In these case studies, we study an incompressible Euler, which means we have the (compressible) Euler equations coupled with a gravity term which manifests in a pressure Poisson equation.

That is, we solve the problem

\( \partial _t Q + \nabla \cdot F(Q) = \left( \begin{array}{c} 0 \\ \rho \nabla \phi \\ - (v, \nabla \phi) \rho \end{array} \right) \)

which gives the Euler equation with \( Q = (\rho, v, E) \in \mathbb{R}^5 \) subject to a modified source term (right-hand side). While \( F(Q) \) stems from the Euler equations in the previous example, a second PDE \( - \Delta \phi = -\pi G/\rho \) adds self-gravitation to this setup.

The resulting problem is a combination of a hyperbolic setup coupled to an elliptic operator. Within ExaHyPE 2, we have developed various approaches how to tackle such a coupled system: We can approximate the right-hand side, map the Poisson problem onto a hyperbolic system with tiny time step sizes, or explicitly solve the Poisson problem's linear equation system per time step.