Comparison of exponential integrators and traditional time integration schemes for the shallow water equations

Brachet, Matthieu; Debreu, Laurent; Eldred, Christopher

doi:10.1016/j.apnum.2022.05.006

Comparison of exponential integrators and traditional time integration schemes for the shallow water equations

Journal Article · 20 May 2022 · Applied Numerical Mathematics

DOI:https://doi.org/10.1016/j.apnum.2022.05.006· OSTI ID:1872031

; Debreu, Laurent ^[1]; ^[2]

University of Grenoble Alpes, Grenoble (France)
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

We report the time integration scheme is probably one of the most fundamental choices in the development of an ocean model. In this paper, we investigate several time integration schemes when applied to the shallow water equations. This set of equations is accurate enough for the modeling of a shallow ocean and is also relevant to study as it is the one solved for the barotropic (i.e. vertically averaged) component of a three dimensional ocean model. We analyze different time stepping algorithms for the linearized shallow water equations. High order explicit schemes are accurate but the time step is constrained by the Courant-Friedrichs-Lewy stability condition. Implicit schemes can be unconditionally stable but, in practice lack accuracy when used with large time steps. In this paper we propose a detailed comparison of such classical schemes with exponential integrators. The accuracy and the computational costs are analyzed in different configurations.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: European Union’s Horizon 2020; French National Research Agency; USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: NA0003525

OSTI ID:: 1872031

Report Number(s):: SAND2022-6879J; 706614

Journal Information:: Applied Numerical Mathematics, Journal Name: Applied Numerical Mathematics Vol. 180; ISSN 0168-9274

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (31)

