Development and implementation of a parallel explicit multirate time stepping strategy for accelerating discontinuous Galerkin computations

This thesis is devoted to the development of robust and efficient time integration methods for ocean modeling which constitutes an important challenge. No singlerate time integration method works well for all physical processes in a complex model, as different subsystems have widely different time scales, dynamic behaviors, and accuracy requirements. If explicit singlerate schemes present attractive properties, such as their simple implementation and their efficient parallelization, they suffer from limiting time steps. The most constrained element, which may be much smaller than the average, determines the effective overall time step. Therefore, variable resolution is also recommended for the time integration. Multirate explicit methods have the vocation of reducing the computational cost by considering different time steps. Two multirate strategies, introduced by Constantinescu and Schlegel, have been adapted to the discontinuous Galerkin framework. The key idea is to gather mesh elements in groups which are stable for a certain range of time steps. The transition between them is accommodated by buffers which should preserve accuracy and conservation properties. The extension of the multirate approach to the parallel framework is challenging because classical mesh partitioning techniques are not adequate as the elements have different workloads. A multi-constraint partitioning strategy is employed to balance equitably the computational work on each processor at each stage of the algorithm. We propose an efficient parallel implementation of this multirate approach which minimizes the computational and communication overheads and maintains multirate speedups up to a significant number of processors.(FSA - Sciences de l) -- UCL, 201

Multirate technique for explicit Nodal Discontinuous Galerkin computations of time domain Maxwell equations on complex geometries

International audienc

Development of a parallel third order explicit multirate scheme

International audienceMultirate schemes aims at circumventing the global stability restriction of classical explicit methods. The idea is to gather mesh elements in groups that satisfy local stability conditions. The transition between groups has to be accommodated to preserve convergence and conservation properties. The extension of these strategies to the parallel framework is challenging since the computational load varies at each stage of the algorithm. Here we focus on the parallel implementation of a third order multirate strategy for discontinuous Galerkin simulations

An efficient parallel implementation of explicit multirate Runge–Kutta schemes for discontinuous Galerkin computations

Although explicit time integration schemes require small computational efforts per time step, their efficiency is severely restricted by their stability limits. Indeed, the multi-scale nature of some physical processes combined with highly unstructured meshes can lead some elements to impose a severely small stable time step for a global problem. Multirate methods offer a way to increase the global efficiency by gathering grid cells in appropriate groups under local stability conditions. These methods are well suited to the discontinuous Galerkin framework. The parallelization of the multirate strategy is challenging because grid cells have different workloads. The computational cost is different for each sub-time step depending on the elements involved and a classical partitioning strategy is not adequate any more. In this paper, we propose a solution that makes use of multi-constraint mesh partitioning. It tends to minimize the inter-processor communications, while ensuring that the workload is almost equally shared by every computer core at every stage of the algorithm. Particular attention is given to the simplicity of the parallel multirate algorithm while minimizing computational and communication overheads. Our implementation makes use of the MeTiS library for mesh partitioning and the Message Passing Interface for inter-processor communication. Performance analyses for two and three-dimensional practical applications confirm that multirate methods preserve important computational advantages of explicit methods up to a significant number of processors

Multirate time stepping for accelerating explicit discontinuous Galerkin computations with application to geophysical flows

This paper presents multirate explicit time-stepping schemes for solving partial differential equations with discontinuous Galerkin elements in the framework of Large-scale marine flows. It addresses the variability of the local stable time steps by gathering the mesh elements in appropriate groups. The real challenge is to develop methods exhibiting mass conservation and consistency. Two multirate approaches, based on standard explicit Runge–Kutta methods, are analyzed. They are well suited and optimized for the discontinuous Galerkin framework. The significant speedups observed for the hydrodynamic application of the Great Barrier Reef confirm the theoretical expectations

Efficient Parallel Multirate Time Stepping with Application to the World Ocean.

