Computing

, Volume 56, Issue 4, pp 371–383

# The exact region of stability for MacCormack scheme

Article

## Abstract

Let the two dimensional scalar advection equation be given by
$$\frac{{\partial u}}{{\partial t}} = \hat a\frac{{\partial u}}{{\partial x}} + \hat b\frac{{\partial u}}{{\partial y}}.$$
We prove that the stability region of the MacCormack scheme for this equation isexactly given by
$$\left( {\hat a\frac{{\Delta _t }}{{\Delta _x }}} \right)^{2/3} + \left( {\hat b\frac{{\Delta _t }}{{\Delta _x }}} \right)^{2/3} \leqslant 1$$
where Δ t , Δ x and Δ y are the grid distances. It is interesting to note that the stability region is identical to the one for Lax-Wendroff scheme proved by Turkel.

65M10

# Der exakte Stabilitätsbereich für das MacCormack Schema

## Zusammenfassung

Wir betrachten die zweidimensionale skalare Advektionsgleichung
$$\frac{{\partial u}}{{\partial t}} = \hat a\frac{{\partial u}}{{\partial x}} + \hat b\frac{{\partial u}}{{\partial y}}$$
und zeigen, daß der Stabilitätsbereich des MacCormack-Schemasgenau durch
$$\left( {\hat a\frac{{\Delta _t }}{{\Delta _x }}} \right)^{2/3} + \left( {\hat b\frac{{\Delta _t }}{{\Delta _x }}} \right)^{2/3} \leqslant 1$$
gegeben ist, wo Δ t , Δ x and δ y die Gitterabstände sind. Interessanterweise ist dieser Stabilitätsbereich identisch mit dem von Turkel für das Lax-Wendroff-Schema bestimmten Stabilitätsbereich.

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