A new class of high order weighted essentially
non-oscillatory (WENO) schemes [J. Comput. Phys., 318 (2016), 110-121]
is applied to solve Euler equations with steady state solutions.
It is known that the classical WENO schemes
[J. Comput. Phys., 126 (1996), 202-228] might suffer from slight
post-shock oscillations. Even though such post-shock oscillations
are small enough in magnitude and do not visually affect the
essentially non-oscillatory property, they are truly responsible
for the residue to hang at a truncation error level instead of
converging to machine zero. With the application of this new class
of WENO schemes, such slight post-shock oscillations are
essentially removed and the residue can settle down to machine
zero in steady state simulations.
This new class of WENO schemes
uses a convex combination of a quartic polynomial with two linear
polynomials on unequal size spatial stencils in one dimension and
is extended to two dimensions in a dimension-by-dimension fashion.
By doing so, such WENO schemes use the same information as the
classical WENO schemes in [J. Comput. Phys., 126 (1996), 202-228]
and yield the same formal order of accuracy in smooth regions,
yet they could converge to steady state solutions with very tiny
residue close to machine zero for our extensive list of
test problems including shocks, contact
discontinuities, rarefaction waves or their interactions,
and with these complex waves
passing through the boundaries of the computational domain.