Basic-ie-newton

Model problem

This example solves a simple version of the time-dependent Richard’s equation using the backward Euler method in time combined with the Newton’s method in each time step. It describes infiltration into an initially dry soil. The example has a exact solution that is given in terms of a Fourier series (see a paper by Tracy). The exact solution is not used here.

We assume the time-dependent Richard’s equation

(1)

where and are given functions of the unknown pressure head , are spatial coordinates, and is time.

equipped with a Dirichlet, given by the initial condition.

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

Defining weak forms

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

CustomWeakFormRichardsIE::CustomWeakFormRichardsIE(double time_step, Solution* h_time_prev) : WeakForm<double>(1)
{
  // Jacobian volumetric part.
  CustomJacobianFormVol* jac_form_vol = new CustomJacobianFormVol(0, 0, time_step);
  jac_form_vol->ext.push_back(h_time_prev);
  add_matrix_form(jac_form_vol);

  // Residual - volumetric.
  CustomResidualFormVol* res_form_vol = new CustomResidualFormVol(0, time_step);
  res_form_vol->ext.push_back(h_time_prev);
  add_vector_form(res_form_vol);
}

Sample results

Solution at t = 0.01 s:

Solution at t = 0.03 s: