Approximation errors asymptotic solutions

Of course, we can never change a physical (or mathematical) problem by simply nondi-mensionalizing variables, no matter what the choice of scale factors. It is only when we attempt to simplify a problem by neglecting some terms compared with others on the basis of nondimensionalization that the correct choice of characteristic scales becomes essential. Fortunately, as we shall see, an incorrect choice of characteristic scales resulting in incorrect approximations of the equations or boundary conditions will always become apparent by the appearance of some inconsistency in the asymptotic-solution scheme. The main cost of incorrect scaling is therefore lost labor (depending on how far we must go to expose the inconsistency for a particular problem), rather than errors in the solution. 

DPCM and IEF are exact (and therefore equivalent) as long as the solute charge lies completely inside the cavity, whereas COSMO is only asymptotically exact in the limit of large dielectric constants. If there is some escaped charge, i.e. if some part of the charge distribution is supported outside the cavity, all these methods are approximations. The error generated by the fact that, in QM calculations, the electronic tail of the solute necessarily spreads outside the cavity, is discussed in Section 1.2.4. 

These formulas represent not only the formal solution, which provides the small residuals in the equations, but also the exact solution of the original problem up to remainder terms. The errors of the approximation can be evaluated (it is a purely mathematical problem). After that, one can write equalities that are called asymptotic expansions of the exact solution. The leading terms of these are... 

Formulas (4.9), (4.10) provide the approximate solution on the slow time scale. From these asymptotics one can see that the second and third components tend to zero, whereas the first one tends to a nonzero constant y = (W/A) at infinity. If we use Taylor expansions of the left sides of Eqs. (4.9) and (4.10), the error estimate of the asymptotic behavior at infinity can easily be derived as follows ... 

This index can be obtained using two grid levels. However, three levels are recommended to properly estimate the order of approximation of convergence and to verify if the solution is in the range of asymptotic convergence. This range is reached when the spacing of the mesh h, for various errors E, results in a constant C, as... 

Comparing the results obtained by the WKB method with the exact solutions for the planar and spherical surface, we find, within 2% error, quantitative agreement in the planar case. For a sphere, we find the same asymptotic dependence of critical adsorption behavior for a wide range of geometries. The main advantage of the WKB method is a unified approach for the various geometries based on the same level of approximations. It can be applied at the same level of complexity to virtually any shape of the polylectrolyte-surface adsorption potential. Recent advances in polyelectrolyte adsorption under confinement [49,167] and adsorption onto low-dielectric interfaces [50] have been presented. 

---