A MANIFOLD-BASED EXPONENTIALLY CONVERGENT ALGORITHM FOR SOLVING NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS

Abstract

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.