Compact Finite Difference Schemes for Ocean Models Blayo, Eric Journal of Computational Physics, Vol. 164, Issue 2 https://doi.org/10.1006/jcph.2000.6565	journal	November 2000
Approximation of the matrix exponential for matrices with a skinny field of values Caliari, Marco; Cassini, Fabio; Zivcovich, Franco BIT Numerical Mathematics, Vol. 60, Issue 4 https://doi.org/10.1007/s10543-020-00809-0	journal	May 2020
Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow. Part I Arakawa, Akio Journal of Computational Physics, Vol. 1, Issue 1 https://doi.org/10.1016/0021-9991(66)90015-5	journal	August 1966
A standard test set for numerical approximations to the shallow water equations in spherical geometry Williamson, David L.; Drake, John B.; Hack, James J. Journal of Computational Physics, Vol. 102, Issue 1 https://doi.org/10.1016/S0021-9991(05)80016-6	journal	September 1992
Bedload transport in shallow water models: Why splitting (may) fail, how hyperbolicity (can) help Cordier, S.; Le, M. H.; Morales de Luna, T. Advances in Water Resources, Vol. 34, Issue 8 https://doi.org/10.1016/j.advwatres.2011.05.002	journal	August 2011
The scaling and modified squaring method for matrix functions related to the exponential Skaflestad, B.; Wright, W. M. Applied Numerical Mathematics, Vol. 59, Issue 3-4 https://doi.org/10.1016/j.apnum.2008.03.035	journal	March 2009
Exponential integrators preserving local conservation laws of PDEs with time-dependent damping/driving forces Bhatt, Ashish; Moore, Brian E. Journal of Computational and Applied Mathematics, Vol. 352 https://doi.org/10.1016/j.cam.2018.12.003	journal	May 2019
Exponential time differencing for the tracer equations appearing in primitive equation ocean models Calandrini, Sara; Pieper, Konstantin; Gunzburger, Max D. Computer Methods in Applied Mechanics and Engineering, Vol. 365 https://doi.org/10.1016/j.cma.2020.113002	journal	June 2020
Domain decomposition based exponential time differencing method for fluid dynamics problems with smooth solutions Liang, Shan; Zhang, Jian; Liu, Xia-Zhen Computers & Fluids, Vol. 194 https://doi.org/10.1016/j.compfluid.2019.104307	journal	November 2019
On the time-splitting scheme used in the Princeton Ocean Model Kamenkovich, V. M.; Nechaev, D. A. Journal of Computational Physics, Vol. 228, Issue 8 https://doi.org/10.1016/j.jcp.2008.12.033	journal	May 2009
Exponential versus IMEX high-order time integrators for thermal convection in rotating spherical shells Garcia, Ferran; Bonaventura, Luca; Net, Marta Journal of Computational Physics, Vol. 264 https://doi.org/10.1016/j.jcp.2014.01.033	journal	May 2014
An efficient exponential time integration method for the numerical solution of the shallow water equations on the sphere Gaudreault, Stéphane; Pudykiewicz, Janusz A. Journal of Computational Physics, Vol. 322 https://doi.org/10.1016/j.jcp.2016.07.012	journal	October 2016
Preconditioned implicit-exponential integrators (IMEXP) for stiff PDEs Luan, Vu Thai; Tokman, Mayya; Rainwater, Greg Journal of Computational Physics, Vol. 335 https://doi.org/10.1016/j.jcp.2017.01.054	journal	April 2017
KIOPS: A fast adaptive Krylov subspace solver for exponential integrators Gaudreault, Stéphane; Rainwater, Greg; Tokman, Mayya Journal of Computational Physics, Vol. 372 https://doi.org/10.1016/j.jcp.2018.06.026	journal	November 2018
Stability analysis of split-explicit free surface ocean models: Implication of the depth-independent barotropic mode approximation Demange, Jérémie; Debreu, Laurent; Marchesiello, Patrick Journal of Computational Physics, Vol. 398 https://doi.org/10.1016/j.jcp.2019.108875	journal	December 2019
Exponential time differencing for mimetic multilayer ocean models Pieper, Konstantin; Sockwell, K. Chad; Gunzburger, Max Journal of Computational Physics, Vol. 398 https://doi.org/10.1016/j.jcp.2019.108900	journal	December 2019
The regional oceanic modeling system (ROMS): a split-explicit, free-surface, topography-following-coordinate oceanic model Shchepetkin, Alexander F.; McWilliams, James C. Ocean Modelling, Vol. 9, Issue 4 https://doi.org/10.1016/j.ocemod.2004.08.002	journal	January 2005
Useful time-stepping methods for the Coriolis term in a shallow water model Walters, Roy A.; Lane, Emily M.; Hanert, Emmanuel Ocean Modelling, Vol. 28, Issue 1-3 https://doi.org/10.1016/j.ocemod.2008.10.004	journal	January 2009
Stability constraints for oceanic numerical models: implications for the formulation of time and space discretizations Lemarié, F.; Debreu, L.; Madec, G. Ocean Modelling, Vol. 92 https://doi.org/10.1016/j.ocemod.2015.06.006	journal	August 2015
Exponential integrators with parallel-in-time rational approximations for the shallow-water equations on the rotating sphere Schreiber, Martin; Schaeffer, Nathanaël; Loft, Richard Parallel Computing, Vol. 85 https://doi.org/10.1016/j.parco.2019.01.005	journal	July 2019
Exponential integrators Hochbruck, Marlis; Ostermann, Alexander Acta Numerica, Vol. 19 https://doi.org/10.1017/S0962492910000048	journal	May 2010
Runge-Kutta methods with minimum error bounds Ralston, Anthony Mathematics of Computation, Vol. 16, Issue 80 https://doi.org/10.1090/S0025-5718-1962-0150954-0	journal	January 1962
A Lawson-type exponential integrator for the Korteweg–de Vries equation Ostermann, Alexander; Su, Chunmei IMA Journal of Numerical Analysis, Vol. 40, Issue 4 https://doi.org/10.1093/imanum/drz030	journal	October 2019
Generalized Runge-Kutta Processes for Stable Systems with Large Lipschitz Constants Lawson, J. Douglas SIAM Journal on Numerical Analysis, Vol. 4, Issue 3 https://doi.org/10.1137/0704033	journal	September 1967
Analysis of Some Krylov Subspace Approximations to the Matrix Exponential Operator Saad, Y. SIAM Journal on Numerical Analysis, Vol. 29, Issue 1 https://doi.org/10.1137/0729014	journal	February 1992
Exponential Rosenbrock-Type Methods Hochbruck, Marlis; Ostermann, Alexander; Schweitzer, Julia SIAM Journal on Numerical Analysis, Vol. 47, Issue 1 https://doi.org/10.1137/080717717	journal	January 2009
Nineteen Dubious Ways to Compute the Exponential of a Matrix Moler, Cleve; Van Loan, Charles SIAM Review, Vol. 20, Issue 4 https://doi.org/10.1137/1020098	journal	October 1978
GPU Accelerated Algorithms for Computing Matrix Function Vector Products with Applications to Exponential Integrators and Fractional Diffusion Farquhar, Megan E.; Moroney, Timothy J.; Yang, Qianqian SIAM Journal on Scientific Computing, Vol. 38, Issue 3 https://doi.org/10.1137/15M1021672	journal	January 2016
On Krylov Subspace Approximations to the Matrix Exponential Operator Hochbruck, Marlis; Lubich, Christian SIAM Journal on Numerical Analysis, Vol. 34, Issue 5 https://doi.org/10.1137/S0036142995280572	journal	October 1997
Algorithm 919: A Krylov Subspace Algorithm for Evaluating the ϕ-Functions Appearing in Exponential Integrators Niesen, Jitse; Wright, Will M. ACM Transactions on Mathematical Software, Vol. 38, Issue 3 https://doi.org/10.1145/2168773.2168781	journal	April 2012
On the use of exponential time integration methods in atmospheric models Clancy, Colm; Pudykiewicz, Janusz A. Tellus A: Dynamic Meteorology and Oceanography, Vol. 65, Issue 1 https://doi.org/10.3402/tellusa.v65i0.20898	journal	May 2013

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Comparison of exponential integrators and traditional time integration schemes for the shallow water equations

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