  This paper proposes a Hermite-kernel realization of the conjugate filter
oscillation reduction (CFOR) scheme for the simulation of fluid flows. The
Hermite kernel is constructed by using the discrete singular convolution (DSC)
algorithm, which provides a systematic generation of low-pass filter and its
conjugate high-pass filters. The high-pass filters are utilized for
approximating spatial derivatives in solving flow equations, while the
conjugate low-pass filter is activated to eliminate spurious oscillations
accumulated during the time evolution of a flow. As both low-pass and high-pass
filters are derived from the Hermite kernel, they have similar regularity,
time-frequency localization, effective frequency band and compact support.
Fourier analysis indicates that the CFOR-Hermite scheme yields a nearly optimal
resolution and has a better approximation to the ideal low-pass filter than
previously CFOR schemes. Thus, it has better potential for resolving natural
high frequency oscillations from a shock. Extensive one- and two-dimensional
numerical examples, including both incompressible and compressible flows, with
or without shocks, are employed to explore the utility, test the resolution,
and examine the stability of the present CFOR-Hermite scheme. Extremely small
ratio of point-per-wavelength (PPW) is achieved in solving the Taylor problem,
advancing a wavepacket and resolving a shock/entropy wave interaction. The
present results for the advection of an isentropic vortex compare very
favorably to those in the literature.
