Parallel-in-Time Multi-Level Integration of the Shallow-Water Equations on the Rotating Sphere

The modeling of atmospheric processes in the context of weather and climate simulations is an important and computationally expensive challenge. The temporal integration of the underlying PDEs requires a very large number of time steps, even when the terms accounting for the propagation of fast atmospheric waves are treated implicitly. Therefore, the use of parallel-in-time integration schemes to reduce the time-to-solution is of increasing interest, particularly in the numerical weather forecasting field. We present a multi-level parallel-in-time integration method combining the Parallel Full Approximation Scheme in Space and Time (PFASST) with a spatial discretization based on Spherical Harmonics (SH). The iterative algorithm computes multiple time steps concurrently by interweaving parallel high-order fine corrections and serial corrections performed on a coarsened problem. To do that, we design a methodology relying on the spectral basis of the SH to coarsen and interpolate the problem in space. The methods are evaluated on the shallow-water equations on the sphere using a set of tests commonly used in the atmospheric flow community. We assess the convergence of PFASST-SH upon refinement in time. We also investigate the impact of the coarsening strategy on the accuracy of the scheme, and specifically on its ability to capture the high-frequency modes accumulating in the solution. Finally, we study the computational cost of PFASST-SH to demonstrate that our scheme resolves the main features of the solution multiple times faster than the serial schemes

Multilevel convergence analysis of multigrid-reduction-in-time

This paper presents a multilevel convergence framework for multigrid-reduction-in-time (MGRIT) as a generalization of previous two-grid estimates. The framework provides a priori upper bounds on the convergence of MGRIT V- and F-cycles, with different relaxation schemes, by deriving the respective residual and error propagation operators. The residual and error operators are functions of the time stepping operator, analyzed directly and bounded in norm, both numerically and analytically. We present various upper bounds of different computational cost and varying sharpness. These upper bounds are complemented by proposing analytic formulae for the approximate convergence factor of V-cycle algorithms that take the number of fine grid time points, the temporal coarsening factors, and the eigenvalues of the time stepping operator as parameters. The paper concludes with supporting numerical investigations of parabolic (anisotropic diffusion) and hyperbolic (wave equation) model problems. We assess the sharpness of the bounds and the quality of the approximate convergence factors. Observations from these numerical investigations demonstrate the value of the proposed multilevel convergence framework for estimating MGRIT convergence a priori and for the design of a convergent algorithm. We further highlight that observations in the literature are captured by the theory, including that two-level Parareal and multilevel MGRIT with F-relaxation do not yield scalable algorithms and the benefit of a stronger relaxation scheme. An important observation is that with increasing numbers of levels MGRIT convergence deteriorates for the hyperbolic model problem, while constant convergence factors can be achieved for the diffusion equation. The theory also indicates that L-stable Runge-Kutta schemes are more amendable to multilevel parallel-in-time integration with MGRIT than A-stable Runge-Kutta schemes.Comment: 26 pages; 17 pages Supplementary Material

Multi-level spectral deferred corrections scheme for the shallow water equations on the rotating sphere

Efficient time integration schemes are necessary to capture the complex processes involved in atmospheric flows over long periods of time. In this work, we propose a high-order, implicit–explicit numerical scheme that combines Multi-Level Spectral Deferred Corrections (MLSDC) and the Spherical Harmonics (SH) transform to solve the wave-propagation problems arising from the shallow-water equations on the rotating sphere. The iterative temporal integration is based on a sequence of corrections distributed on coupled space–time levels to perform a significant portion of the calculations on a coarse representation of the problem and hence to reduce the time-to-solution while preserving accuracy. In our scheme, referred to as MLSDC-SH, the spatial discretization plays a key role in the efficiency of MLSDC, since the SH basis allows for consistent transfer functions between space–time levels that preserve important physical properties of the solution. We study the performance of the MLSDC-SH scheme with shallow-water test cases commonly used in numerical atmospheric modeling. We use this suite of test cases, which gradually adds more complexity to the nonlinear system of governing partial differential equations, to perform a detailed analysis of the accuracy of MLSDC-SH upon refinement in time. We illustrate the stability properties of MLSDC-SH and show that the proposed scheme achieves up to eighth-order convergence in time. Finally, we study the conditions in which MLSDC-SH achieves its theoretical speedup, and we show that it can significantly reduce the computational cost compared to single-level Spectral Deferred Corrections (SDC)

Multi-level spectral deferred corrections scheme for the shallow water equations on the rotating sphere

Efficient time integration schemes are necessary to capture the complex processes involved in atmospheric flows over long periods of time. In this work, we propose a high-order, implicit–explicit numerical scheme that combines Multi-Level Spectral Deferred Corrections (MLSDC) and the Spherical Harmonics (SH) transform to solve the wave-propagation problems arising from the shallow-water equations on the rotating sphere. The iterative temporal integration is based on a sequence of corrections distributed on coupled space–time levels to perform a significant portion of the calculations on a coarse representation of the problem and hence to reduce the time-to-solution while preserving accuracy. In our scheme, referred to as MLSDC-SH, the spatial discretization plays a key role in the efficiency of MLSDC, since the SH basis allows for consistent transfer functions between space–time levels that preserve important physical properties of the solution. We study the performance of the MLSDC-SH scheme with shallow-water test cases commonly used in numerical atmospheric modeling. We use this suite of test cases, which gradually adds more complexity to the nonlinear system of governing partial differential equations, to perform a detailed analysis of the accuracy of MLSDC-SH upon refinement in time. We illustrate the stability properties of MLSDC-SH and show that the proposed scheme achieves up to eighth-order convergence in time. Finally, we study the conditions in which MLSDC-SH achieves its theoretical speedup, and we show that it can significantly reduce the computational cost compared to single-level Spectral Deferred Corrections (SDC)