Differential‐equation‐based representation of truncation errors for accurate numerical simulation

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Abstract
High‐order compact finite difference schemes for two‐dimensional convection‐diffusion‐type differential equations with constant and variable convection coefficients are derived. The governing equations are employed to represent leading truncation terms, including cross‐derivatives, making the overall O(h4) schemes conform to a 3 × 3 stencil. We show that the two‐dimensional constant coefficient scheme collapses to the optimal scheme for the one‐dimensional case wherein the finite difference equation yields nodally exact results. The two‐dimensional schemes are tested against standard model problems, including a Navier‐Stokes application. Results show that the two schemes are generally more accurate, on comparable grids, than O(h2) centred differencing and commonly used O(h) and O(h3) upwinding schemes.

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