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)#\[a(T,\varphi) = \left(\rho C_p \frac{\partial T}{\partial t}, \varphi \right) + \left(\rho C_p \mathbf{u} \cdot \nabla T, \varphi \right) + \left( k \nabla T, \nabla \varphi \right) = (F,\varphi) = f(\varphi).\]