A Manifold-Based Exponentially Convergent Algorithm for Solving Non-Linear Partial Differential Equations

Chein-Shan Liu

《Journal of Marine Science and Technology》 20卷4期 (2012/08) Pp. 441-449

https://doi.org/10.6119/JMST-011-0421-1

並列摘要

For solving a non-linear system of algebraic equations of the type: F(subscript i)(x(subscript j)) = 0, i, j = 1, …, n, a Newton-like algorithm is still the most popular one; however, it had some drawbacks as being locally convergent, sensitive to initial guess, and time consumption in finding the inversion of the Jacobian matrix ∂F(subscript i) /∂x(subscript j). Based-on a manifold defined in the space of (x(subscript i), t) we can derive a system of non-linear Ordinary Differential Equations (ODEs) in terms of the fictitious time-like variable t, and the residual error is exponentially decreased to zero along the path of x(t) by solving the resultant ODEs. We apply it to solve 2D non-linear PDEs, and the vector-form of the matrix-type non-linear algebraic equations (NAEs) is derived. Several numerical examples of non-linear PDEs show the efficiency and accuracy of the present algorithm. A scalar equation is derived to find the adjustive fictitious time stepsize, such that the irregular bursts appeared in the residual error curve can be overcome. We propose a future direction to construct a really exponentially convergent algorithm according to a manifold setting.

並列關鍵字

non-linear algebraic equations (NAEs) ； non-linear partial differential equations ； manifold-based exponentially convergent algorithm (MBECA) ； fictitious time integration method (FTIM)

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