A Positive-Definite, WENO-Limited, High-Order Finite Volume Solver for 2-D Transport on the Cubed Sphere Using an ADER Time Discretization
Abstract
Modern computer architectures reward added computation if it reduces algorithmic dependence, reduces data movement, increases accuracy/robustness, and improves memory accesses. The driving motive for this study is to develop a numerical algorithm that respects these constraints while improving accuracy and robustness. This study introduces the ADER-DT (Arbitrary DERivatives in time and space-differential transform) time discretization to positive-definite, weighted essentially nonoscillatory (WENO)-limited, finite volume transport on the cubed sphere in lieu of semidiscrete integrators. The cost of the ADER-DT algorithm is significantly improved from previous implementations without affecting accuracy. A new function-based WENO implementation is also detailed for use with the ADER-DT time discretization. While ADER-DT costs about 1.5 times more than a fourth-order, five-stage strong stability preserving Runge-Kutta (SSPRK4) method, it is far more computationally dense (which is advantageous on accelerators such as graphics processing units), and it has a larger effective maximum stable time step. ADER-DT errors converge more quickly with grid refinement than SSPRK4, giving 6.5 times less error in the L∞ norm than SSPRK4 at the highest refinement level for smooth data. For nonsmooth data, ADER-DT resolves C0 discontinuities more sharply. For a complex flow field, ADER exhibits less phase error than SSPRK4. In conclusion, improving bothmore »
- Authors:
-
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
- National Center for Atmospheric Research, Boulder, CO (United States)
- Publication Date:
- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Oak Ridge Leadership Computing Facility (OLCF)
- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
- OSTI Identifier:
- 1461937
- Grant/Contract Number:
- AC05-00OR22725
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Advances in Modeling Earth Systems
- Additional Journal Information:
- Journal Volume: 10; Journal Issue: 7; Journal ID: ISSN 1942-2466
- Publisher:
- American Geophysical Union (AGU)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; WENO; cubed sphere; transport; ADER; accelerators; HPC
Citation Formats
Norman, Matthew R., and Nair, R. D. A Positive-Definite, WENO-Limited, High-Order Finite Volume Solver for 2-D Transport on the Cubed Sphere Using an ADER Time Discretization. United States: N. p., 2018.
Web. doi:10.1029/2017MS001247.
Norman, Matthew R., & Nair, R. D. A Positive-Definite, WENO-Limited, High-Order Finite Volume Solver for 2-D Transport on the Cubed Sphere Using an ADER Time Discretization. United States. https://doi.org/10.1029/2017MS001247
Norman, Matthew R., and Nair, R. D. Fri .
"A Positive-Definite, WENO-Limited, High-Order Finite Volume Solver for 2-D Transport on the Cubed Sphere Using an ADER Time Discretization". United States. https://doi.org/10.1029/2017MS001247. https://www.osti.gov/servlets/purl/1461937.
@article{osti_1461937,
title = {A Positive-Definite, WENO-Limited, High-Order Finite Volume Solver for 2-D Transport on the Cubed Sphere Using an ADER Time Discretization},
author = {Norman, Matthew R. and Nair, R. D.},
abstractNote = {Modern computer architectures reward added computation if it reduces algorithmic dependence, reduces data movement, increases accuracy/robustness, and improves memory accesses. The driving motive for this study is to develop a numerical algorithm that respects these constraints while improving accuracy and robustness. This study introduces the ADER-DT (Arbitrary DERivatives in time and space-differential transform) time discretization to positive-definite, weighted essentially nonoscillatory (WENO)-limited, finite volume transport on the cubed sphere in lieu of semidiscrete integrators. The cost of the ADER-DT algorithm is significantly improved from previous implementations without affecting accuracy. A new function-based WENO implementation is also detailed for use with the ADER-DT time