Hyperbolic Balance Laws: modeling, analysis, and numerics (hybrid meeting)

This workshop brought together leading experts, as well as the most promising young researchers, working on nonlinear hyperbolic balance laws. The meeting focused on addressing new cutting-edge research in modeling, analysis, and numerics. Particular topics included ill-/well-posedness, randomness and multiscale modeling, flows in a moving domain, free boundary problems, games and control

Communication-Avoiding Algorithms for a High-Performance Hyperbolic PDE Engine

The study of waves has always been an important subject of research. Earthquakes, for example, have a direct impact on the daily lives of millions of people while gravitational waves reveal insight into the composition and history of the Universe. These physical phenomena, despite being tackled traditionally by different fields of physics, have in common that they are modelled the same way mathematically: as a system of hyperbolic partial differential equations (PDEs). The ExaHyPE project (“An Exascale Hyperbolic PDE Engine") translates this similarity into a software engine that can be quickly adapted to simulate a wide range of hyperbolic partial differential equations. ExaHyPE’s key idea is that the user only specifies the physics while the engine takes care of the parallelisation and the interplay of the underlying numerical methods. Consequently, a first simulation code for a new hyperbolic PDE can often be realised within a few hours. This is a task that traditionally can take weeks, months, even years for researchers starting from scratch. My main contribution to ExaHyPE is the development of the core infrastructure. This comprises the development and implementation of ExaHyPE’s solvers and adaptive mesh refinement procedures, it’s MPI+X parallelisation as well as high-level aspects of ExaHyPE’s application-tailored code generation, which allows to adapt ExaHyPE to model many different hyperbolic PDE systems. Like any high-performance computing code, ExaHyPE has to tackle the challenges of the coming exascale computing era, notably network communication latencies and the growing memory wall. In this thesis, I propose memory-efficient realisations of ExaHyPE’s solvers that avoid data movement together with a novel task-based MPI+X parallelisation concept that allows to hide network communication behind computation in dynamically adaptive simulations

An Arbitrary-Lagrangian-Eulerian hybrid finite volume/finite element method on moving unstructured meshes for the Navier-Stokes equations

We present a novel second-order semi-implicit hybrid finite volume / finite element (FV/FE) scheme for the numerical solution of the incompressible and weakly compressible Navier-Stokes equations on moving unstructured meshes using an Arbitrary-Lagrangian-Eulerian (ALE) formulation. The scheme is based on a suitable splitting of the governing PDE into subsystems and employs staggered grids, where the pressure is defined on the primal simplex mesh, while the velocity and the remaining flow quantities are defined on an edge-based staggered dual mesh. The key idea of the scheme is to discretize the nonlinear convective and viscous terms using an explicit FV scheme that employs the space-time divergence form of the governing equations on moving space-time control volumes. For the convective terms, an ALE extension of the Ducros flux on moving meshes is introduced, which is kinetic energy preserving and stable in the energy norm when adding suitable numerical dissipation terms. Finally, the pressure equation of the Navier-Stokes system is solved on the new mesh configuration using a continuous FE method, with $\mathbb{P}_1$ Lagrange elements. The ALE hybrid FV/FE method is applied to several incompressible test problems ranging from non-hydrostatic free surface flows over a rising bubble to flows over an oscillating cylinder and an oscillating ellipse. Via the simulation of a circular explosion problem on a moving mesh, we show that the scheme applied to the weakly compressible Navier-Stokes equations is able to capture weak shock waves, rarefactions and moving contact discontinuities. We show that our method is particularly efficient for the simulation of weakly compressible flows in the low Mach number limit, compared to a fully explicit ALE schem

High-order conservative finite difference GLM-MHD schemes for cell-centered MHD

We present and compare third- as well as fifth-order accurate finite difference schemes for the numerical solution of the compressible ideal MHD equations in multiple spatial dimensions. The selected methods lean on four different reconstruction techniques based on recently improved versions of the weighted essentially non-oscillatory (WENO) schemes, monotonicity preserving (MP) schemes as well as slope-limited polynomial reconstruction. The proposed numerical methods are highly accurate in smooth regions of the flow, avoid loss of accuracy in proximity of smooth extrema and provide sharp non-oscillatory transitions at discontinuities. We suggest a numerical formulation based on a cell-centered approach where all of the primary flow variables are discretized at the zone center. The divergence-free condition is enforced by augmenting the MHD equations with a generalized Lagrange multiplier yielding a mixed hyperbolic/parabolic correction, as in Dedner et al. (J. Comput. Phys. 175 (2002) 645-673). The resulting family of schemes is robust, cost-effective and straightforward to implement. Compared to previous existing approaches, it completely avoids the CPU intensive workload associated with an elliptic divergence cleaning step and the additional complexities required by staggered mesh algorithms. Extensive numerical testing demonstrate the robustness and reliability of the proposed framework for computations involving both smooth and discontinuous features.Comment: 32 pages, 14 figure, submitted to Journal of Computational Physics (Aug 7 2009

Contributions to the numerical solution of heterogeneous fluid mechanics models

A high order projection hybrid finite volume – finite element method is developed to solve incompressible and compressible low Mach number flows. Furthermore, turbulent regimes are also considered thanks to the k–ε model. The unidimensional advection-diffusion-reaction equation is used to construct, analyze and assess high order finite volume schemes. Two families of methods are studied: Kolgan-type schemes and ADER methodology. A modification of the last one is proposed providing a new numerical method called Local ADER. The designed method is extended to solve the transport-diffusion stage of the three-dimensional projection method. Within the projection stage the pressure correction is computed by a piecewise linear finite element method. Numerical results are presented, aimed at verifying the formal order of accuracy of the schemes and to assess the performance of the method on several realistic test problems