Lagrangian schemes with topology changes and well balanced techniques for the solution of hyperbolic partial differential equations

Elena Gaburro
Inria Bordeaux
Jeudi, 18 Février, 2021 - 11:00 - 12:00
In our work we develop new numerical methods able to preserve also at the discrete level some adjoint
constraints of the PDE systems connected to the physics of the studied problem or to its geometry, such
as equilibria, asymptotic limits or material interfaces.
Our strategy considers in particular two directions: the use of well balanced techniques able to preserve
up to machine precision the equilibria of the studied systems, and the Lagrangian schemes where the
mesh moves with the fluid flows to reduce the errors due to convection and precisely track material
In particular, in this talk we present a new family of very high order accurate direct Arbitrary-
Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous Galerkin (DG) schemes for the
solution of general nonlinear hyperbolic PDE systems on moving Voronoi meshes that are regenerated
at each time step and which explicitly allow topology changes in time, in order to benefit simultaneously
from high order methods, high quality grids and substantially reduced numerical dissipation [1, 2].
A variant of this algorithm [4] is also coupled with well balanced techniques for the Euler equations
of gasdynamics with gravity leading to an extremely low dissipative scheme able to capture physical
instabilities of a Keplerian disk without them being hidden by spurious numerical oscillations [3].
The talk is closed with an outlook to future application of well balancing in the field of general relativity
though the fully coupled Einstein-Euler system (FO-CCZ4+GRMHD), where the matter source terms
are given in a self-consistent manner by solving the GRMHD equations, coupled with the FO-CCZ4
formulation of the Einstein field equations for evolving the metric of the space-time [5].
[1] E. Gaburro, Archives of Computational Methods in Engineering (2020).
[2] E. Gaburro, W. Boscheri, S. Chiocchetti, C. Klingenberg, V. Springel, M. Dumbser, Journal of Computa-
tional Physics (2020).
[3] E. Gaburro, M.J. Castro, M. Dumbser, Monthly Notices of the Royal Astronomical Society (2018).
[4] E. Gaburro, M. Dumbser and M.J. Castro, Computer and Fluids (2017).
[5] E. Gaburro, M.J. Castro, M. Dumbser, Well balanced methods for general relativity, in preparation (2021).