Higher Order Accurate Difference Solutions of Fluid Mechanics Problems by a Compact Differencing Technique

A general fourth order differencing scheme proposed by Professor H.-O. Kreiss of Uppsala University is developed and applied to three viscous test problems to verify the accuracy and applicability of the technique. The procedure is atypical since only three nodes are necessary to obtain the desired fourth order accuracy. This is accomplished by a differencing technique which considers the function and all necessary derivatives as unknowns. The relations for these derivatives yield simple tridiagonal equations which can be easily solved. This differencing can be combined with either explicit or implicit time differencing with little difficulty and only a few changes in the solution procedure. The test problems solved are Burgers' equation, Howarth's retarded boundary layer flow, and the incompressible driven cavity. Comparisons of the fourth order results with those computed using second order methods are presented for each test case and clearly indicate that the accuracy achieved by these fourth order computations is always significantly better than current second order procedures.