster by an order of magnitude than ode45 in all the numerical examples, especially when the nonlinear terms are very complicated and difficult to evaluate.