For example, in our recently published articles, me and my collaborators followed the approach outlined below:
- start with a finite difference scheme with backward Euler time discretization of the Richards equation in mixed saturation-pressure formulation (i.e., the implicit finite difference scheme approximating the pressure 𝜓𝑖,𝑘 at the i-th lattice site and time 𝑘);
- add to the left-hand side of the scheme a stabilization term 𝐿(𝜓𝑖,𝑘𝑠+1−𝜓𝑖,𝑘𝑠) to obtain an explicit scheme solving for 𝜓𝑖,𝑘𝑠+1;
- solve the nonlinearities of the water content 𝜃(𝜓𝑖,𝑘𝑠) and conductivity 𝐾(𝜓𝑖,𝑘𝑠) by iterating the explicit scheme until the norm ||𝜓𝑖,𝑘𝑠+1−𝜓𝑖,𝑘𝑠|| falls below a given tolerance.
Such explicit schemes solve problems for unsaturated flow regime and transition to saturated regime ten times faster than a two-point flux approximation finite volume method used as reference. Instead, the computation time in case of strictly saturated regime can be up to two orders of magnitude larger than for the finite volume scheme.
Are there efficient explicit schemes able to solve the parabolic-elliptic degenerate Richards equation for both saturated and unsaturated regime, as well as for transition between the two regimes?
Thanks
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