TY - JOUR
T1 - A spectral deferred correction method applied to the shallow water equations on a sphere
AU - Jia, Jun
AU - Hill, Judith C.
AU - Evans, Katherine J.
AU - Fann, George I.
AU - Taylor, Mark A.
PY - 2013
Y1 - 2013
N2 - 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.
AB - 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.
KW - Differential equations
KW - Error analysis
KW - Numerical analysis/modeling
UR - http://www.scopus.com/inward/record.url?scp=84884961289&partnerID=8YFLogxK
U2 - 10.1175/MWR-D-12-00048.1
DO - 10.1175/MWR-D-12-00048.1
M3 - Article
AN - SCOPUS:84884961289
SN - 0027-0644
VL - 141
SP - 3435
EP - 3449
JO - Monthly Weather Review
JF - Monthly Weather Review
IS - 10
ER -