Discretization of Differential Equations
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.
KeywordsComputational Fluid Dynamic Finite Volume Method Collocation Method Trial Function Point Collocation
- 18.G. Beer, I. Smith, C. Duenser, The Boundary Element Method with Programming: For Engineers and Scientists (Springer, Berlin, 2008) Google Scholar
- 79.R. Eymard, T. Galloue, R. Herbin, Finite volume methods, in Handbook of Numerical Analysis, vol. 7, ed. by P.G. Ciarlet, J.L. Lions (Elsevier, Amsterdam, 2000), pp. 713–1020 Google Scholar
- 98.B.G. Galerkin, On electrical circuits for the approximate solution of the Laplace equation. Vestnik Inzh. 19, 897–908 (1915) Google Scholar
- 200.J. Peiro, S. Sherwin, in Handbook of Materials Modeling, vol. 1, ed. by S. Yip (Springer, Berlin, 2005), pp. 1–32 Google Scholar
- 276.L.C. Wrobel, M.H. Aliabadi, The Boundary Element Method (Wiley, New York, 2002) Google Scholar
- 277.G. Wunsch, Feldtheorie (VEB Technik, Berlin, 1973) Google Scholar