A Simple Local Variational Iteration Method and Related Algorithm for Nonlinear Science and Engineering

TL;DR

This paper introduces a simple, efficient local variational iteration method for solving nonlinear differential equations, leveraging Chebyshev collocation and parallel processing to achieve high accuracy and speed, especially for complex nonlinear problems.

Contribution

The paper presents a novel local variational iteration method that avoids Jacobian inversion and enables parallel computation, improving efficiency and applicability for nonlinear science problems.

Findings

LVIM is highly accurate compared to ode45.

LVIM is significantly faster, especially for complex nonlinear problems.

The method is easily implementable with sparse matrix operations.

Abstract

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.