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A Taylor series analysis of the upwind scheme discussed above will show that it is first-order accurate in space and time. Modified wavenumber analysis shows that the first-order upwind scheme introduces severe numerical diffusion /dissipation in the solution where large gradients exist due to necessity of high wavenumbers to represent sharp ...
Solution in the central difference scheme fails to converge for Peclet number greater than 2 which can be overcome by using an upwind scheme to give a reasonable result. [1]: Fig. 5.5, 5.13 Therefore the upwind differencing scheme is applicable for Pe > 2 for positive flow and Pe < −2 for negative flow. For other values of Pe, this scheme ...
In order to find the cell face value a quadratic function passing through two bracketing or surrounding nodes and one node on the upstream side must be used. In central differencing scheme and second order upwind scheme the first order derivative is included and the second order derivative is ignored.
This simulation result contrasts extremely well against the above first-order upwind and second-order central difference results shown above. This scheme also provides good results when applied to sets of equations - see results below for this scheme applied to the Euler equations.
For large Peclet numbers (|Pe| > 2) it uses the Upwind difference scheme, which first order accurate but takes into account the convection of the fluid. As it can be seen in figure 4 that for Pe = 0, it is a linear distribution and for high Pe it takes the upstream value depending on the flow direction.
Unlike first-order upwind scheme, the MacCormack does not introduce diffusive errors in the solution. However, it is known to introduce dispersive errors ( Gibbs phenomenon ) in the region where the gradient is high.
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However, for large Peclet numbers (generally > 2) this approximation gave inaccurate results. It was recognized independently by several investigators [1] [2] that the less expensive but only first order accurate upwind scheme can be employed but that this scheme produces results with false diffusion for multidimensional cases. Many new schemes ...