§3.7 移动网格方法

Substantial effort has gone into reformulating Eulerian methods as described in §3.4 into Lagrangian mesh approaches. More details on the idea of hydrodynamics on moving mesh for cosmological application can be found in the pioneering work by ^[101] and references therein. This early approach started from a regular mesh which then, by following the flow of the fluid, was deformed. The Euler equations were evolved by calculating the fluxes across the cell borders. An example of the resulting mesh for a cosmological simulation can be seen in the left part of Fig. 6. One disadvantage (or challenge) of this technique in practical applications is that individual cells can be extensively deformed and stretched. Modern schemes circumvent this problem by constructing an unstructured mesh based on a Voronoi or Delaunay tessellation (see ^[102] and references therein). The relevant geometry of the cells, based on the mesh generating points $\vec{r}_i$ and $\vec{r}_j$ is illustrated in the right part of Fig. 6. The fluxes then have to be calculated at the centroid of the interface (note that this is not necessarily on the straight line between the two mesh generating points, as indicated by the dotted line) $$ \mathbf{Q}_i^{(n+1)} = \mathbf{Q}_i^{(n)} - \Delta t \sum_j A_{ij}\hat{\mathbf{F}}_{ij}^{(n+1/2)}. $$ The motion $\vec{w}$ of this interface is uniquely defined by the velocities $\vec{w}_i$ and $\vec{w}_j$, and the fluxes have to be calculated with the Riemann solver in the rotated frame ($x',y'$). A detailed description of this technique along with its performance in test problems can be found in ^[102] .