10.3.3 Relaxation method

The relaxation method [336,247] starts by first discretizing the governing equations by finite differences on a mesh with points. The computation begins with an initial guess and improves the solution iteratively or in other words relaxes to the true solution. Let us consider an arc length parametrized curve connecting and on the surface with a mesh of points satisfying . We approximate the first order differential equations by the trapezoidal rule [117]

with boundary conditions

Here the -vectors , are meant to approximate and . has known components, while has known components. This discrete approximation will be accurate to the order of ( ). Equation (10.47) forms a system of nonlinear equations with unknowns . The remaining equations come from boundary conditions (10.48). Let us refer to (10.47) as

where , and refer to (10.48) as

then we have nonlinear equations

This system of nonlinear equations can be solved by quadratically convergent Newton iteration, if a sufficiently accurate starting vector is provided. The Newton iteration scheme is given by

where superscripts denote - iteration and is the by Jacobian matrix of with respect to .

Since the corrections are based on a first order Taylor approximation, the usual Newton method may not be sufficient for a complex nonlinear problem unless a good initial approximation is provided. If the vector norm of the correction vector is large, then it is an indication that the problem is highly nonlinear and may produce a divergent iteration. To achieve more stability we can employ a step correction procedure

where chosen so that , where is a scaled vector norm and defined as

where , , and are the scale factors for each variable. Maekawa [247] used = =1 and = =10, since the magnitude of and are roughly ten times larger that of and as numerical experiments have shown. If the equation reduces to the usual Newton's method, while if the rate of convergence will be less than quadratic. Newton's method terminates when the norm of the solution vector is smaller than the pre-specified tolerance . The order of should be proportional to , since we are using the trapezoidal rule (see (10.47)).