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)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 given functions of the unknown pressure head h, \bm{x}=(x,z) are spatial coordinates, and t is time.

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{\partial h}{\partial t} v d\bm{x} + \int_{\Omega} K(h) \nabla h \cdot \nabla v d\bm{x} - \int_{\Omega} K'(h) \frac{\partial h}{\partial z} v d\bm{x} = 0.

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);

  // Residual - volumetric.
  CustomResidualFormVol* res_form_vol = new CustomResidualFormVol(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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