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In this paper, we discuss some limitations of the modified equations approachas a tool for stability analysis for a class of explicit linear schemes toscalar partial derivative equations. We show that the infinite series obtainedby Fourier transform of the modified equation is not always convergent and thatin the case of divergence, it becomes unrelated to the scheme. Based on theseresults, we explain when the stability analysis of a given truncation of amodified equation may yield a reasonable estimation of a stability conditionfor the associated scheme. We illustrate our analysis by some examples ofschemes namely for the heat equation and the transport equation.

Stability theory for some scalar finite difference schemes : Validity of the modified equations approach