On stable and explicit numerical methods for the advection–diffusion equation

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In this paper two stable and explicit numerical methods to integrate the one-dimensional (1D) advection–diffusion equation are presented. These schemes are stable by design and follow the main general concept behind the semi-Lagrangian method by constructing a virtual grid where the explicit method becomes stable. It is shown that the new schemes compare well with analytic solutions and are often more accurate than implicit schemes. In particular, the diffusion-only case is explored in some detail. The error produced by the stable and explicit method is a function of the ratio between the standard deviation σ 0 of the initial Gaussian state and the characteristic virtual grid distance Δ S. Larger values of this ratio lead to very accurate results when compared to implicit methods, while lower values lead to less accuracy. It is shown that the σ 0/Δ S ratio is also significant in the advection–diffusion problem: it determines the maximum error generated by new methods, obtained with a certain combination of the advection and diffusion values. In addition, the error becomes smaller when the problem becomes more advective or more diffusive.