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

A hyperbolic reformulation of the Serre-Green-Naghdi model for general bottom topographies

We present a novel hyperbolic reformulation of the Serre-Green-Naghdi (SGN) model for the description of dispersive water waves. Contrarily to the classical Boussinesq-type models, it contains only first order derivatives, thus allowing to overcome the numerical difficulties and the severe time step restrictions arising from higher order terms. The proposed model reduces to the original SGN model when an artificial sound speed tends to infinity. Moreover, it is endowed with an energy conservation law from which the energy conservation law associated with the original SGN model is retrieved when the artificial sound speed goes to infinity. The governing partial differential equations are then solved at the aid of high order ADER discontinuous Galerkin finite element schemes. The new model has been successfully validated against numerical and experimental results, for both flat and non-flat bottom. For bottom topographies with large variations, the new model proposed in this paper provides more accurate results with respect to the hyperbolic reformulation of the SGN model with the mild bottom approximation recently proposed in "C. Escalante, M. Dumbser and M.J. Castro. An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous Galerkin schemes, Journal of Computational Physics 2018"

On improving the efficiency of ADER methods

The (modern) arbitrary derivative (ADER) approach is a popular technique for the numerical solution of differential problems based on iteratively solving an implicit discretization of their weak formulation. In this work, focusing on an ODE context, we investigate several strategies to improve this approach. Our initial emphasis is on the order of accuracy of the method in connection with the polynomial discretization of the weak formulation. We demonstrate that precise choices lead to higher-order convergences in comparison to the existing literature. Then, we put ADER methods into a Deferred Correction (DeC) formalism. This allows to determine the optimal number of iterations, which is equal to the formal order of accuracy of the method, and to introduce efficient $p$-adaptive modifications. These are defined by matching the order of accuracy achieved and the degree of the polynomial reconstruction at each iteration. We provide analytical and numerical results, including the stability analysis of the new modified methods, the investigation of the computational efficiency, an application to adaptivity and an application to hyperbolic PDEs with a Spectral Difference (SD) space discretization