TY - JOUR

T1 - Combination of the variational iteration method and numerical algorithms for nonlinear problems

AU - Wang, Xuechuan

AU - Xu, Qiuyi

AU - Atluri, Satya N.

PY - 2020/3

Y1 - 2020/3

N2 - 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.

AB - 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.

KW - Chebyshev polynomial

KW - Collocation method

KW - Local variational iteration method

KW - Nonlinear differential equation

UR - http://www.scopus.com/inward/record.url?scp=85075375458&partnerID=8YFLogxK

U2 - 10.1016/j.apm.2019.10.034

DO - 10.1016/j.apm.2019.10.034

M3 - Article

AN - SCOPUS:85075375458

VL - 79

SP - 243

EP - 259

JO - Applied Mathematical Modelling

JF - Applied Mathematical Modelling

SN - 0307-904X

ER -