Upwind Scheme - Second-order Upwind Scheme

Second-order Upwind Scheme

The spatial accuracy of the first-order upwind scheme can be improved by including 3 data points instead of just 2, which offers a more accurate finite difference stencil for the approximation of spatial derivative. For the second-order upwind scheme, becomes the 3-point backward difference in equation (3) and is defined as

\qquad \qquad u_x^- = \frac{3u_i^n - 4u_{i-1}^n + u_{i-2}^n}{2\Delta x}

and is the 3-point forward difference, defined as

\qquad \qquad u_x^+ = \frac{-u_{i+2}^n + 4u_{i+1}^n - 3u_i^n}{2\Delta x}

This scheme is less diffusive compared to the first-order accurate scheme and is called linear upwind differencing (LUD) scheme.

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