discretization. While ADER-DT costs about 1.5 times more than a fourth-order, five-stage strong stability preserving Runge-Kutta (SSPRK4) method, it is far more computationally dense (which is advantageous on accelerators such as graphics processing units), and it has a larger effective maximum stable time step. ADER-DT errors converge more quickly with grid refinement than SSPRK4, giving 6.5 times less error in the L∞ norm than SSPRK4 at the highest refinement level for smooth data. For nonsmooth data, ADER-DT resolves C0 discontinuities more sharply. For a complex flow field, ADER exhibits less phase error than SSPRK4. In conclusion, improving both accuracy and robustness as well as better respecting modern computational efficiency requirements, we believe the method presented herein is competitive for efficiently transporting tracers over the sphere for applications targeting modern computing architectures.},
doi = {10.1029/2017MS001247},
journal = {Journal of Advances in Modeling Earth Systems},
number = 7,
volume = 10,
place = {United States},
year = {Fri Jun 15 00:00:00 EDT 2018},
month = {Fri Jun 15 00:00:00 EDT 2018}
}
Web of Science
Works referenced in this record:
Cascade interpolation for semi-Lagrangian advection over the sphere
journal, April 1999
- Nair, Ramachandran; Côté, Jean; Staniforth, Andrew
- Quarterly Journal of the Royal Meteorological Society, Vol. 125, Issue 556
Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems
journal, February 2007
- Dumbser, Michael; Käser, Martin
- Journal of Computational Physics, Vol. 221, Issue 2
A spectral element semi-Lagrangian (SESL) method for the spherical shallow water equations
journal, September 2003
- Giraldo, F. X.; Perot, J. B.; Fischer, P. F.
- Journal of Computational Physics, Vol. 190, Issue 2
The Mass-Conservative Cell-Integrated Semi-Lagrangian Advection Scheme on the Sphere
journal, March 2002
- Nair, Ramachandran D.; Machenhauer, Bennert
- Monthly Weather Review, Vol. 130, Issue 3
Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points
journal, August 2005
- Henrick, Andrew K.; Aslam, Tariq D.; Powers, Joseph M.
- Journal of Computational Physics, Vol. 207, Issue 2
High-order finite volume shallow water model on the cubed-sphere: 1D reconstruction scheme
journal, September 2015
- Katta, Kiran K.; Nair, Ramachandran D.; Kumar, Vinod
- Applied Mathematics and Computation, Vol. 266
The “Cubed Sphere”: A New Method for the Solution of Partial Differential Equations in Spherical Geometry
journal, March 1996
- Ronchi, C.; Iacono, R.; Paolucci, P. S.
- Journal of Computational Physics, Vol. 124, Issue 1
A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes
journal, February 2008
- Capdeville, G.
- Journal of Computational Physics, Vol. 227, Issue 5
“Yin-Yang grid”: An overset grid in spherical geometry: SPHERICAL OVERSET GRID “YIN-YANG GRID”
journal, September 2004
- Kageyama, Akira; Sato, Tetsuya
- Geochemistry, Geophysics, Geosystems, Vol. 5, Issue 9
A standard test case suite for two-dimensional linear transport on the sphere
journal, January 2012
- Lauritzen, P. H.; Skamarock, W. C.; Prather, M. J.
- Geoscientific Model Development, Vol. 5, Issue 3
Arbitrarily High-Order-Accurate, Hermite WENO Limited, Boundary-Averaged Multi-Moment Constrained Finite-Volume (BA-MCV) Schemes for 1-D Transport
journal, January 2015
- Norman, Matthew R.
- Procedia Computer Science, Vol. 51
A standard test case suite for two-dimensional linear transport on the sphere: results from a collection of state-of-the-art schemes
journal, January 2013
- Lauritzen, P. H.; Ullrich, P. A.; Jablonowski, C.
- Geoscientific Model Development Discussions, Vol. 6, Issue 3
Finite volume evolution Galerkin (FVEG) methods for three-dimensional wave equation system
journal, September 2007
- Lukáčová-Medvid'ová, M.; Warnecke, G.; Zahaykah, Y.
- Applied Numerical Mathematics, Vol. 57, Issue 9
Flux-Corrected Transport
journal, August 1997
- Boris, Jay P.; Book, David L.
- Journal of Computational Physics, Vol. 135, Issue 2
Fully multidimensional flux-corrected transport algorithms for fluids
journal, June 1979
- Zalesak, Steven T.
