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# P.4.3 Apply a Taylor series expansion to a mixed backward formula for the first derivative: (Ux)i = 1 Ax (aui-2+ bui-1 + cu +

## P.4.3 Apply a Taylor series expansion to a mixed backward formula for the first derivative: (Ux)i = 1 Ax (aui-2+ bui-1 + cu + dui+1) Derive the family of second order accurate formulas and the corresponding truncation error in function of the coefficient d. Obtain the unique third order accurate upwind-biased scheme and determine the corresponding truncation error. Hint: Show that (d = 1/2-a; b = -3a-1/2; c= 3a) and that the second order truncation error is equal to Axuxxx (1/6 a). The unique third order scheme is obtained for a = 1/6: (ux)i = 1 6Ax Ax (24u (ui-2-6ui-1 + 3u; + 2ui+1) + 12 24

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