For solving a non-linear system of algebraic equations of the type: Fi(xj) = 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 ∂Fi /∂xj. Based-on a manifold defined in the space of (xi, 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.
"A MANIFOLD-BASED EXPONENTIALLY CONVERGENT ALGORITHM FOR SOLVING NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS,"
Journal of Marine Science and Technology: Vol. 20:
4, Article 12.
Available at: https://jmstt.ntou.edu.tw/journal/vol20/iss4/12