Abstract
The purpose of this chapter is to study potentially useful schemes and discretizations of the linear advection equation. We discuss various stable and consistent schemes such as the leap-frog scheme, the upstream scheme (or upwind scheme), the Lax–Wendroff scheme, and the semi-Lagrangian scheme. The conditions under which they are stable are also discussed, along with ways to avoid the initial problem in centered-in-time schemes. We consider problems like numerical dispersion, numerical diffusion, and computational modes, including ways to minimize their effects, and we discuss flux corrective schemes, which improve the ubiquitous numerical diffusion inherent in low-order schemes like the upstream scheme. Finally, we describe the Courant–Friedrich–Levy (CFL) condition and its physical interpretation.
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Notes
- 1.
Solving (5.3) for a limited domain, say \(x\in [0,L]\), there is an upper bound to the wavelength with a corresponding lower bound on \(\alpha _m\).
- 2.
Let \(A=a+\mathrm {i}b\) be an imaginary number with real part a and imaginary part b. Then \(|A| = \sqrt{AA^*} = \sqrt{a^2 + b^2}\), where \(A^* = a-\mathrm {i}b\) is the complex conjugate of A.
- 3.
In describing this model, people often write “while a second-order scheme is employed for advective terms”.
- 4.
The requirement \(\partial _xu = 0\) is satisfied here since \(u=u_0=\) const.
- 5.
For capillary waves, \(\partial _{\alpha }c > 0\), implying that a capillary wave travels more slowly than its energy.
- 6.
The CFL condition for stability is assumed to be satisfied, so \(\sqrt{1-\lambda ^2}\) is a real number.
- 7.
- 8.
If \(u_0\) is a function of space and time, then the characteristics are curved lines with \({\mathrm{D}^*x}/{\mathrm {d}t}\) defining the slope at any point in the x, t plane.
- 9.
If \(u_0 < 0\), then \(u_0 = -|u_0|\) and \(x_Q > x_j\), and the upstream scheme is \(\theta _j^{n+1} = (1-C)\theta _j^n + C\theta _{j+1}^n\).
- 10.
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Røed, L.P. (2019). Advection Problem. In: Atmospheres and Oceans on Computers. Springer Textbooks in Earth Sciences, Geography and Environment. Springer, Cham. https://doi.org/10.1007/978-3-319-93864-6_5
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