Invited Lecture
Staggered discretisation of the Euler equation: how to achieve local conservation
University of Zurich, Switzerland
Abstract
This paper is focused on the approximation of the Euler equation of compressible fluid dynamics on a staggered mesh.
To this aim, the flow parameter are described by the velocity, the density and the internal energy.
The thermodynamic quantities are described on the elements of the mesh, and this the approximation is only $L^2$,
while the kinematic quantities are globally continuous. The method is general in the sense that the thermodynamical
and kinetic parameters are described by arbitrary degree polynomials, in practice the difference between the degrees
of the kinematic parameters and the thermodynamical ones is equal to $1$. The integration in time is done using
a defect correction method.
As such, there is no hope that the limit solution, if it exists, will be a weak solution of the problem.
In order to guaranty this property, we introduce a general correction method in the spirit of the Lagrangian staggered
method described in [1] and [2],
and we prove a Lax Wendroff theorem. The proof is valid for multidimensional version of the scheme,
though all the numerical illustrations, on classical benchmark problems,
are all one dimensional because we have an easy access to the exact solution for comparison. We also provide two dimensional
simulations on triangular and quad meshes. We conclude by explanning that the method is general and can be used in a different
setting as the specific one used here, for example finite volume, of discontinuous Galerkin methods.
References
[1] Abgrall R., Tokareva S.,
Staggered Grid Residual Distribution Scheme for Lagrangian Hydrodynamics,
SIAM SISC, vol 39(5), 2017
[2] Abgrall R., Lipnikov K., Morgan N., Tokareva S.,
Multidimensional staggered grid residual distribution scheme for lagrangian hydrodynamics,
SIAM SISC vol 42, 2020.