Multigrid Scheme for the Euler Equations

Abstract

The steady Euler equations are numerically solved by a pseudo unsteady type method in which the energy equation is replaced by the Bernoulli relation. This so called H-system can be written:

$$ \begin{gathered} \,\frac{{\partial \rho }}{{\partial t}} + div\;\overline Q = 0 \hfill \\ \frac{{\partial \overline Q }}{{\partial t}} + div\;\overline{\overline \prod } = 0 \hfill \\ \end{gathered} $$

((1))

with $\overline{\overline \Pi } = \overline Q \otimes \overline V + p\overline{\overline 1}$ and $\overline Q = \rho \overline V$ The static pressure p can be derived from the density ρ and from the fluid velocity V̄ through the Bernoulli relation:

$$ p = \frac{{\gamma - 1}}{\gamma }\;\rho \;({H_0} - \frac{{{V^2}}}{2}). $$

For the solution of system (1) we have used a second order scheme based on a Lax-Wendroff type discretization accelerated by a multigrid method [1]. This method is based on the multigrid scheme proposed by Ni in 1981 [2].The well known local time stepping technique is also used to accelerate the convergence.