Tobias Weinzierl

Michael Dumbser, Francesco Fambri, Maurizio Tavelli et al.

In this paper we discuss a new and very efficient implementation of high order accurate ADER discontinuous Galerkin (ADER-DG) finite element schemes on modern massively parallel supercomputers. The numerical methods apply to a very broad class of nonlinear systems of hyperbolic partial differential equations. ADER-DG schemes are by construction communication avoiding and cache blocking and are furthermore very well-suited for vectorization, so that they appear to be a good candidate for the future generation of exascale supercomputers. We introduce the numerical algorithm and show some applications to a set of hyperbolic equations with increasing level of complexity, ranging from the compressible Euler equations over the equations of linear elasticity and the unified Godunov-Peshkov-Romenski (GPR) model of continuum mechanics to general relativistic magnetohydrodynamics (GRMHD) and the Einstein field equations of general relativity. We present strong scaling results of the new ADER-DG schemes up to 180,000 CPU cores. To our knowledge, these are the largest runs ever carried out with high order ADER-DG schemes for nonlinear hyperbolic PDE systems. We also provide a detailed performance comparison with traditional Runge-Kutta DG schemes.

Maurizio Tavelli, Michael Dumbser, Dominic Etienne Charrier et al.

In most classical approaches of computational geophysics for seismic wave propagation problems, complex surface topography is either accounted for by boundary-fitted unstructured meshes, or, where possible, by mapping the complex computational domain from physical space to a topologically simple domain in a reference coordinate system. In this paper we propose a completely different strategy. We address the problem of geometrically complex free surface boundary conditions with a novel diffuse interface method on adaptive Cartesian meshes that consists in the introduction of a characteristic function $ 0\leqα\leq 1$ which identifies the location of the solid medium and the surrounding air and thus implicitly defines the location of the free surface boundary. Our new approach completely avoids the problem of mesh generation, since all that is needed for the definition of the complex surface topography is to set a scalar color function to unity inside the regions covered by the solid and to zero outside. An analysis of the eigenvalues of the PDE system shows that the complexity of the geometry has no influence on the admissible time step size due to the CFL condition. The model reduces to the classical linear elasticity equations inside the solid medium where the gradients of $α$ are zero, while in the diffuse interface zone at the free surface boundary the governing PDE system becomes nonlinear. We can prove that the solution of the Riemann problem with arbitrary data and a jump in $α$ from unity to zero yields a Godunov-state at the interface that satisfies the free-surface boundary condition exactly. In order to reduce numerical dissipation, we use high order DG finite element schemes on adaptive AMR grids together with a high resolution shock capturing subcell finite volume (FV) limiter in the diffuse interface region.

Dominic E. Charrier, Tobias Weinzierl

We present a communication- and data-sensitive formulation of ADER-DG for hyperbolic differential equation systems. Sensitive here has multiple flavours: First, the formulation reduces the persistent memory footprint. This reduces pressure on the memory subsystem. Second, the formulation realises the underlying predictor-corrector scheme with single-touch semantics, i.e., each degree of freedom is read on average only once per time step from the main memory. This reduces communication through the memory controllers. Third, the formulation breaks up the tight coupling of the explicit time stepping's algorithmic steps to mesh traversals. This averages out data access peaks. Different operations and algorithmic steps are ran on different grid entities. Finally, the formulation hides distributed memory data transfer behind the computation aligned with the mesh traversal. This reduces pressure on the machine interconnects. All techniques applied by our formulation are elaborated by means of a rigorous task formalism. They break up ADER-DG's tight causal coupling of compute steps and can be generalised to other predictor-corrector schemes.

Philipp Samfass, Tobias Weinzierl, Anne Reinarz et al.

Resilient algorithms in high-performance computing are subject to rigorous non-functional constraints. Resiliency must not increase the runtime, memory footprint or I/O demands too significantly. We propose a task-based soft error detection scheme that relies on error criteria per task outcome. They formalise how ``dubious'' an outcome is, i.e. how likely it contains an error. Our whole simulation is replicated once, forming two teams of MPI ranks that share their task results. Thus, ideally each team handles only around half of the workload. If a task yields large error criteria values, i.e.~is dubious, we compute the task redundantly and compare the outcomes. Whenever they disagree, the task result with a lower error likeliness is accepted. We obtain a self-healing, resilient algorithm which can compensate silent floating-point errors without a significant performance, I/O or memory footprint penalty. Case studies however suggest that a careful, domain-specific tailoring of the error criteria remains essential.

Marion Weinzierl, Tobias Weinzierl

We present a family of spacetree-based multigrid realizations using the tree's multiscale nature to derive coarse grids. They align with matrix-free geometric multigrid solvers as they never assemble the system matrices which is cumbersome for dynamically adaptive grids and full multigrid. The most sophisticated realizations use BoxMG to construct operator-dependent prolongation and restriction in combination with Galerkin/Petrov-Galerkin coarse-grid operators. This yields robust solvers for nontrivial elliptic problems. We embed the algebraic, problem- and grid-dependent multigrid operators as stencils into the grid and evaluate all matrix-vector products in-situ throughout the grid traversals. While such an approach is not literally matrix-free---the grid carries the matrix---we propose to switch to a hierarchical representation of all operators. Only differences of algebraic operators to their geometric counterparts are held. These hierarchical differences can be stored and exchanged with small memory footprint. Our realizations support arbitrary dynamically adaptive grids while they vertically integrate the multilevel operations through spacetree linearization. This yields good memory access characteristics, while standard colouring of mesh entities with domain decomposition allows us to use parallel manycore clusters. All realization ingredients are detailed such that they can be used by other codes.