This approach attempts to improve the accuracy and speed of the Ewald sum calculation by using a polynomial approximation rather than tabulation. The idea is to find a polynomial approximation to the Ewald potential that is cheaper to evaluate and then differentiate it analytically to get the forces. There exist several polynomials to approximate the Ewald sum ranging from the simple method of Brush et al. [13] to the more accurate cubic harmonics of Von der Lage and Bethe [52].
Hansen's work [28] for Monte Carlo simulations used an anisotropic approximation
where 
was split into an isotropic part and a cubically symmetric term while
utilizing exact cubic harmonics for better convergence.
The error reported in Hansen's method for the total potential energy was less than 
.
Similar work was done by Slaterry et al. [48], but the solution was expressed as a sum
of Poisson's equation with cubic symmetry.
Adams and Dubey [2] offer a variety of approximations for charge-charge, charge-dipole,
and dipole-dipole systems. In one accurate approximation, the Ewald potential
was expanded in powers of r and then the expansion was approximated with functions
of cubic symmetry to obtain the coefficients.
The resulting sum expansion has this form:
where 
, l is even; 
is zero except for 
; and S is the
``self term''. The 
and 
coefficients are tabulated in [2].
Adams and Dubey report an RMS error of 
for l=6, and 
with l=14. Our experimentation showed that the above approximations are expensive for moderate to
high accuracy calculations. Alternatively, one can use a polynomial approximation to the 
function that is faster to evaluate if tabulation is expensive, e.g. see [1].
However, for high accuracy computations or long time simulations,
such approximations will perform poorly as errors accumulate.
In summary, approximations methods, analogous to tabulation, offer a fast alternative to Ewald
simulations with limited accuracy.