- Journal of Computational Physics, Vol. 31, Issue 3
SLICE-S: A Semi-Lagrangian Inherently Conserving and Efficient scheme for transport problems on the Sphere
journal, October 2004
- Zerroukat, Mohamed; Wood, Nigel; Staniforth, Andrew
- Quarterly Journal of the Royal Meteorological Society, Vol. 130, Issue 602
Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics
journal, April 2009
- Balsara, Dinshaw S.; Rumpf, Tobias; Dumbser, Michael
- Journal of Computational Physics, Vol. 228, Issue 7
Moving Vortices on the Sphere: A Test Case for Horizontal Advection Problems
journal, February 2008
- Nair, Ramachandran D.; Jablonowski, Christiane
- Monthly Weather Review, Vol. 136, Issue 2
A conservative semi-Lagrangian multi-tracer transport scheme (CSLAM) on the cubed-sphere grid
journal, March 2010
- Lauritzen, Peter H.; Nair, Ramachandran D.; Ullrich, Paul A.
- Journal of Computational Physics, Vol. 229, Issue 5
ADER schemes for scalar non-linear hyperbolic conservation laws with source terms in three-space dimensions
journal, January 2005
- Toro, E. F.; Titarev, V. A.
- Journal of Computational Physics, Vol. 202, Issue 1
The Flux-Form Semi-Lagrangian Spectral Element (FF-SLSE) method for tracer transport: The FF-SLSE Method for Tracer Transport
journal, July 2013
- Ullrich, Paul A.; Norman, Matthew R.
- Quarterly Journal of the Royal Meteorological Society, Vol. 140, Issue 680
High-Order Finite-Volume Transport on the Cubed Sphere: Comparison between 1D and 2D Reconstruction Schemes
journal, July 2015
- Katta, Kiran K.; Nair, Ramachandran D.; Kumar, Vinod
- Monthly Weather Review, Vol. 143, Issue 7
Optimization-based limiters for the spectral element method
journal, June 2014
- Guba, Oksana; Taylor, Mark; St-Cyr, Amik
- Journal of Computational Physics, Vol. 267
Multidimensional Flux-Form Semi-Lagrangian Transport Schemes
journal, September 1996
- Lin, Shian-Jiann; Rood, Richard B.
- Monthly Weather Review, Vol. 124, Issue 9
A conservative multi-tracer transport scheme for spectral-element spherical grids
journal, January 2014
- Erath, Christoph; Nair, Ramachandran D.
- Journal of Computational Physics, Vol. 256
An Incremental Remapping Transport Scheme on a Spherical Geodesic Grid
journal, August 2005
- Lipscomb, William H.; Ringler, Todd D.
- Monthly Weather Review, Vol. 133, Issue 8
High order multi-moment constrained finite volume method. Part I: Basic formulation
journal, June 2009
- Ii, Satoshi; Xiao, Feng
- Journal of Computational Physics, Vol. 228, Issue 10
A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes
journal, September 2008
- Dumbser, Michael; Balsara, Dinshaw S.; Toro, Eleuterio F.
- Journal of Computational Physics, Vol. 227, Issue 18
A WENO-limited, ADER-DT, finite-volume scheme for efficient, robust, and communication-avoiding multi-dimensional transport
journal, October 2014
- Norman, Matthew R.
- Journal of Computational Physics, Vol. 274
Hermite WENO schemes and their application as limiters for Runge–Kutta discontinuous Galerkin method: one-dimensional case
journal, January 2004
- Qiu, Jianxian; Shu, Chi-Wang
- Journal of Computational Physics, Vol. 193, Issue 1
The Spectral Element Atmosphere Model (SEAM): High-Resolution Parallel Computation and Localized Resolution of Regional Dynamics
journal, March 2004
- Fournier, Aimé; Taylor, Mark A.; Tribbia, Joseph J.
