
2.3
Sept 10, 2015 17:33:35 GMT
Post by matthorr on Sept 10, 2015 17:33:35 GMT
An approach to combine some of the best features of upwind differencing and central differencing is to bias the scheme in the upwind direction. To examine the accuracy of such upwindbiased schemes construct a Taylor table for a mixed backward formula approximation to the first derivative:
\( (u_x)_j = \frac{1}{\Delta x}\left(au_{j +1} + bu_j + cu_{j −1} + du_{j −2} \right)\quad + ? \)
and find the optimum value for the parameters and the resulting Taylor series error, \(er_t\) . What is the order of the method? Without performing a Taylor Series analysis what do you think is the order of accuracy of a backward difference scheme that includes points \( j − 2, j −1\) and \(j\)? A central difference scheme that includes points \(j − 2, j −1, j, j +1\) and \(j + 2\) ?

