The problem of spurious oscillations in the numerical solution of the equations of gas dynamics

Abstract

One of the purposes of the present study was to show how the analysis of the equivalent system corresponding to the general class of schemes, ℒ ^∝_β is useful for determining the relative merits of these schemes ; hence, it is possible to find a scheme well-suited to a specific problem. The schemea ℒ^IV is the best among all ℒ ^∝_β for representing compression or shock waves even if it may give oscillations when the dispersion is too large. A similar conclusion (α # 2) was obtained in [8] from numerical experiments with ℒ ^2/∝₁ . On the other hand, this advantage becomes a disadvantage in a rarefaction wave ; but the oscillations are damped with time because the gradients tend to decrease in such a wave (compare fig. 2 and fig. 3). Moreover, since the couple $( \propto = 1 + \tfrac{{\sqrt 5 }}{2},\beta = \tfrac{1}{2})$ minimizes the $\mathop {Max}\limits_{\eta \in [ - 1,1]}$ E₂. with the constraint E₂⩾O , the scheme ℒ^IV is, among the ℒ ^∝_β which are dissipative in compression and shock waves, also the best one to represent rarefaction waves.