Abstract
We now turn to numerical methods that can be used to approximate the solution of the heat equation. We develop the finite difference method in great detail, with particular emphasis on stability issues, which are delicate. We concentrate on the heat equation in one dimension of space, with homogeneous Dirichlet boundary conditions. We also give some indications about finite difference (in time)-finite element (in space) approximation.
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Notes
- 1.
For consistency with the notation used in the stationary case, we should denote these vectors \(U^j_{h,k}\). We drop the h, k index for brevity, but of course, these vectors crucially depend on h and k.
- 2.
Here also, the vectors \(F^j\) and \(U_0\) depend respectively on h and k, and on h.
- 3.
This is the same computation as in Chap. 2.
- 4.
Well, actually it may well be known somewhere in the literature, but let us assume it is not known for the sake of the argument.
- 5.
- 6.
For an easier visualization, we also plot a linear interpolation of the finite difference discrete values.
- 7.
Beware of the notation: up to now \(V^j\) meant the jth vector in the sequence, but here \(\mathscr {A}^j\) means the jth power of the matrix \(\mathscr {A}\).
- 8.
This is the only relevant case.
- 9.
For any matrix A, \(A^*\) denotes the adjoint matrix of A, i.e., its conjugate transpose.
- 10.
Again, beware of the notation: \(u^j\) is the jth element in the sequence, whereas \(\mathscr {T}^j\) is the jth iterate of the operator \(\mathscr {T}\).
- 11.
Again, beware of the notation: here \(\mathscr {G}^j\) is the jth iterate of \(\mathscr {G}\) and \(a^j\) is the function a to the power j.
- 12.
Here again, G depends on h and k even though the notation does not make it plain.
- 13.
Again, beware of the notation: \(u^j\) is the jth function in the sequence, whereas \(G^j\) is the jth iterate of the operator G and \(a^j\) is the function a to the power j.
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© 2016 Springer International Publishing Switzerland
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Le Dret, H., Lucquin, B. (2016). The Finite Difference Method for the Heat Equation. In: Partial Differential Equations: Modeling, Analysis and Numerical Approximation. International Series of Numerical Mathematics, vol 168. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-27067-8_8
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DOI: https://doi.org/10.1007/978-3-319-27067-8_8
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Publisher Name: Birkhäuser, Cham
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