Spatial discretization of the governing equations

We discretize both the Stokes and heat diffusion equations using a Finite Differences approach on a staggered grid (ref Taras book here). Pressure, temperature and the diagonal stress/strain-rate components (Cᵢᵢ) live at cell centers, the off-diagonal components (Cᵢⱼ) at cell corners, and velocity/flux components on the cell faces they cross:

Time and pseudo-time discretization of the APT equations

We discretize both the Stokes and heat diffusion equations using a Finite Differences approach on a staggered grid (ref Taras book here).

Heat diffusion

Upon convergence we recover

Stokes equations

Conservation of momentum

Conservation of mass

Constitutive equation