Ulrich Rüde

Lorenz John, Ulrich Rüde, Barbara Wohlmuth et al.

In this article, we discuss several classes of Uzawa smoothers for the application in multigrid methods in the context of saddle point problems. Beside commonly used variants, such as the inexact and block factorization version, we also introduce a new symmetric method, belonging to the class of Uzawa smoothers. For these variants we unify the analysis of the smoothing properties, which is an important part in the multigrid convergence theory. These methods are applied to the Stokes problem for which all smoothers are implemented as pointwise relaxation methods. Several numerical examples illustrate the theoretical results.

Simon Bogner, Regina Ammer, Ulrich Rüde

In this paper we analyze the boundary treatment of the lattice Boltzmann method (LBM) for simulating 3D flows with free surfaces. The widely used free surface boundary condition of Körner et al. (2005) is shown to be first order accurate. The article presents a new free surface boundary scheme that is suitable for second order accurate simulations based on the LBM. The new method takes into account the free surface position and its orientation with respect to the computational lattice. Numerical experiments confirm the theoretical findings and illustrate the different behavior of the original method and the new method.

Markus Huber, Lorenz John, Petra Pustejovska et al.

This article proposes modifications to standard low order finite element approximations of the Stokes system with the goal of improving both the approximation quality and the parallel algebraic solution process. Different from standard finite element techniques, we do not modify or enrich the approximation spaces but modify the operator itself to ensure fundamental physical properties such as mass and energy conservation. Special local a~priori correction techniques at re-entrant corners lead to an improved representation of the energy in the discrete system and can suppress the global pollution effect. Local mass conservation can be achieved by an a~posteriori correction to the finite element flux. This avoids artifacts in coupled multi-physics transport problems. Finally, hardware failures in large supercomputers may lead to a loss of data in solution subdomains. Within parallel multigrid, this can be compensated by the accelerated solution of local subproblems. These resilient algorithms will gain importance on future extreme scale computing systems.

Markus Huber, Ulrich Rüde, Christian Waluga et al.

The Stokes system with constant viscosity can be cast into different formulations by exploiting the incompressibility constraint. For instance the strain in the weak formulation can be replaced by the gradient to decouple the velocity components in the different coordinate directions. Thus the discretization of the simplified problem leads to fewer nonzero entries in the stiffness matrix. This is of particular interest in large scale simulations where a reduced memory bandwidth requirement can help to significantly accelerate the computations. In the case of a piecewise constant viscosity, as it typically arises in multi-phase flows, or when the boundary conditions involve traction, the situation is more complex, and one has to treat the cross derivatives in the original Stokes system with care. A naive application of the standard vectorial Laplacian results in a physically incorrect solution, while formulations based on the strain increase the computational effort everywhere, even when the inconsistencies arise only from an incorrect treatment in a small fraction of the computational domain. Here we propose a new approach that is consistent with the strain-based formulation and preserves the decoupling advantages of the gradient-based formulation in isoviscous subdomains. The modification is equivalent to locally changing the discretization stencils, hence the more expensive discretization is restricted to a lower dimensional interface, making the additional computational cost asymptotically negligible. We demonstrate the consistency and convergence properties of the method and show that in a massively parallel setup, the multigrid solution of the resulting discrete systems is faster than for the classical strain-based formulation. Moreover, we give an application example which is inspired by geophysical research.

Sebastian Eibl, Ulrich Rüde

Creating a highly parallel and flexible discrete element software requires an interdisciplinary approach, where expertise from different disciplines is combined. On the one hand domain specialists provide interaction models between particles. On the other hand high-performance computing specialists optimize the code to achieve good performance on different hardware architectures. In particular, the software must be carefully crafted to achieve good scaling on massively parallel supercomputers. Combining all this in a flexible and extensible, widely usable software is a challenging task. In this article we outline the design decisions and concepts of a newly developed particle dynamics code MESA-PD that is implemented as part of the waLBerla multi-physics framework. Extensibility, flexibility, but also performance and scalability are primary design goals for the new software framework. In particular, the new modular architecture is designed such that physical models can be modified and extended by domain scientists without understanding all details of the parallel computing functionality and the underlying distributed data structures that are needed to achieve good performance on current supercomputer architectures. This goal is achieved by combining the high performance simulation framework waLBerla with code generation techniques. All code and the code generator are released as open source under GPLv3 within the publicly available waLBerla framework (www.walberla.net).

Markus Huber, Björn Gmeiner, Ulrich Rüde et al.

Fault tolerant algorithms for the numerical approximation of elliptic partial differential equations on modern supercomputers play a more and more important role in the future design of exa-scale enabled iterative solvers. Here, we combine domain partitioning with highly scalable geometric multigrid schemes to obtain fast and fault-robust solvers in three dimensions. The recovery strategy is based on a hierarchical hybrid concept where the values on lower dimensional primitives such as faces are stored redundantly and thus can be recovered easily in case of a failure. The lost volume unknowns in the faulty region are re-computed approximately with multigrid cycles by solving a local Dirichlet problem on the faulty subdomain. Different strategies are compared and evaluated with respect to performance, computational cost, and speed up. Especially effective are strategies in which the local recovery in the faulty region is executed in parallel with global solves and when the local recovery is additionally accelerated. This results in an asynchronous multigrid iteration that can fully compensate faults. Excellent parallel performance on a current peta-scale system is demonstrated.