Abstract
This chapter is devoted to methods of time integration. First, the methods of solving ordinary differential equations are described, including basic methods, predictor-corrector and multipoint methods, and Runge–Kutta methods. The application of these methods to the generic unsteady transport equation is described next, including analysis of stability and accuracy. Implicit second-order schemes, which are most widely used in commercial CFD-software, are described in detail, including the handling of non-uniform time steps. The features of some basic methods are demonstrated on two examples.
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- 1.
The quadrature approximations used above assume a uniform time step; if the time step is allowed to vary, the coefficients multiplying the function values at different time levels become complicated functions of step sizes, as we saw in Chap. 3 for finite differences in space.
- 2.
In many geophysical applications, certain motions, e.g., acoustic waves, move very rapidly relative to others, e.g., winds or ocean currents. It is useful then to actually split the computation and carry out separate calculations for the rapid and slow parts of the system (Klemp et al. 2007, or Blumberg and Mellor 1987).
- 3.
This number is often called the CFL number, where CFL stands for the initials of R. Courant, K. Friedrichs and H. Lewy, who first defined it in their paper from 1928.
- 4.
Recall that the procedure is (1) expand the terms in the difference scheme in a two-dimensional (x, t) Taylor series about a single point (in this case, \(x_i, t_{n}\)) to obtain an expanded partial differential equation, (2) use this partial differential equation itself (and derivatives of it) to replace all of the higher-order and mixed time terms with spatial derivative terms in the expanded PDE, and (3) re-arrange to have the original PDE on the left-hand side and the remaining terms on the right-hand side. Warming and Hyett (1974)’s Procedure Table is useful when doing the method by hand. The remaining terms are the truncation error, i.e., the difference between what the difference equation solves and the original equation. Keep the lowest-order ones to see the most important physical effects.
- 5.
For diffusion (dissipation) to occur, the coefficients of the even-order derivatives in the modified equation must have alternating signs, the one for the second-order term being positive and the one for the fourth-order term being negative, and so on. Note that second-order physical diffusion is present in the original equation here.
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Ferziger, J.H., Perić, M., Street, R.L. (2020). Methods for Unsteady Problems. In: Computational Methods for Fluid Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-99693-6_6
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DOI: https://doi.org/10.1007/978-3-319-99693-6_6
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