Upwind Scheme - Third-order Upwind Scheme

Third-order Upwind Scheme

For the third-order upwind scheme, in equation (3) is defined as

\qquad \qquad u_x^- = \frac{2u_{i+1} + 3u_i - 6u_{i-1} + u_{i-2}}{6\Delta x}

and is defined as

\qquad \qquad u_x^+ = \frac{-u_{i+2} + 6u_{i+1} - 3u_i - 2u_{i-1}}{6\Delta x}

This scheme is less diffusive compared to the second-order accurate scheme. However, it is known to introduce slight dispersive errors in the region where the gradient is high.

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