The Accelerated Pseudo-Transient (APT) method consists in augmenting the right-hand-side of the target PDE with a pseudo-time derivative (where 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 APT heat-diffusion equation is:

We use a second order APT scheme were continuation is also done on the flux, so that:

Stokes equations

For example, the APT formulation of the Stokes equations yields:

Constitutive equations

A APT continuation is also done on the constitutive law:

where the wide tile denotes the effective damping coefficients and is the pseudo-time step. These are defined as in Räss et al. (2022):

and

where the P-wave   is the characteristic velocity scale for Stokes, and is the Reynolds number.

Free-surface stabilization

For a vertically stratified 2D model, the free-surface stabilization used by the standard Stokes and DYREL solvers adds

to the vertical momentum residual. This velocity-dependent term contributes the local operator diagonal

The standard APT update uses   for the stabilized vertical pseudo-time step. DYREL adds the same to its vertical diagonal preconditioner and Gershgorin eigenvalue bound. This keeps the iterative coefficients consistent with the stabilized residual. A non-positive standard APT denominator is rejected through the solver's non-finite residual check because it indicates that the physical timestep is too large for the local density inversion.

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Physical parameters

Symbol	Parameter
	Temperature
	Flux
	Deviatoric stress
	Deviatoric strain rate
	Velocity
	External forces
	Pressure
	Viscosity
	Density
	Compressibility
	Shear modulus
	Thermal expansivity
	Heat capacity
	Heat conductivity

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Pseudo-transient parameters

Symbol	Parameter
	Pseudo time step
	Pseudo bulk modulus
	Pseudo shear modulus
	Characteristic velocity scale
	Pseudo density
	Relaxation time
	Reynolds number