Moving mesh simulation of Euler equations

The plots below illustrate the difference between computations on a moving mesh (left) and a fixed grid (right), both with the same number of grid points. The partial differential differential equations being solved are the Euler equations of gasdynamics:

with piecewise constant initial conditions (also known as Sod's shock tube problem). The discontinuous initial data gives rise to an expansion fan that travels to the left, as well as a shock wave and contact discontinuity, both travelling to the right.

The plots of density, , clearly demonstrate that the moving mesh method resolves all three solution features more accurately using the same number of grid points. The moving mesh method clearly displays much better shock resolution, and the plot of the mesh trajectories (at bottom) indicates how mesh points smoothly cluster together as the discontinuity develops, and then follow along with the shock. The mesh contours at the bottom indicate how the mesh points track the shock and contact line.

Moving mesh solution (N=60) Fixed mesh solution (N=60)

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