A spectral deferred correction method applied to the shallow water equations on a sphere

Jun Jia, Judith C. Hill, Katherine J. Evans, George I. Fann, Mark A. Taylor

Research output: Contribution to journalArticlepeer-review

14 Scopus citations

Abstract

Although significant gains have been made in achieving high-order spatial accuracy in global climate modeling, less attention has been given to the impact imposed by low-order temporal discretizations. For long-time simulations, the error accumulation can be significant, indicating a need for higher-order temporal accuracy. A spectral deferred correction (SDC) method is demonstrated of even order, with second- to eighth-order accuracy and A-stability for the temporal discretization of the shallow water equations within the spectral-element High-Order Methods Modeling Environment(HOMME). Because this method is stable and of high order, larger time-step sizes can be taken while still yielding accurate long-time simulations. The spectral deferred correction method has been tested on a suite of popular benchmark problems for the shallow water equations, and when compared to the explicit leapfrog, five-stage Runge-Kutta, and fully implicit (FI) second-order backward differentiation formula (BDF2) time-integration methods, it achieves higher accuracy for the same or larger time-step sizes. One of the benchmark problems, the linear advection of a Gaussian bell height anomaly, is extended to run for longer time periods to mimic climate-length simulations, and the leapfrog integration method exhibited visible degradation for climate length simulations whereas the second-order and higher methods did not. When integrated with higher-order SDC methods, a suite of shallow water test problems is able to replicate the test with better accuracy.

Original languageEnglish
Pages (from-to)3435-3449
Number of pages15
JournalMonthly Weather Review
Volume141
Issue number10
DOIs
StatePublished - 2013

Keywords

  • Differential equations
  • Error analysis
  • Numerical analysis/modeling

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