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Expanded Stability through Higher Temporal Accuracy for Time-Centered Advection Schemes

Richard J. BabarskyDepartment of Mathematics, James Madison University, Harrisonburg, Virginia

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Robert SharpleyDepartment of Mathematics, University of South Carolina, Columbia, South Carolina

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Abstract

Applying standard explicit time-differencing to hyperbolic equations (i.e., which characterize convection-dominated atmospheric flows) invariably results in rather severe stability restrictions. The primary problem appears to be attributable to the differencing approximation of the time derivative term. In this study the authors show that, for explicit, time-centered advection schemes, achieving higher-order temporal accuracy results in schemes with significantly improved stability properties compared with conventional leapfrog methods. Linear results show that marked improvement is possible in the stability properties by including in the differencing scheme a crucial term approximating the time derivative of third order. The critical CFL number for this time-centered Taylor (TCT) scheme is shown to exceed that of second-order leapfrog by nearly a factor of 2. Similar results hold for the corresponding fourth-order schemes. A solid-body rotation test confirms the findings of the two-dimensional stability analysis and compares these time-centered schemes with popular forward-in-time methods. One-dimensional nonlinear results corroborate the fundamental stablizing effect of the TCT approach with the TCT algorithm offering significant improvements in nonlinear stability over leapfrog methods as well as MacCormack’s scheme—a popular nonlinear, dissipative differencing scheme.

Corresponding author address: Dr. Richard Babarsky, BDM Federal, Inc., 1501 BDM Way, McLean, VA 22102-3204.

Abstract

Applying standard explicit time-differencing to hyperbolic equations (i.e., which characterize convection-dominated atmospheric flows) invariably results in rather severe stability restrictions. The primary problem appears to be attributable to the differencing approximation of the time derivative term. In this study the authors show that, for explicit, time-centered advection schemes, achieving higher-order temporal accuracy results in schemes with significantly improved stability properties compared with conventional leapfrog methods. Linear results show that marked improvement is possible in the stability properties by including in the differencing scheme a crucial term approximating the time derivative of third order. The critical CFL number for this time-centered Taylor (TCT) scheme is shown to exceed that of second-order leapfrog by nearly a factor of 2. Similar results hold for the corresponding fourth-order schemes. A solid-body rotation test confirms the findings of the two-dimensional stability analysis and compares these time-centered schemes with popular forward-in-time methods. One-dimensional nonlinear results corroborate the fundamental stablizing effect of the TCT approach with the TCT algorithm offering significant improvements in nonlinear stability over leapfrog methods as well as MacCormack’s scheme—a popular nonlinear, dissipative differencing scheme.

Corresponding author address: Dr. Richard Babarsky, BDM Federal, Inc., 1501 BDM Way, McLean, VA 22102-3204.

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