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 upwind-biased schemes construct a Taylor table for a mixed backward formula approximation to the first derivative:
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 \(j + 2\) ?