Advection Stabilization
Advection Stabilization#
ASPECT implements several advection schemes for the temperature and compositional field equations. Specifically, the parameter Parameter name: Stabilization method allows using one of the following methods:
Entropy Viscosity Stabilization
SUPG Stabilization
Both add additional terms to the temperature (or compositional field) equation. We will discuss the case for the temperature equation here. The compositional fields only differ in having a zero conductivity, fewer right-hand side terms, and \(\rho C_p=1\). The strong form of the temperature equation reads
\[\rho C_p \frac{\partial T}{\partial t} + \rho C_p \mathbf{u} \cdot \nabla T - \nabla \cdot k\nabla T = F,\]
where \(F\) is the combination of source and reaction terms, while the weak form - with test function \(\varphi\) and L2 inner product \((\cdot,\cdot)\) - is
(28)#\[a(T,\varphi) =
\left(\rho C_p \frac{\partial T}{\partial t}, \varphi \right)
+ \left(\rho C_p \mathbf{u} \cdot \nabla T, \varphi \right)
+ \left( k \nabla T, \nabla \varphi \right) = (F,\varphi) = f(\varphi).\]