Abstract
This chapter is an introduction to finite difference approximation methods. Key concepts like local truncation error, numerical stability and convergence of approximate solutions are developed in a one-dimensional setting. This chapter establishes the theoretical framework that is used to analyse the convergence of finite difference approximations in later chapters.
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Notes
- 1.
More precisely, suppose that z(h) is a quantity that depends on h. We say that \(z(h)=\mathcal {O} (h^{p})\) if there is a constant C, independent of h, such that \(|z(h)|\le Ch^{p}\) as \(h\rightarrow 0\).
- 2.
The shorthand version \(\mathcal {L}_{h} U_{m}\) to denote the value of \(\mathcal {L}_{h}U\) at the mth grid point.
- 3.
The order of convergence p is usually an integer but exceptions to this rule are not uncommon, so beware.
- 4.
The subscript h will often be omitted and we write \(\mathcal {R}\) and \(\mathscr {R}\) since they have no continuous counterparts and to avoid the notation becoming too onerous.
- 5.
Inequalities of the form \(V\ge 0\), where V is a grid function, mean that \(V_{m}\ge 0\) for all \(m=0, 1{,\ldots ,}\,M\) (or \(m=1, 2{,\ldots ,}\, M-1\), depending on context).
- 6.
This condition is satisfied if, for example, a Dirichlet BC is applied at one or both ends of the interval: \(\mathscr {L}_{h}U_{0}:=U_{0}\) would lead to \(b_{0}=1>a_{0}+c_{0}=0\).
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© 2015 Springer International Publishing Switzerland
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Griffiths, D.F., Dold, J.W., Silvester, D.J. (2015). Finite Difference Methods in \(\mathbb {R}^1\) . In: Essential Partial Differential Equations. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-22569-2_6
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DOI: https://doi.org/10.1007/978-3-319-22569-2_6
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