Abstract
In this chapter, finite difference (FD) methods are described for the generic scalar transport equation. Here we present methods of approximating first, second, and mixed derivatives, using Taylor series expansion, and polynomial fitting. Derivation of higher-order methods, and treatment of nonlinear terms and boundaries is discussed. Attention is also paid to the effects of grid non-uniformity on truncation error and to the estimation of discretization errors. Application of some of the basic methods to several examples are described for Cartesian grids. Spectral methods are also briefly introduced, both as tools for analysis and methods for solving differential equations.
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Notes
- 1.
For example, an equation such as Eq. (3.43) and the pictured molecules would be generated by discretization of the Poisson equation \( \varvec{\nabla \cdot } (\varvec{\nabla } \Phi ) = f\).
- 2.
It is called modified wavenumber by some authors.
- 3.
Chapter 3 of Boyd (2001) points out that this pseudo-spectral constraint can be obtained by using the Dirac delta-function, \(\delta (x-x_i)\) as a test function in the weighted-residual method described above.
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Ferziger, J.H., Perić, M., Street, R.L. (2020). Finite Difference Methods. In: Computational Methods for Fluid Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-99693-6_3
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DOI: https://doi.org/10.1007/978-3-319-99693-6_3
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Publisher Name: Springer, Cham
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Online ISBN: 978-3-319-99693-6
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