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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 ...
In this paper he constructed the first high-order, total variation diminishing (TVD) scheme where he obtained second order spatial accuracy. The idea is to replace the piecewise constant approximation of Godunov's scheme by reconstructed states, derived from cell-averaged states obtained from the previous time-step. For each cell, slope limited ...
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 ...
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.
The order of differencing can be reversed for the time step (i.e., forward/backward followed by backward/forward). For nonlinear equations, this procedure provides the best results. For linear equations, the MacCormack scheme is equivalent to the Lax–Wendroff method .
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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.
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