Differential Equations Lecture Work Solutions 296

Differential Equations Lecture Work Solutions 296 - u yy i...

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The error term can be found when computing the next term in Taylor series ( B h 3 1 6 C h 3 2 6 ) u yyy i j = 1 3 h 2 1 h 1 + h 2 h 2 2 h 1 + h 2 ! u yyy i j The denominator is of order h and numerator of order h 2 so the method is of order h u yy i j = 2 h 1 h 2 u ij + 2 h 1 ( h 1 + h 2 ) u ij +1 + 2 h 2 ( h 1 + h 2 ) u ij 1 Let’s check the special case of uniformly spaced points, i.e. h 1 = h 2 = h. The above equation becomes the well known
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Unformatted text preview: u yy i j = 2 h 2 u i j + 1 h 2 u i j +1 + 1 h 2 u i j 1 The error term listed above is 1 3 h 2 2 h h 2 2 h ! u yyy i j = 0 and so we have to go to the next term in Taylor and thus we get a second order (as expected for uniformly spaced points). 296...
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