Abstract : Many complex physical phenomena may be modeled by means of non-linear hyperbolic systems. When approximating such systems, one requires the use of efficient, accurate and stable numerical schemes. On the one hand, the use of high-order methods will be necessary in order to reduce the numerical diffusion inherent to the numerical approach. On the other hand, it is common for these systems the existence of some particular steady-state solutions that should be preserved, which will need the use of a well-balanced scheme.
The goal of this mini-symposium is the discussion and presentation of state-of-the-art computational and numerical methods of high-order well-balanced schemes with applications to hyperbolic systems.
Organizer(s) : Tomas Morales de Luna, Ernesto Guerrero-Fernandez
00528 (1/2) : 1C @G606 [Chair: Tomas Morales de Luna]
[01682] High order well-balanced finite volume and discontinuous Galerkin schemes for a first order hyperbolic reformulation of the coupled Einstein-Euler system in 3+1 general relativity
Format : Talk at Waseda University
Author(s) :
Michael Dumbser (University of Trento)
Abstract : We present new well-balanced finite volume and discontinuous Galerkin schemes for the solution of a new first order hyperbolic Z4 formulation of the Einstein-Euler system of general relativity. Nonlinear involutions are accounted for via a covariant GLM cleaning technique. We introduce a new, simple and efficient type of well-balancing that automatically applies to any numerical discretization and arbitrary equilibria in multiple space dimensions. We show numerical results for vacuum spacetimes and for a TOV star.
[01699] Numerical approximation of non-convex relativistic hydrodynamics
Format : Talk at Waseda University
Author(s) :
Susana Serna (Universitat Autonoma de Barcelona)
Antonio Marquina (Universidad de Valencia)
Abstract : We explore the rich and complex dynamics that a phenomenological equation of state (EoS) with non-convex regions in the pressure-density plane may develop as a result of genuinely relativistic effects. We study the parameter space of the EoS to ensure its causality and thermodynamical consistency. We approximate the non-conventional dynamics developed in the evolution of relativistic blast waves by means of a high order shock capturing scheme.
[01720] Recovering primitive variables in special relativistic hydrodynamics
Format : Talk at Waseda University
Author(s) :
Antonio Marquina (Universidad de Valencia)
Susana Serna (Universitat Autonoma de Barcelona)
Jose M Ibanez (Universidad de Valencia)
Abstract : We study an iterative procedure based on fixed-point strategy to recover primitive variables
in each time step of the evolution of Special Relativistic Hydrodynamic equations. Given a set of three conserved values we start the iteration by prescribing an initial zero pressure so that if the first iterate
is strictly positive, then, the fixed-point iteration monotonically converges to the unique
pressure associated to the conserved variables.
[01290] Well-Balanced High-Order Discontinuous Galerkin Methods for Systems of Balance Laws
Format : Talk at Waseda University
Author(s) :
Ernesto Guerrero Fernández (National Oceanic and Atmospheric Administration (NOAA))
Cipriano Escalante Sanchez (Universidad de Málaga)
Manuel Castro Díaz (Universidad de Málaga)
Abstract : This work introduces a general strategy to develop well-balanced high-order Discontinuous Galerkin (DG) numerical schemes for systems of balance laws. The essence of our approach is a local projection step that guarantees the exactly well-balanced character of the resulting numerical method for smooth stationary solutions. The strategy can be adapted to some well-known different time marching DG discretisations. Particularly, in this article, Runge--Kutta DG and ADER DG methods are studied. Additionally, a limiting procedure based on a modified WENO approach is described to deal with the spurious oscillations generated in the presence of non-smooth solutions, keeping the well-balanced properties of the scheme intact. The resulting numerical method is then exactly well-balanced and high-order in space and time for smooth solutions. Finally, some numerical results are depicted using different systems of balance laws to show the performance of the introduced numerical strategy.
[01687] Structure preserving high order discontinuous Galerkin schemes for general relativity
Format : Talk at Waseda University
Author(s) :
Elena Gaburro (Inria)
Michael Dumbser (University of Trento)
Ilya Peshkov (University of Trento)
Olindo Zanotti (University of Trento)
Manuel J. Castro (University of Malaga)
Abstract : In this talk we present a novel first order hyperbolic reformulation of the coupled Einstein-Euler system allowing robust and long-time stable simulations for the joint evolution of matter and space-time in general relativity. Among our numerical results we have long-time evolution of TOV neutron stars and accretion disks and a black holes collision. This is obtained through high order discontinuous Galerkin schemes endowed with subcell finite volume limiter, well balanced techniques and GLM curl cleaning.
[01663] Multidimensional approximate Riemann solvers for hyperbolic nonconservative systems
Format : Talk at Waseda University
Author(s) :
José M. Gallardo (University of Málaga)
Abstract : This work deals with the development of efficient incomplete multidimensional Riemann solvers for hyperbolic systems. We present a general strategy for constructing genuinely two-dimensional Riemann solvers, that can be applied for solving systems including source and coupling terms. Two-dimensional effects are taken into account through the approximate solutions of 2d Riemann problems arising at the vertices of the computational mesh. Applications to magnetohydrodynamics and shallow water equations are presented.
[01841] A fully-well-balanced hydrodynamic reconstruction
Format : Talk at Waseda University
Author(s) :
Christophe Berthon (Nantes Université)
Victor Michel-Dansac (INRIA)
Abstract : The present work concerns the numerical approximation of the weak solutions of the shallow-water model. To address such an issue, the well-known hydrostatic reconstruction is adopted. Such a relevant technique easily gives numerical schemes able to exactly capture the steady states at rest. Here, necessary conditions are stated on the reconstruction process in order to also capture the moving steady states. An example of suitable hydrodynamic reconstruction is presented and tested.
[01690] A well-balanced scheme for landslide models
Format : Talk at Waseda University
Author(s) :
Manuel J. Castro (Universidad de Malaga)
Cipriano Escalante Sanchez (Universidad de Málaga)
José Garres-Díaz (Universidad de Córdoba, Spain)
Tomas Morales de Luna (Universidad de Malaga)
Abstract : When landslide models in the shallow water framework, special care has to be taken with the stationary solutions. Indeed, motion of the material only begins when the slope of the material is bigger than that of the repose angle. Preserving such steady states is not a trivial task. We present here different strategies to design well-balance high-order finite volume schemes.
