Abstract
The simplest approach to discretize a differential equation replaces differential quotients by quotients of finite differences. For the space variables this method works best on a regular grid. Finite volume methods, which are very popular in computational fluid dynamics, take averages over small control volumes and can be easily used with irregular grids. Finite elements and finite volumes belong to the general class of finite element methods which are prominent in the engineering sciences and use an expansion in piecewise polynomials with small support. Spectral methods, on the other hand, expand the solution as a linear combination of global basis functions. A general concept is the method of weighted residuals. Most popular is Galerkin’s method. The simpler point collocation and sub-domain collocation methods fulfill the differential equation only at certain points or averaged over certain control volumes. The more demanding least-squares method has become popular in computational fluid dynamics and computational electrodynamics.
If the Green’s function is available for a problem, the method of boundary elements is an interesting alternative. It reduces the dimensionality and is, for instance, very popular in chemical physics to solve the Poisson-Boltzmann equation.
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Notes
- 1.
Dirichlet b.c. concern the function values, Neumann b.c. the derivative, Robin b.c. a linear combination of both, Cauchy b.c. the function value and the normal derivative and mixed b.c. have different character on different parts of the boundary.
- 2.
Differential equations which are higher order in time can be always brought to first order by introducing the time derivatives as additional variables.
- 3.
If A is not Hermitian we have to distinguish left- and right-eigenvectors.
- 4.
Also called expansion functions.
- 5.
Generalization to systems of equations is straightforward.
- 6.
One or more linear differential operators, usually a combination of the function and its first derivatives.
- 7.
This requirement can be replaced by additional equations for the u i , for instance with the tau method [195].
- 8.
Also called test functions.
- 9.
The triangulation is not determined uniquely by the nodes.
- 10.
The order of the indices does matter.
- 11.
Higher accuracy can be achieved, for instance, by Gaussian integration.
- 12.
This is only possible if the fundamental solution or Green’s function is available.
- 13.
The minus sign is traditionally used.
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Scherer, P.O.J. (2013). Discretization of Differential Equations. In: Computational Physics. Graduate Texts in Physics. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00401-3_11
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