We investigate the parallel performance of Parallel Spectral Deferred corrections, a numerical approach that provides small-scale parallelism for the numerical solution of initial value problems. The scheme is applied to the shallow-water equation and uses an implicit-explicit splitting that, in order to be efficient, integrates fast modes implicitly and slow modes explicitly. We describe parallel OpenMP-based implementations of parallel Spectral Deferred Corrections for two well established simulation codes: the finite volume based operational ocean model ICON-O and the spherical harmonics based research code SWEET. We also develop a performance model and benchmark our implementations on a single node of the JUSUF (SWEET) and JUWELS (ICON-O) system at Jülich Supercomputing Centre. A reduction of time-to-solution across a range of accuracies is demonstrated. For ICON-O, we show speedup over the currently used Adams–Bashforth-2 integrator with OpenMP loop parallelization. For SWEET, we show speedup over serial Spectral Deferred Corrections and a second order implicit-explicit integrator.
BickartTA (1977) An efficient solution process for implicit Runge–Kutta methods. SIAM Journal on Numerical Analysis14(6): 1022–1027.
2.
ButcherJC (1976) On the implementation of implicit Runge-Kutta methods. BIT Numerical Mathematics16(3): 237–240.
3.
ČaklovićGLunetTGötschelS, et al. (2025) Improving efficiency of parallel across the method spectral deferred corrections. SIAM Journal on Scientific Computing47(1): A430–A453.
4.
Caldas SteinstraesserJGda Silva PeixotoPSchreiberM (2024) Parallel-in-time integration of the shallow water equations on the rotating sphere using Parareal and MGRIT. Journal of Computational Physics496: 112591.
5.
CampinJMAdcroftAHillC, et al. (2004) Conservation of properties in a free-surface model. Ocean Modelling6(3): 221–244.
6.
DanilovSSidorenkoDWangQ, et al. (2017) The finite-volume sea ice–ocean model (FESOM2). Geoscientific Model Development10(2): 765–789.
7.
EmmettMMinionML (2012) Toward an efficient parallel in time method for partial differential equations. Communications in Applied Mathematics and Computational Science7: 105–132.
8.
FalgoutRDFriedhoffSKolevTV, et al. (2014) Parallel time integration with multigrid. SIAM Journal on Scientific Computing36: C635–C661.
9.
GalewskyJScottRKPolvaniLM (2004) An initial-value problem for testing numerical models of the global shallow-water equations. Tellus A: Dynamic Meteorology and Oceanography56(5): 429–440.
10.
GanderMJ (2015) 50 years of time parallel time integration. In: Multiple Shooting and Time Domain Decomposition. Springer, 69–113.
11.
GanderMJLunetT (2024) Time Parallel Time Integration. SIAM.
12.
GearCW (1988) Parallel methods for ordinary differential equations. CALCOLO25(1-2): 1–20.
13.
HackJJJakobR (1992) Description of a global shallow water model based on the spectral transform method. In: Technical Report No. NCAR/TN-343+STR. University Corporation for Atmospheric Research.
HamonFPSchreiberMMinionML (2020) Parallel-in-time multi-level integration of the shallow-water equations on the rotating sphere. Journal of Computational Physics407: 109210.
16.
IserlesANørsettSP (1990) On the theory of parallel Runge-Kutta methods. IMA Journal of Numerical Analysis10(4): 463–488.
17.
JacksonKR (1991) A survey of parallel numerical methods for initial value problems for ordinary differential equations. IEEE Transactions on Magnetics27: 3792–3797.
18.
JiaJHillJCEvansKJ, et al. (2014) A spectral deferred correction method applied to the shallow water equations on a sphere. Monthly Weather Review141: 3435–3449.
19.
Jülich Supercomputing Centre (2021) JUWELS cluster and booster: exascale pathfinder with modular supercomputing Architecture at Juelich supercomputing centre. Journal of large-scale research facilities7(A138): 13.
KornP (2017) Formulation of an unstructured grid model for global ocean dynamics. Journal of Computational Physics339: 525–552.
22.
KornPLinardakisL (2018) A conservative discretization of the shallow-water equations on triangular grids. Journal of Computational Physics375: 871–900.
23.
KornPBrüggemannNJungclausJH, et al. (2022) ICON-O: the ocean component of the ICON earth system model—global simulation characteristics and local telescoping capability. Journal of Advances in Modeling Earth Systems14(10): e2021MS002952.
24.
LionsJLMadayYTuriniciG (2001) A “Parareal” in time discretization of PDE’s. Comptes Rendus de l'Académie des Sciences - Series I - Mathematics332: 661–668.
25.
OngBWSchroderJB (2020) Applications of time parallelization. Computing and Visualization in Science23(1-4): 11.
26.
OngBWHaynesRDLaddK (2016) Algorithm 965: RIDC methods: a family of parallel time integrators. ACM Transactions on Mathematical Software43(1): 8:1–8:13.
27.
PhilippiBSlawigT (2022) The Parareal algorithm applied to the FESOM 2 ocean circulation model. arXiv:2208.07598v1 [math.NA]. https://arxiv.org/abs/2208.07598v1
28.
PhilippiBSlawigT (2023) A micro-macro Parareal implementation for the ocean-circulation model FESOM2. arXiv:2306.17269v1 [math.NA]. https://arxiv.org/abs/2306.17269v1
29.
RuprechtDSpeckR (2016) Spectral deferred corrections with fast-wave slow-wave splitting. SIAM Journal on Scientific Computing38(4): A2535–A2557.
SchreiberMPeixotoPSHautT, et al. (2018) Beyond spatial scalability limitations with a massively parallel method for linear oscillatory problems. The International Journal of High Performance Computing Applications32(6): 913–933.
32.
SpeckR (2018) Parallelizing spectral deferred corrections across the method. Computing and Visualization in Science19: 75–83.
33.
SpeckRRuprechtDMinionM, et al. (2016) Inexact spectral deferred corrections. In: DickopfTGanderMJHalpernL, et al. (eds) Domain Decomposition Methods in Science and Engineering XXII, Lect. Springer, Vol. 104, 389–396.
34.
van der HouwenPJSommeijerBP (1991) Iterated Runge–Kutta methods on parallel computers. SIAM Journal on Scientific and Statistical Computing12(5): 1000–1028.
35.
WanHGiorgettaMAZänglG, et al. (2013) The ICON-1.2 hydrostatic atmospheric dynamical core on triangular grids – part 1: formulation and performance of the baseline version. Geoscientific Model Development6(3): 735–763.
36.
WeiserM (2014) Faster SDC convergence on non-equidistant grids by DIRK sweeps. BIT Numerical Mathematics55(4): 1219–1241.
37.
ZänglGReinertDRípodasP, et al. (2015) The ICON (ICOsahedral Non-hydrostatic) modelling framework of DWD and MPI-M: description of the non-hydrostatic dynamical core. Quarterly Journal of the Royal Meteorological Society141(687): 563–579.
