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formatting eq.
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aelligp authored Jul 30, 2024
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# Pseudo-transient iterative method
The pseudo-transient method consists in augmenting the right-hand-side of the target PDE with a pseudo-time derivative (where $\psi$ is the pseudo-time) of the primary variables.
The pseudo-transient method consists in augmenting the right-hand-side of the target PDE with a pseudo-time derivative (where $\psi$ is the pseudo-time) of the primary variables. We then solve the resulting system of equations with an iterative method. The pseudo-time derivative is then gradually reduced, until the original PDE is solved and the changes in the primary variables are below a preset tolerance.

## Heat diffusion
The pseudo-transient heat-diffusion equation is:
Expand All @@ -26,13 +26,10 @@ $\frac{1}{2\widetilde{G}} \frac{\partial\boldsymbol{\tau}}{\partial\psi}+ \frac{

where the wide tile denotes the effective damping coefficients and $\psi$ is the pseudo-time step. These are defined as in [Räss et al. (2022)](https://gmd.copernicus.org/articles/15/5757/2022/):

$ \widetilde{\rho} = Re\frac{\eta}{\widetilde{V}L}, \qquad
\widetilde{G} = \frac{\widetilde{\rho} \widetilde{V}^2}{r+2}, \qquad
\widetilde{K} = r \widetilde{G}
$
$\widetilde{\rho} = Re\frac{\eta}{\widetilde{V}L}, \qquad \widetilde{G} = \frac{\widetilde{\rho} \widetilde{V}^2}{r+2}, \qquad \widetilde{K} = r \widetilde{G}$

and
$
\widetilde{V} = \sqrt{ \frac{\widetilde{K} +2\widetilde{G}}{\widetilde{\rho}}}, \qquad
r = \frac{\widetilde{K}}{\widetilde{G}}, \qquad
Re = \frac{\widetilde{\rho}\widetilde{V}L}{\eta}
$

$\widetilde{V} = \sqrt{ \frac{\widetilde{K} +2\widetilde{G}}{\widetilde{\rho}}}, \qquad r = \frac{\widetilde{K}}{\widetilde{G}}, \qquad Re = \frac{\widetilde{\rho}\widetilde{V}L}{\eta}$

where the P-wave $\widetilde{V}=V_p$ is the characteristic velocity scale for Stokes, and $Re$ is the Reynolds number.

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