We investigate the stability of numerical schemes based on a polynomial expansion method. There exist no stable polynomial schemes with higher-order accuracy in case of advection equations according to the Godunov theory. We show that a stable polynomial scheme with third-order accuracy exists in case of advection-diffusion equations. We construct a third-order polynomial scheme with positive coefficients under an allowable condition among the Courant numbers and the diffusion numbers for advection-diffusion equations. We extend the present method into two-dimensional and three-dimensional equations. Numerical experiments of initial shock propagation show good solution with the present scheme. Finally, we apply the present scheme to two-dimensional airfoil analysis.
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