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

Analyse numérique - Equations aux dérivées partielles

Lieu:

En visio

Orateur:

Elena Gaburro

Affiliation:

Inria Bordeaux

Dates:

Jeudi, 18 Février, 2021 - 11:00 - 12:00

Résumé:

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

interfaces.

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].

References

[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).