Solution

Expand the two central second-difference operators. The given stencil identity yields

Γ_{9} u = h^{2} Δ u + \frac{h ^{4}}{12} Δ^{2} u + h^{6} [\frac{u _{xxxxxx} + u _{yyyyyy}}{360} + \frac{u _{xxxx yy} + u _{xx yyyy}}{72}] + O (h^{8}) .

(1)

Consequently its unnormalized local residual is

Γ_{9} u - h^{2} f = (h^{4} /12) Δ f + O (h^{6})

. The normalized Laplacian approximation has leading error $(h^{2} /12) Δ f$ and is second order. These Taylor statements require the indicated smooth derivatives; boundary accuracy is a separate matter.

 Ancestors (11)

i
39B
Paper 2
ii
2009
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
List of universities
 Home

 View article source

 Discussion (0) New discussion

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version

Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 2 / 39B / i / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1) New discussion

 Articles by others on the same topic (0)

 Discussion (0) New discussion