Higher Order Numerical Methods for Evolutionary PDEs: Applied Mathematics Meets Astrophysical Applications (15w5134)

Arriving in Banff, Alberta Sunday, May 10 and departing Friday May 15, 2015


(Wurzburg University)

(Brown University)

Volker Springel (Heidelberg University)


This workshop shall concern itself with the numerical methods for evolutionary partial differential equations describing physical fluid flows. Examples of the underlying physical laws being modeled are conservation and thermodynamical principles. These give rise to time-dependent nonlinear partial differential equations that typically come with constraints, say an equation of state or variants of the second law of thermodynamics. These models may involve microscopic features, hinting at the complexity of these models. One area of application where these models are used is astrophysics. In astrophysics we have entered an era where we have an unprecedented wealth of observations. This data calls for modeling and numerical simulation with ever increasing sophistication. Astrophysical simulations account for about one third of the computational time used on supercomputers indicating the importance alloted to this field. The development of supercomputers goes towards ever more cores that communicate less and less. This favors numerical algorithms that are accurate by relying on communication of a processor with only its neighbors. This has to be taken into account when developing schemes that are more accurate. In this workshop we want to see what this demand means for the numerical methods. In particular we want to investigate higher order methods that are very local.In the last decades finite volume methods have come to fruition where with so called total variation diminishing methods one can produce second order methods that work quite well. The step beyond second order is challenging. Existing approaches, like WENO methods, DG methods, residual distribution methods etc. suffer from either not being very local or they tend to produce spurious oscillations near shocks. We hope that the interchange between applied mathematicians and astrophysicists will shed some light on how to move forward. Examples of questions to be discussed are: how much higher order is really needed, numerical issues related to higher order in time, the issue of spurious oscillations near shocks for DG schemes, what to do about singularities arising in in 3-d flows where there are more issues to deal with than shocks, the question of Galilean invariance of models and schemes.Among the ever more sophisticated models that astrophysicists are turning to are compressible flows involving magnetic fields. We want to take this into account by inviting some specialists from this area, both from the mathematical and the physics side. Issues that arise here are among others on how to deal with constraints caused by the magnetic field, like anisotropic diffusion, or how to enforcing that there are no magnetic monopoles.
This is intended as a workshop on numerical methods that interacts with computational astrophysicists. We expect this to give an impetus on the development of numerical methods. Vice-versa, we expect the astrophysicists to use methods not common in their community before, which later than give feedback to the mathematicians on features of these methods.