We develop new error estimates for the one-dimensional advection equation, considering general space-time discretization schemes based on Runge–Kutta methods and finite difference discretizations. We then derive conditions on the number of points per wavelength for a given error tolerance from these new estimates. Our analysis also shows the existence of synergistic space-time discretization methods that permit to gain one order of accuracy at a given CFL number. Our new error estimates can be used to analyze the choice of space-time discretizations considered when testing Parallel-in-Time methods.
This review article serves to summarize the many advances in time-parallel computations since the excellent review article by Gander, "50 years of Time Parallel Integration" (Gander, in: 50 years of time parallel time integration. Multiple shooting and time domain decomposition, Springer, Berlin, 2015). We focus, when possible, on applications of time parallelism and the observed speedup and efficiency, highlighting the challenges and benefits of parallel time computations. The applications covered range from numerous PDE-based simulations (both hyperbolic and parabolic), to PDE-constrained optimization, powergrid simulations, and machine learning. The time-parallel methods covered range from various iterative schemes (multigrid, waveform, multiple shooting, domain decomposition) to direct time-parallel methods.
This paper describes a space-time parallel algorithm with space-time adaptive mesh refinement (AMR). AMR with subcycling is added to multigrid reduction-in-time (MGRIT) in order to provide solution efficient adaptive grids with a reduction in work performed on coarser grids. This algorithm is achieved by integrating two software libraries: XBraid (Parallel time integration with multigrid. https://computation.llnl.gov/projects/parallel-timeintegration-multigrid ) and Chombo (Chombo software package for AMR applications—design document, 2014). The former is a parallel time integration library using multigrid and the latter is a massively parallel structured AMR library. Employing this adaptive space-time parallel algorithm is Chord (Comput Fluids 123:202–217, 2015), a computational fluid dynamics (CFD) application code for solving compressible fluid dynamics problems. For the same solution accuracy, speedups are demonstrated from the use of space-time parallelization over the time-sequential integration on Couette flow and Stokes’ second problem. On a transient Couette flow case, at least a $$1.5\times $$ speedup is achieved, and with a time periodic problem, a speedup of up to $$13.7\times $$ over the time-sequential case is obtained. In both cases, the speedup is achieved by adding processors and exploring additional parallelization in time. The numerical experiments show the algorithm is promising for CFD applications that can take advantage of the time parallelism. Future work will focus on improving the parallel performance and providing more tests with complex fluid dynamics to demonstrate the full potential of the algorithm.
The code d3f++ developed under the leadership of GRS for simulating 3D flow and solute transport includes an advanced formulation for fracture flow and transport in lower-dimensional elements that has still been in need of qualification. A variety of in-situ experiments in the granitic rock at the Swedish Hard Rock Laboratory Äspö offers an excellent basis for this purpose. Particularly well suited is the work performed in the TRUE Block Scale Project where a hydrostructural model comprising 30 large-scale fractures in a cube-shaped domain with a side length of 200 m has been set up for simulating groundwater flow which in turn forms the basis for reproducing the results of the injection–extraction tracer test “C2”. The present paper deals with modelling 3D flow and solute transport with the code d3f++ as a qualification exercise. The modelling results reveal an excellent representation of the contrasts in flow velocity across the fractures. Also the jumps in tracer concentration along intersecting fractures where waters with different tracer concentrations mix are handled well by the code. Calibration has nevertheless been required in order to achieve a reasonable fit of measured and calculated breakthrough curves.
