Ca. Kennedy et Mh. Carpenter, SEVERAL NEW NUMERICAL-METHODS FOR COMPRESSIBLE SHEAR-LAYER SIMULATIONS, Applied numerical mathematics, 14(4), 1994, pp. 397-433
An investigation is conducted of several numerical schemes for use in
the computation of two-dimensional, spatially evolving, laminar, varia
ble-density compressible shear layers. Schemes with various temporal a
ccuracies and arbitrary spatial accuracy for both inviscid and viscous
terms are presented and analyzed. All integration schemes make use of
explicit or compact finite-difference derivative operators. Three cla
sses of schemes are considered: an extension of MacCormack's original
second-order temporally accurate method, a new third-order temporally
accurate variant of the coupled space-time schemes proposed by Rusanov
and by Kutler et al. (RKLW), and third- and fourth-order Runge-Kutta
schemes. The RKLW scheme offers the simplicity and robustness of the M
acCormack schemes and gives the stability domain and the nonlinear thi
rd-order temporal accuracy of the Runge-Kutta method. In each of the s
chemes, stability and formal accuracy are considered for the interior
operators on the convection-diffusion equation U(t) + aU(x) = a(v)U(xx
) for which a and alpha(v) are constant. Both spatial and temporal acc
uracies are verified on the equation U(t) = [b(x)U(x)]x, as well as on
U(t) + F(x) = 0. Numerical boundary treatments of various orders of a
ccuracy are chosen and evaluated for asymptotic stability. Formally ac
curate boundary conditions are derived for explicit sixth-order, penta
diagonal sixth-order, and explicit, tridiagonal, and pentadiagonal eig
hth-order central-difference operators when used in conjunction with R
unge-Kutta integrators. Damping of high wavenumber, nonphysical inform
ation is accomplished for all schemes with the use of explicit filters
, derived up to sixth order on the boundaries and twelfth order in the
interior. Several schemes are used to compute variable-density compre
ssible shear layers, where regions of large gradients of flowfield var
iables arise near and away from the shear-layer centerline. Results in
dicate that in the present simulations, the effects of differences in
temporal and spatial accuracy between the schemes were less important
than the filtering effects. Extended MacCormack schemes were very robu
st, but were inefficient because of restrictive CFL limits. The third-
order temporally accurate RKLW schemes were less dissipative, but had
shorter run times. Runge-Kutta integrators did not possess sufficient
dissipation to be useful candidates for the computation of variable-de
nsity compressible shear layers at the levels of resolution used in th
e current work.