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|Title: ||Time Splitting Methods Applied To A Nonlinear Advective Equation|
|Authors: ||Shrivathsa, B|
|Advisors: ||Nanjundiah, R S|
|Keywords: ||Nonlinear Differential Equations|
Meterology - Time Splitting Methods
Nonlinear Advective Equation - Splitting Methods
|Submitted Date: ||Jul-2006|
|Series/Report no.: ||G20954|
|Abstract: ||Time splitting is a numerical procedure used in solution of partial diﬀerential equations whose solutions allow multiple time scales. Numerical schemes are split for handling the stiﬀness in equations, i.e. when there are multiple time scales with a few time scales being smaller than the others. When there are
such terms with smaller time scales, due to the Courant number restriction, the computational cost becomes high if these terms are treated explicitly.
In the present work a nonlinear advective equation is solved numerically using diﬀerent techniques based on a generalised framework for splitting methods.
The nonlinear advective equation was chosen because it has an analytical solution making comparisons with numerical schemes amenable and also because its nonlinearity mimics the equations encountered in atmospheric
modelling. Using the nonlinear advective equation as a test bed, an analysis of the splitting methods and their inﬂuence on the split solutions has been made.
An understanding of inﬂuence of splitting schemes requires knowledge of behaviour of unsplit schemes beforehand. Hence a study on unsplit methods has also been made.
In the present work, using the nonlinear advective equation, it shown that the three time level schemes have high phase errors and underestimate energy (even though they have a higher order of accuracy in time). It is also found that the leap-frog method, which is used widely in atmospheric modelling, is the worst among examined unsplit methods. The semi implicit method, again a popular splitting method with atmospheric modellers is the worst among examined split methods.
Three time-level schemes also need explicit ﬁltering to remove the computational mode. This ﬁltering can have a signiﬁcant impact on the obtained numerical solutions, and hence three-time level schemes appear to be
unattractive in the context of the nonlinear convective equation. Based on this experience, splitting methods for the two-time level schemes is proposed. These schemes realistically capture the phase and energy of the nonlinear advective equation.|
|Appears in Collections:||Centre for Atmospheric and Oceanic Sciences (caos)|
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