An efficient two-stage form of the Crank- Nicolson scheme with application to an HDG method for poroelasticity

Resumo

We propose an alternative form of the Crank-Nicolson scheme devised for linear space operators,
though its properties, such as convergence and stability, also hold for nonlinear ones. The discretization mimics
the implicit Euler scheme by adding a decoupled weighted average equation as a second stage for each time step.
This creates a two-stage method in contrast to the classical one-stage one, but it also evaluates the space operator
at only the middle point, therefore generating a scheme that is computationally more efficient for highly expensive
space operators. The scheme also simplifies the time update of the solution, and its implementation easily extends
from the implementation of the implicit Euler scheme. In order to verify the scheme’s stability and convergence,
we apply it to a series of time dependent one-dimensional problems, and also to a two-dimensional poroelasticity
problem, also known as Biot‘s Consolidation problem. Our application to the Consolidation problem uses an HDG
method to approximate the space operator, which exemplifies yet another feature of our scheme: it does not rely
on Lagrange multipliers to update the solution at the interior of elements.