Advection is at the heart of fluid dynamics and is responsible for many interesting phenomena. Unfortunately, it is also the source of the non-linearity of fluid dynamics. As such, its numerical treatment is challenging and often suboptimal. One way to more effectively deal with advection is by using a Lagrangian formulation instead of the conventional Eulerian view.

This work aims to show that the Lagrangian formulation, and its underlying geometric and physical character, are fundamental in overcoming the challenges posed by (non-)linear advection. The Mimetic Spectral Element Method ensures that this geometric and physical character is kept when the equations are discretised. The advection term can then be dealt with exactly. The method is put to the test by solving the inviscid Burgers and isentropic Euler equations in one spatial dimension. The results confirm the exactness of the advection term and good overall accuracy.