We analyze the random fluctuations of several multiscale algorithms, such as the multiscale finite element method (MsFEM) and the finite element heterogeneous multiscale method (HMM), that have been developed to solve partial differential equations with highly heterogeneous coefficients. Such multiscale algorithms are often shown to correctly capture the homogenization limit when the highly oscillatory random medium is stationary and ergodic. This paper is concerned with the random fluctuations of the solution about the deterministic homogenization limit. We consider the simplified setting of the one-dimensional elliptic equation, where the theory of random fluctuations is well understood. We develop a fluctuation theory for the multiscale algorithms in the presence of random environments with short-range and long-range correlations. For a given mesh size h, we show that the fluctuations converge in distribution in the space of continuous paths to Gaussian processes as the correlation length ε→0. We next derive the limit of such Gaussian processes as h→0 and compare this limit with the distribution of the random fluctuations of the continuous model. When such limits agree, we conclude that the multiscale algorithm captures the random fluctuations accurately and passes the corrector test. This property serves as an interesting benchmark to assess the behavior of the multiscale algorithm in practical situations where the assumptions necessary for the theory of homogenization are not met. What we find is that the computationally more expensive methods MsFEM, and HMM with a choice of parameter δ=h, correctly capture the random fluctuations both for short-range and long-range oscillations in the medium. The less expensive method HMM with δ<h correctly captures the fluctuations for long-range oscillations and strongly amplifies their size in media with short-range oscillations. We present a modified scheme with an intermediate computational cost that captures the random fluctuations in all cases.

In this paper, we develop third-order conservative sign-preserving and steady-state-preserving time integrations and seek their applications in multispecies and multireaction chemical reactive flows. In this problem, the density and pressure are nonnegative, and the mass fraction for the ith species, denoted as zi, 1<=i<=M, should be between 0 and 1, where M is the total number of species. There are four main difficulties in constructing high-order bound-preserving techniques for multispecies and multireaction detonations. First of all, most of the bound-preserving techniques available are based on Euler forward time integration. Therefore, for problems with stiff source, the time step will be significantly limited. Secondly, the mass fraction does not satisfy a maximum-principle and hence it is not easy to preserve the upper bound 1. Thirdly, in most of the previous works for gaseous denotation, the algorithm relies on second-order Strang splitting methods where the ux and stiff source terms can be solved separately, and the extension to high-order time discretization seems to be complicated. Finally, most of the previous ODE solvers for stiff problems cannot preserve the total mass and the positivity of the numerical approximations at the same time. In this paper, we will construct third-order conservative sign-preserving Rugne-Kutta and multistep methods to overcome all these difficulties. The time integrations do not depend on the Strang splitting, i.e. we do not split the flux and the stiff source terms. Moreover, the time discretization can handle the stiff source with large time step and preserves the steady-state. Numerical experiments will be given to demonstrate the good performance of the bound-preserving technique and the stability of the scheme for problems with stiff source terms.

We study semi-Lagrangian discontinuous Galerkin (SLDG) and Runge-Kutta discontinuous Galerkin (RKDG) schemes for some front propagation problems in the presence of an obstacle term, modeled by a nonlinear Hamilton-Jacobi equation of the form $\min(u_t + c u_x, u - g(x))=0$, in one space dimension. New convergence results and error bounds are obtained for Lipschitz regular data. These ``low regularity" assumptions are the natural ones for the solutions of the studied equations. Numerical tests are given to illustrate the behavior of our schemes.

In this paper, we develop and analyze an arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method with a time-dependent approximation space for one dimensional conservation laws, which satisfies the geometric conservation law. For the semi-discrete ALE-DG method, when applied to nonlinear scalar conservation laws, a cell entropy inequality, L2 stability and error estimates are proven. More precisely, we prove the sub-optimal (k+1/2) convergence for monotone fluxes, and optimal (k+1) convergence for an upwind flux, when a piecewise P^k polynomial approximation space is used. For the fully-discrete ALE-DG method, the geometric conservation law and the local maximum principle are proven. Moreover, we state conditions for slope limiters, which ensure total variation stability of the method. Numerical examples show the capability of the method.

In this paper we investigate staggered discontinuous Galerkin method for the Helmholtz equation with large wave number on general quadrilateral and polygonal meshes. The method is highly flexible by allowing rough grids such as the trapezoidal grids and highly distorted grids, and at the same time, is numerical flux free. Furthermore, it allows hanging nodes, which can be simply treated as additional vertices. By exploiting a modified duality argument, the stability and convergence can be proved under the condition that \kappa h is sufficiently small, where \kappa h is the wave number and \kappa h is the mesh size. Error estimates for both the scalar and vector variables in \kappa h norm are established. Several numerical experiments are tested to verify our theoretical results and to present the capability of our method for capturing singular solutions.