Finding Solutions that Converge

The key problem is to find a method for making the new guess that converges rapidly to the correct answer. There are a host of techniques. Unfortunately there is no best method for all equations. Some methods that converge very rapidly for some equations will diverge for other equations i.e., the series of new guesses will oscillate around the correct solution with ever-increasing deviations. This is one kind of numerieal instability. 

In attempts to develop conventional electrodynamic models of the individual photon, it is difficult to finding axisymmetric solutions that both converge at the photon center and vanish at infinity. This was already realized by Thomson [22] and later by other investigators [23]. 

Root finding in (2.1) invariably proceeds by iteration (refs. 1-3), constructing a sequence x, x2>... of approximate solutions that are expected to converge to a root r of (2.1). Naturally you would like to terminate the... 

The procedure shown in (7.29) has the same shortcomings as the one-dimensional case. It is, moreover, quite difficult to find a proper formulation that converges to the solution. 

A numerical method is said to be direct when it finds a solution within a given precision and, with a given accuracy, in an initially known number of operations. The time required to solve a differential equation is then well known a priori, and it is independent of the initial or boundary conditions of the problem. Iterative methods, on the other hand, are based on a sequence of approximations to the required solution, starting from an initial guess that converges to the solution. The number of operations, and the time required by these latter methods, are initially unknown because they depend on the initial guess and may vary dramatically as a function of the parameters of the problem. 

The above examples illustrate the fact that, in the case of singularly perturbed elliptic and parabolic equations, the use of classical finite difference schemes does not enable us to find the approximate solutions and the normalized diffusion fluxes with e-uniform accuracy. To find approximate solutions and normalized fluxes that converge e-uniformly, it is necessary to develop special numerical methods, in particular, special finite difference schemes. 

The solution procedure is to assume a value of e2. Substitute into Eq (1) and try values of e1 until the RHS equals the LHS. Substitute that value of e1 into the RHS of Eq (2) and find another value of e2. Repeat until convergence. The final values are... 

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