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Introduction to Modeling Convection in Planets and StarsMagnetic Field, Density Stratification, Rotation$
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Gary A. Glatzmaier

Print publication date: 2013

Print ISBN-13: 9780691141725

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691141725.001.0001

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Time Integration Schemes

Time Integration Schemes

Chapter:
Chapter Eight Time Integration Schemes
Source:
Introduction to Modeling Convection in Planets and Stars
Author(s):

Gary A. Glatzmaier

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691141725.003.0008

This chapter focuses on time integration schemes, including fourth-order accurate Runge–Kutta and predictor-corrector schemes as well as schemes that allow larger time steps (and therefore fewer steps for a given amount of simulated time) by treating the linear diffusion terms implicitly. The nonlinear terms, however, couple all the modes and so would be extremely expensive to treat implicitly; therefore they are usually treated explicitly. Such “semi-implicit” schemes considerably improve the efficiency of the computer code. The chapter also describes the Crank–Nicolson scheme and concludes by showing how the current numerical model can easily be modified to study mantle convection (also called “geodynamics”) using the vorticity equation in the limit of an infinite Prandtl number.

Keywords:   time integration schemes, Runge–Kutta scheme, predictor-corrector scheme, nonlinear terms, semi-implicit scheme, computer code, Crank–Nicolson scheme, numerical model, mantle convection, infinite Prandtl number

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