One example that came to mind after reading the title of your question only, is the Stefan-Boltzmann radiation law applied to a heat conduction problem (radiated heat flux is proportional to the fourth power of temperature). Your particular non-linearity doesn't ring a bell though.

As a really artificial example, it could be thought of as a first-order approximation to a voltage-dependent contact resistance in an electrical potential problem. Something on the lines of:

V+(R+alpha*V) J \dot n=V_source,

where V is the electrical potential (voltage), J is current density = -sigma \grad V (sigma is the conductivity), R and alpha define the first-order approximation of the voltage-dependent resistance, and finally V_source is the source voltage.

Antti, Thank you! This is a nice example.

It is not possible to read the form of the non-linear Robin BC you mention, but you might take a look at my paper: Moutsopoulos, K. N. (2013), Solutions of the Boussinesq equation subject to a non-linear Robin boundary condition, Water Resour. Res., vol. 49, 7–18, doi:10.1029/2012WR012221.

In the paper above, the case of inflow to an aquifer or porous medium inhibited by the presence of a semipervious (clogging) layer is investigated.

The investigated PDE is the non-linear Boussinesq equation (Eq. 1), and the non-linear Robin BC is Eq. (5).

Dear Prof. Moutsopoulos, I have uploaded a picture showing the form of the nonlinear Robin BC I am trying to tackle. By a quick glance into your paper, it seems that your equation (4a) is of the above form. However, the more general form in equation (5) has an extra quadratic term. Which one did you actually use? Also, I have the impression that your spatial domain is 1D, i.e. h only depends on x, neither y nor z, am I right?

I used Eq. (5) which for the specific problem I investigated in my paper (flow to a unconfined aquifer, inhibited by the presence of a clogging layer) is the accurate form of the BC.

It is true that while the second term on the left hand side of Eq, (5) is the same to  the third term on the left hand side term of your relation (now I can read it!), both the the first and the second left hand side terms of your equation does not appear in Eq. (5). Instead, as you mentioned, an extra quadratic term is included in Eq. (5).

Yes, it is true that the spatial domain is 1D, nevertheless beside a FDM solution also a semi-analytical solution is provided  in my paper.

Prof. Moutsopoulos, Thank you for the clarification. In my case, the domain is a 2D rectangle: the left boundary has a Dirichlet BC, the right and top Neumann BCs, the bottom the nonlinear Robin BC. I have worked out an explicit formula for finite volume methods to fill the ghost cells for operator evaluation with accuracy as the fourth-, fifth-, and the sixth-orders.

Then I used a fourth-order finite volume method I developed (SIAM Journal on Scientific Computing 2012; 34: B179-B201)  to solve the advection-diffusion equation.  When the nonlinear Robin BC is absent and all the boundary conditions are linear, I have fourth-order accuracy both in time and in space. However, when the nonlinear Robin BC is used for the bottom boundary, the convergence rates reduces to the second order for all of the 1-, 2-, and max-norms. For more details, see a related question:

https://www.researchgate.net/post/Who_knows_literature_on_the_analysis_of_the_advection-diffusion_equation_with_a_nonlinear_Robin_boundary_condition

Have you observed similar order reduction in your 1D FDM case when the non-linearity is high?

Dear Professor Zhang,

The aim of my study was mainly to develop a semi-analytical solutions. The numerical solutions were derived to check their accuracy.

Furthermore only the case of the strongest non-linearities was considered (partially dry aquifer). Therefore I cannot answer your question, because problems with less pronounced non-linearities were not examined.

The comparison of the semi-analytical and numerical results was statisfactory. However, a small time step-size was necessary to achieve   convergence of the FDM results (see section 4.1 of the paper).

I wish that you have success with your work. 

Dear Prof. Moutsopoulos: thank you very much for the update, best luck to you!