Compact high-order schemes for the Euler equations

An implicit approximate factorization (AF) algorithm is constructed, which has the following characteristics.• In two dimensions: The scheme is unconditionally stable, has a 3×3 stencil and at steady state has a fourth-order spatial accuracy. The temporal evolution is time accurate either to first or second order through choice of parameter.• In three dimensions: The scheme has almost the same properties as in two dimensions except that it is now only conditionally stable, with the stability condition (the CFL number) being dependent on the “cell aspect ratios,”Δy/Δx andΔz/Δx. The stencil is still compact and fourth-order accuracy at steady state is maintained.Numerical experiments on a two-dimensional shock-reflection problem show the expected improvement over lower-order schemes, not only in accuracy (measured by theL2 error) but also in the dispersion. It is also shown how the same technique is immediately extendable to Runge-Kutta type schemes, resulting in improved stability in addition to the enhanced accuracy.