Dispersion-fitted finite differences

Interestingly, if one Taylor series expands Eq. (36) and equates the terms of the same order in kj with Eq. (37) one can derive the standard Lagrangian FD approximations (i.e., require the coefficient of kj to be —1, and require the coefficient of all other orders in kj up to the desired order of approximation to be 0.) A more global approach is to attempt to fit Eq. (36) to Eq. (37) over some range of Kj = kjA values that leads to a maximum absolute error between Eq. (36) and Eq. (37) less than or equal to some prespecrfied value, E. This is the essential idea of the dispersion-fitted finite difference method [25]. 

Gray. S.K. and Goldfield. E.M. (2001) Dispersion fitted finite difference method with applications to molecular quantum mechanics J. Chem. Phys. 115, 8331-8344. 

Figure 14.1-3 is an oversimplification. In practice, mass transfer and dispersion effects smooth out the shock wave into an S-shaped curve. Equation (14.1-3) with finite differences does give a good prediction of where the center of the S-shaped curve will be. When equilibrium theory predicts a shock wave, a constant pattern is observed and constant pattern solutions can be used. Except at the two comets, the diffuse wave is often a good fit to the experimental data. This allows the calculation of nonlinear equilibrium constants from diffuse waves. ... 

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