In this paper, we present a constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) for parabolic equations with multiscale coefficients, arising from applications in porous media. We will present the construction of CEM-GMsFEM and rigorously analyze its convergence for the parabolic equations. The convergence rate is characterized by the coarse grid size and the eigenvalue decay of local spectral problems, but is independent of the scale length and contrast of the media. The analysis shows that the method has a first order convergence rate with respect to coarse grid size in the energy norm and second order convergence rate with respect to coarse grid size in L^2 norm under some appropriate assumptions. For the temporal discretization, finite difference techniques are used and the convergence analysis of full discrete scheme is given. Moreover, a posteriori error estimator is

In this paper, we generalize the so-called inverse Lax-Wendroff boundary treatment \cite{ilw} for the inflow boundary of a linear hyperbolic problem discretized by the recently introduced central compact schemes. The outflow boundary is treated by the classical extrapolation and a stability analysis for the resulting scheme is provided. To ensure the stability of the considered schemes provided with the chosen boundaries, the G-K-S theory is used, first in the semidiscrete case then in the fully discrete case with the third-order TVD Runge-Kutta time discretization. Afterwards, due to the high algebraic complexity of the G-K-S theory, the stability is analyzed by visualizing the eigenspectrum of the discretized operators. We show in this paper that the results obtained with these two different approaches are perfectly consistent. We also illustrate the high accuracy of the presented schemes on simple test problems.

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We use a suite of cosmological hydrodynamic simulations, run by two fixed grid codes, in the context of LCDM model to investigate the properties of solenoidal and dilatational, i.e, curl and compressive, motions of the intergalactic medium (IGM), and the impact of numerical viscosity on turbulence. The codes differ only in the spatial difference discretization, one is a second-order TVD scheme and the other is a fifth-order positivity-preserving WENO scheme. We find that (1) The vortical motion grows rapidly since $z=2$, and reaches $\sim 10 km/s -75 km/s$, depending on the gas density, at low redshifts. The small-scale compressive ratio $r_{CS}$ drops from $0.8$ to $0.5$ during the same period. (2) Power spectra of the solenoidal velocity possess two regimes, $\propto k^{-0.89}$ and $\propto k^{-2.02}$, while the total velocity and dilatational velocity follow the scaling $k^{-1.88}$ and $k^{-2.20}$ respectively all through the turbulent scale $\sim 2-0.2 h^{-1}Mpc$. The IGM turbulence may contain two different phases, the supersonic phase and the post-supersonic phase, simultaneously along the hierarchical structure formation history. (3) The non-thermal pressure support, measured by the vortical kinetic energy, is comparable with the thermal pressure for $\rho_b \sim 10$, or $T \sim 10^5 K$ at $z=0.25$. The deviation of the baryon fraction from the cosmic mean shows a positive correlation with the turbulence pressure support. (4) A relatively higher numerical viscosity would artificially dissipate both the compressive and vortical motions in the IGM more seriously, resulting in less developed density fluctuation and vorticity, and leading to remarkably shortened turbulence scale. Shocks in regions out of the clusters are outstandingly suppressed by the numerical viscosity since $z=2$, which could directly cause the different levels of turbulence in the IGM between the codes. We also discuss the impact of ingredients not included in our simulation.

Elliptic Bellman equations with coefficients independent of the variable x are considered. Error bounds for certain types of finite-difference schemes are obtained. These estimates are sharper than the earlier results in Krylov's article of 1997.