We design a scheme for the Euler equations under gravitational fields based on our subcell hydrostatic reconstruction framework.
To give a proper definition of the nonconservative product terms due to the gravitational potential, we first separate the singularity to be an infinitely thin layer, on where the potential is smoothed by defining an intermediate potential without disturbing its monotonicity ; then the physical variables are extended and controlled to be consistent with the Rayleigh-Taylor stability, which contribute the positivity-preserving property to keep the nonnegativity of both gas density and pressure even with vacuum states. By using the hydrostatic equilibrium state variables the well-balanced property is obtained to maintain the steady state even with vacuum fronts. In addition, we proved the full discrete discrete entropy inequality, which preserve the convergence of the solution to the physical solution, with an error term which tends to zero as the mesh size approaches to zero if the potential is Lipschitz continuous. The new scheme is very natural to understand and easy to implement.
The numerical experiments demonstrate the scheme's robustness to resolve the nonlinear waves and vacuum fronts.
In this paper, a non-asymptotic pseudo-state estimator for a class of commensurate fractional order linear systems is designed
in noisy environment. Different from existing modulating functions methods, the proposed method is based on the system
model with fractional sequential derivatives by introducing fractional order modulating functions. By applying the fractional
order integration by parts formula and thanks to the properties of the fractional order modulating functions, a set of fractional
derivatives and fractional order initial values of the output are analogously obtained by algebraic integral formulas. Then,
an explicit formula of the pseudo-state is accomplished by using the fractional sequential derivatives of the output computed
based on the previous results. This formula does not contain any source of errors in continuous noise-free case, and can be used
to non-asymptotically estimate the pseudo-state in discrete noisy case. The construction of the fractional order modulating
functions is also shown, which is independent of the time. Finally, simulations and comparison results demonstrate the efficiency
and robustness of the proposed method