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                <term xml:lang="en">very high-order</term>
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              <p>A new finite volume method for one dimensional convection diffusion problems is designed. The scheme is based on the Polynomial Reconstruction Operator (PRO-scheme) where a 5-degree polynomial representation is provided for each cell using the mean-values of the neighboring cells. Convective and diffusive numerical fluxes derive from the polynomial reconstruction and lead to a finite volume scheme of sixth-order of accuracy. Numerical examples are proposed to show the effectiveness of the method and the ability to manage the convection and the diffusion independently, {\it i.e.} the scheme is still functional even with vanishing viscosity.</p>
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