Recent trends and New directions

NumHyp17 will be on recent developments and directions in the general area of numerical methods for hyperbolic and convection dominated partial differential equations (PDEs). These PDEs arise in a large number of models in physics and engineering. Prominent examples include the compressible and incompressible Euler and Navier-Stokes equations, the shallow water equations, the Magneto-Hydrodynamics equations. Example of applications area are: aerodynamics, oceanography, plasma physics, solid mechanics.

These PDEs have been subjected to extensive analytical and numerical studies over the last decades. It is widely known that their solutions can exhibit very complex behavior including the presence of singularities such as shock waves, sensitive dependence to initial conditions, presence of multiple spatio-temporal scales and the appearance of chaotic or turbulent motions. The theoretical questions of existence, uniqueness, regularity, stability and asymptotic behavior are still poorly understood. Nevertheless, a large number of numerical methods have been developed over the past three to four decades in order to simulate them. Issues of critical importance such as proofs of convergence, convergence rate and treatment of source terms, multiple scales, turbulent regions still remain unsolved. There is no consensus on an optimal numerical method.

NumHyp17 will focus on some of the key unresolved issues:

• Uncertainty quantification for hyperbolic PDEs (Monte Carlo type methods, Deterministic methods such as stochastic collocation and stochastic spectral methods
• Numerical approximation of measure valued and statistical solutions for unstable and turbulent flows,
• Model order reduction and optimal control,
• Treatment of complex geometries: Robust discretizations on unstructured grids, high order approximation, both for time and space, low order and high order meshing, mesh adaptation, and efficiency.
• Numerical methods for problems with multiple scales,
• Efficient treatment of source terms, well balanced schemes, residual schemes.
• Applications of hyperbolic and convection dominated PDEs: geosciences,fluids,solid mechanics

NumHyp17 is the fifth in a series of biannual conferences entitled that began with a meeting in Castro Urdiales, Spain in 2009. Further editions of this conference were held at Roscoff, France in 2011, Aachen, Germany in 2013 and in Cortona, Italy in 2015. NumHyp17 will be a key activity of the European innovative training network entitled ModCompShock ( Modeling and computation of shocks and interfaces).