This milestone will conclude when a higher order scheme for the Solute Transport equation is fully proposed, implemented, and tested within DORiE.

• Propose a numeric scheme.
• Implement it in DORiE.
• Validate and test it against analytic and other numeric solutions.

To achieve this, Prof. Peter Bastian proposed to do it in the following sequence of steps:

1. Implement the flux reconstruction with \mathcal{RT}_0 finite elements:

   • Implement an easy and straightforward low order reconstruction.

2. Solution with Finite Volume Method:

   • Move the dune-modelling code to dorie.
   • Prepare few benchmarks to test the code in order to compare later with the dG method.

3. Implement the dG method for \mathcal{Q}_1/\mathcal{P}_1:

   • Firstly try with the averaged saturation (\theta \in \mathcal{P}_0, C_w\in\mathcal{Q}_1/\mathcal{P}_1), compensate the balance of mass, and investigate its behaviour.
   • Later, try with the continuous saturation (\theta \in \mathcal{P}_k, C_w\in\mathcal{Q}_1/\mathcal{P}_1), and check whether we get numerical problems in the time discretization.

4. Increase the order dG.

   • Do the flux reconstruction in higher order, that is, j_w^*\in\mathcal{RT}_s, s\in \{k,k-1\}.
   • Solve with for polynomials of order k, that is, C_w\in\mathcal{Q}_k/\mathcal{P}_k.