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A Manifold-Based Exponentially Convergent Algorithm For Solving Non-Linear Partial Differential Equations, Chein-Shan Liu

Journal of Marine Science and Technology

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 …