After implementation, we can see that the algorithms that individually model the first and second equations are very correct. We have obtained the expected results through error calculation and visualization on paraview. In the case of the coupled system, the problem encountered in the algorithm has not been solved for the moment, therefore no results could be obtained for the system for the moment. Afterwards, the goal will be to fix the bug in order to complete the implementation, and use the algorithm to solve the two test cases present in the validation part.