Example 1: nonlinear diffusion in a star

Files

• Comprehensive test files:

  • explicit diffusion coefficient : main.cpp
  • implicit diffusion coefficient : main.cpp

• Reference results for comparison:

  • explicit diffusion coefficient :time_specialized.csv
  • implicit diffusion coefficient :time_specialized.csv

Statement of the problem

This example code is based on the "Example 16: Time Dependent Heat Conduction", on MFEM's website. It solves a diffusion equation in a 2D star $\Omega$ (GMSH mesh coming from MFEM examples).

$$\begin{align} \frac{\partial c}{\partial t}&=[\nabla \cdot{} \displaystyle(\kappa+\alpha c) \nabla c ]\text{ in }\Omega \end{align}$$

where the concentration $c$ in the diffusion coefficient can be evaluated at the previous time step, as done in the MFEM example, or at the current time step to form a fully nonlinear system.

Initial condition

The initial condition is set to $2$ inside a circle of radius $0.5$, $1$ outside.

Parameters used for the test

For this test, the following parameters are considered:

Parameter	Symbol	Value
Primary coefficient	$\kappa$	$0.5$
Secondary coefficient	$\alpha$	$10^{-2}$

Boundary conditions

Neumann boundary conditions are prescribed on the boundary of the domain:

$$\begin{align} {\bf{n}} \cdot{} \lambda \nabla c&=0 \text{ on }\partial\Omega \end{align}$$

Numerical scheme

• Time integration: Euler Implicit over the interval $t\in[0,0.5]$, $\delta t=0.01$
• Spatial discretization: built from GMSH + quadratic FE + $2$ uniform levels of refinement (it could be extended further)

Results

The figure 1 shows the nonlinear diffusion inside the star.

Figure 1: nonlinear diffusion in a star obtained with $5$ level of refinement.