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Title: 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 » 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.« less

Authors:
ORCiD logo [1]; ORCiD logo [2]
  1. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
  2. 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) (SC-21)
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. doi: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. doi: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 = {2018},
month = {6}
}

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