- Monthly Weather Review, Vol. 132, Issue 3
The Constrained Interpolation Profile Method for Multiphase Analysis
journal, May 2001
- Yabe, Takashi; Xiao, Feng; Utsumi, Takayuki
- Journal of Computational Physics, Vol. 169, Issue 2, p. 556-593
Finite-volume transport on various cubed-sphere grids
journal, November 2007
- Putman, William M.; Lin, Shian-Jiann
- Journal of Computational Physics, Vol. 227, Issue 1
A Conservative Semi-Lagrangian Discontinuous Galerkin Scheme on the Cubed Sphere
journal, January 2014
- Guo, Wei; Nair, Ramachandran D.; Qiu, Jing-Mei
- Monthly Weather Review, Vol. 142, Issue 1
A low communication and large time step explicit finite-volume solver for non-hydrostatic atmospheric dynamics
journal, February 2011
- Norman, Matthew R.; Nair, Ramachandran D.; Semazzi, Fredrick H. M.
- Journal of Computational Physics, Vol. 230, Issue 4
Localized Artificial Viscosity Stabilization of Discontinuous Galerkin Methods for Nonhydrostatic Mesoscale Atmospheric Modeling
journal, December 2015
- Yu, M. L.; Giraldo, F. X.; Peng, M.
- Monthly Weather Review, Vol. 143, Issue 12
Semi-Lagrangian Integration Schemes for Atmospheric Models—A Review
journal, September 1991
- Staniforth, Andrew; Côté, Jean
- Monthly Weather Review, Vol. 119, Issue 9
On High Order Strong Stability Preserving Runge–Kutta and Multi Step Time Discretizations
journal, October 2005
- Gottlieb, Sigal
- Journal of Scientific Computing, Vol. 25, Issue 1
A high order characteristic discontinuous Galerkin scheme for advection on unstructured meshes
journal, November 2016
- Lee, D.; Lowrie, R.; Petersen, M.
- Journal of Computational Physics, Vol. 324
Weighted Essentially Non-oscillatory Schemes
journal, November 1994
- Liu, Xu-Dong; Osher, Stanley; Chan, Tony
- Journal of Computational Physics, Vol. 115, Issue 1
Finite Volume Evolution Galerkin Methods for Nonlinear Hyperbolic Systems
journal, December 2002
- Lukáčová-Medvid'ová, M.; Saibertová, J.; Warnecke, G.
- Journal of Computational Physics, Vol. 183, Issue 2
A Hermite upwind WENO scheme for solving hyperbolic conservation laws
journal, February 2008
- Capdeville, G.
- Journal of Computational Physics, Vol. 227, Issue 4
Multi-moment ADER-Taylor methods for systems of conservation laws with source terms in one dimension
journal, August 2012
- Norman, M. R.; Finkel, H.
- Journal of Computational Physics, Vol. 231, Issue 20
Hermite WENO limiting for multi-moment finite-volume methods using the ADER-DT time discretization for 1-D systems of conservation laws
journal, February 2015
- Norman, Matthew R.
- Journal of Computational Physics, Vol. 282
Discontinuous Galerkin Transport on the Spherical Yin–Yang Overset Mesh
journal, January 2013
- Hall, David M.; Nair, Ramachandran D.
- Monthly Weather Review, Vol. 141, Issue 1
A spectral finite volume transport scheme on the cubed-sphere
journal, September 2007
- Cheruvu, Vani; Nair, Ramachandran D.; Tufo, Henry M.
- Applied Numerical Mathematics, Vol. 57, Issue 9
A Discontinuous Galerkin Transport Scheme on the Cubed Sphere
journal, April 2005
- Nair, Ramachandran D.; Thomas, Stephen J.; Loft, Richard D.
- Monthly Weather Review, Vol. 133, Issue 4
Algorithmic improvements for schemes using the ADER time discretization
journal, June 2013
- Norman, M. R.
- Journal of Computational Physics, Vol. 243
Derivative Riemann solvers for systems of conservation laws and ADER methods
journal, February 2006
- Toro, E. F.; Titarev, V. A.
- Journal of Computational Physics, Vol. 212, Issue 1
Nodal High-Order Discontinuous Galerkin Methods for the Spherical Shallow Water Equations
journal, September 2002
- Giraldo, F. X.; Hesthaven, J. S.; Warburton, T.
- Journal of Computational Physics, Vol. 181, Issue 2