Model problem

This example is similar to basic-ie-newton except it uses the Picard’s method in each time step (not Newton’s method).

We assume the time-dependent Richard’s equation

(1)C(h) \frac{\partial h}{\partial t} - \nabla \cdot (K(h) \nabla h) - K'(h) \frac{\partial h}{\partial z}= 0

where C and K are non-differentiable and thus in some cases the Newton’s method has convergence problems. For this reason, the Picard’s method is a popular approach. If Picards is used, then Andersson acceleration should be employed.

equipped with a Dirichlet, given by the initial condition.

x*(100. - x)/2.5 * y/100 - 1000. + H\underline{\ }OFFSET

The pressure head ‘h’ is between -1000 and 0. For convenience, we increase it by an offset H_OFFSET = 1000. In this way we can start from a zero coefficient vector.

Weak formulation

The corresponding weak formulation reads

\int_{\Omega} C(h) \frac{h^{n+1}_{k+1} - h^{n}}{\triangle t} v d\bm{x} + \int_{\Omega} K(h^{n+1}_{k}) \nabla h^{n+1}_{k+1} \cdot \nabla v d\bm{x} - \int_{\Omega} K'(h^{n+1}_{k}) \frac{\partial h^{n+1}_{k+1}}{\partial z} v d\bm{x} = 0.

Here n refers to time steps and k to Picard’s iterations within one time step.

Defining weak forms

The weak formulation is a combination of custom Jacobian and Residual weak forms:

CustomWeakFormRichardsIEPicard::CustomWeakFormRichardsIEPicard(double time_step, Solution* h_time_prev, Solution* h_iter_prev) : WeakForm<double>(1)
  // Jacobian.
  CustomJacobian* matrix_form = new CustomJacobian(0, 0, time_step);

  // Residual.
  CustomResidual* vector_form = new CustomResidual(0, time_step);

Sample results

Solution at t = 0.01 s:

sample result

Solution at t = 0.03 s:

sample result

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