The development of suitable and fast time integration methods for ocean modeling constitutes an important challenge. No single time-discretisation works well for all physical processes in a complex marine model, as different subsystems have widely different characteristics in terms of time scales, dynamic behaviour, and accuracy requirements. We believe that building appropriate time stepping strategies for multi-scale computations will enable us to gain an order of magnitude. Indeed, unstructured-mesh generation processes are complex and, even though it is possible to control average element sizes in specific regions of the domain, it is not the case for each element size. The smallest element is usually much more smaller than the criterion that was prescribed a priori and it determines the stable time step for the entire model. Therefore, the computational efficiency of explicit time-stepping methods may be drastically low. Multirate schemes represent a class of methods that use various time steps on different grid cells. The strategy consists in splitting the domain in a smart way. Grid cells are gathered in different groups that satisfy the local CFL stability conditions for a certain range of time steps. Standard explicit RungeKutta methods are applied on independent partitions while buffer groups have to be introduced between them, with adapted methods, in order to accommodate the transitions between them. These methods are especially suited for the Discontinuous Galerkin spatial discretization. Nevertheless, development of such methods is still challenging. Both, stability requirements and conservation properties should be satisfied. Two approaches are explored. Constantinescu introduced a conservative 2nd order scheme while Schlegel proposed a 3rd method that is, unfortunately, not conservative. Large-scale applications like the Great Barrier Reef require the use of parallel computers. Some kind of load balancing strategy has to be supplied to accomodate multirate schemes: indeed, small elements have a higher cost than large elements in such a strategy. Moreover, small elements at inter-processor interfaces will require more frequent updates. The key idea consists in creating an optimized mesh partition in a way that the amount of grid cells of the different multirate groups is ideally the same on each computer core. However, a compromise should also be found between the effective work on each processor and the amount of communications between them

Development of a parallel third order explicit multirate scheme

Multirate schemes aims at circumventing the global stability restriction of classical explicit methods. The idea is to gather mesh elements in groups that satisfy local stability conditions. The transition between groups has to be accommodated to preserve convergence and conservation properties. The extension of these strategies to the parallel framework is challenging since the computational load varies at each stage of the algorithm. Here we focus on the parallel implementation of a third order multirate strategy for discontinuous Galerkin simulations

Efficient parallel multirate time stepping for accelerating explicit discontinuous Galerkin computations.

The development of suitable and fast time integration methods for ocean modeling con- stitutes an important challenge. No single time-discretisation works well for all physical processes in a complex marine model, as dierent subsystems have widely dierent charac- teristics in terms of time scales, dynamic behaviour, and accuracy requirements. We believe that building appropriate time stepping strategies for multi-scale computations will enable us to gain an order of magnitude. Indeed, unstructured-mesh generation processes are complex and, even though it is possible to control average element sizes in specic regions of the domain, it is not the case for each element size. The smallest element is usually much more smaller than the criterion that was prescribed a priori and it determines the stable time step for the entire model. Therefore, the computational eciency of explicit time-stepping methods may be drastically low. Multirate schemes represent a class of methods that use various time steps on dierent grid cells. The strategy consists in splitting the domain in a smart way. Grid cells are gathered in dierent groups that satisfy the local CFL stability conditions for a certain range of time steps. Standard explicit Runge- Kutta methods are applied on independent partitions while buer groups have to be introduced between them, with adapted methods, in order to accommodate the transitions between them. These methods are especially suited for the Discontinuous Galerkin spatial discretization. Nevertheless, development of such methods is still challenging. Both, stability requirements and conservation properties should be satised. Two approaches are explored. Constantinescu introduced a conservative 2nd order scheme while Schlegel proposed a 3rd method that is, unfortunately, not conservative. Large-scale applications like the Great Barrier Reef require the use of parallel computers. Some kind of load balancing strategy has to be supplied to accomodate multirate schemes: indeed, small elements have a higher cost than large elements in such a strategy. Moreover, small elements at inter-processor interfaces will require more frequent updates. The key idea consists in creating an optimized mesh partition in a way that the amount of grid cells of the dierent multirate groups is ideally the same on each computer core. However, a compromise should also be found between the eective work on each processor and the amount of communications between them