Relaxation coefficients

The relaxation coefficients in Eq. 107 are defined following P. Marbaix and van Ypersele (2003) as

$$K_r = {\frac{{K_r}^\ast \,f_R }{2 \Delta t}}$$ (107)

$$\kappa_r = {\frac{{\kappa_r}^\ast d^2 \,f_R }{2 \Delta t}}$$ (108)

Here, the coefficients approach their maximum values over $M_R$ (5 or 9) points using one of the following function options:

$$f_R = {\left( {\frac{M_R - i + 1 }{ M_R}}\right)}^{a_R}$$ (109)

$$= {\rm exp}\left[-a_R \left( {\frac{i-1 }{ M_R}}\right)\right]$$ (110)

$$= {\rm exp}\Bigg\lbrace -{1}{ 2} {\left[ a_R \left( {\frac{i-1 }{ M_R}}\right) \right] }^2 \Bigg\rbrace$$ (111)

$$= {1}{ 2} \Bigg\lbrace 1 \,+\, {\rm cos}\left[ \pi \, \left( {\frac{i - 1 }{ M_R}}\right) \right] \Bigg\rbrace$$ (112)

where

$$f_R = (i \geq M_R+1)$$ 0 (113)

$$f_R = 1 (i=1)$$ (114)

$$(0 < f_R < 1) ( 2 \leq i \leq M_R)$$ (115)

so that $f_R$ decreases from 1 (at the boundary) to 0 within the sponge zone. $a_R$ is a parameter which controls the rate at which $f_R$ goes to zero away from the lateral boundary. There really doesn't seem to be a consensus in the literature on the best form for $f_R$, which is why the multiple options for $f_R$ are available for testing in ASP. P. Marbaix and van Ypersele (2003) did a rather complete survey of relaxation methods, and they recommended an exponential function for $f_R$, however from tests using ASP, the Gaussian (exponential) form (Eq. 112) has been found to give the best results and is thus the current default function. In terms of the size of the sponge zone, the model currently uses $M_R=9$ as P. Marbaix and van Ypersele (2003) showed that this configuration produces slight better results than the 5-point form: indeed, tests with this model have indicated the same conclusion, although generally the results using the two values of $M_R$ are fairly similar with a slightly smoother transition for the 9-point case. ASP currently uses ${K_r}^\ast=0.6$ (when $M_R=5$) or ${K_r}^\ast=0.9$ (when $M_R=9$).