  A very simple and efficient local variational iteration method 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.
