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A direct numerical simulation of thermal convection between horizontal plane boundaries has been performed, at a Rayleigh number Ra=9800Ra c , where Ra c is the critical Rayleigh number for the onset of convection. The flow is found to be fully turbulent, and analysis of the probability distributions for temperature fluctuations indicates that this is within the “hard turbulence” regime, as defined by the Chicago group. Good agreement is shown to exist between their experiments and the present simulation.

We derive thermodynamically consistent models for diblock copolymer solutions coupled with the electric and magnetic field, respectively. These models satisfy the second law of thermodynamics and are therefore thermodynamically consistent. We then design a set of 2nd order, linear, semi-discrete schemes for the models using the energy quadratization method and the supplementary variable method, respectively, which preserve energy dissipation rates of the models. The spatial discretization is carried out subsequently using 2nd order finite difference methods, leading to fully discrete, energy-dissipation-rate preserving algorithms that are thermodynamically consistent. Convergence rates are numerically confirmed through mesh refinement tests and several numerical examples are given to demonstrate the role of mobility in pattern formation, defect removing effect of both electric and magnetic fields as well as the hysteresis effect with respect to applied external fields in copolymer solutions.

In this paper, we are concerned with Jacobi polynomials $$P_n^{(\alpha ,\beta )}(x)$$ on the Bernstein ellipse with motivation mainly coming from recent studies of convergence rate of spectral interpolation. An explicit representation of $$P_n^{(\alpha ,\beta )}(x)$$ is derived in the variable of parametrization. This formula further allows us to show that the maximum value of $$\left| P_n^{(\alpha ,\beta )}(z)\right| $$ over the Bernstein ellipse is attained at one of the endpoints of the major axis if $$\alpha +\beta \ge -1$$ . For the minimum value, we are able to show that for a large class of Gegenbauer polynomials (i.e., $$\alpha =\beta $$ ), it is attained at two endpoints of the minor axis. These results particularly extend those previously known only for some special cases. Moreover, we obtain a more refined asymptotic estimate for Jacobi polynomials on the Bernstein ellipse.

In this paper, a linear and decoupled Euler finite element scheme is proposed for solving the 3D incompressible Navier–Stokes equations with mass diffusion numerically by the mini element for the velocity equation and the $$P_2$$ conforming element for the density equation. When the time step size $$\tau $$ and the mesh size h both are sufficiently small, the proposed FEM algorithm is unconditionally stable at the full discrete level, which is a key issue in designing the efficient algorithm for the multi-physical field problem. Furthermore, optimal temporal-spatial error estimates are presented for the velocity in $$\mathbf{L} ^2$$ -norm and the density in $$H^1$$ -norm without any constraint of $$\tau $$ and h by using the technique of error splitting.

The Residual Smoothing Scheme (RSS) have been introduced in Averbuch et al. (A fast and accurate multiscale scheme for parabolic equations, unpublished) as a backward Euler’s method with a simplified implicit part for the solution of parabolic problems. RSS have stability properties comparable to those of semi-implicit schemes while giving possibilities for reducing the computational cost. A similar approach was introduced independently in Costa (Time marching techniques for the nonlinear Galerkin method, 1998), Costa et al. (SIAM J Sci Comput 23(1):46–65, 2001) but from the Fourier point of view. We present here a unified framework for these schemes and propose practical implementations and extensions of the RSS schemes for the long time simulation of nonlinear parabolic problems when discretized by using high order finite differences compact schemes. Stability results are presented in the linear and the nonlinear case. Numerical simulations of 2D incompressible Navier–Stokes equations are given for illustrating the robustness of the method.

Turbulence-degraded image frames are distorted by both turbulent deformations and space–time-varying blurs. To suppress these effects, we propose a multi-frame reconstruction scheme to recover a latent image from the observed distorted image sequence. Recent approaches are commonly based on registering each frame to a reference image, by which geometric turbulent deformations can be estimated and a sharp image can be restored. A major challenge is that a fine reference image is usually unavailable, as every turbulence-degraded frame is distorted. A high-quality reference image is crucial for the accurate estimation of geometric deformations and fusion of frames. Besides, it is unlikely that all frames from the image sequence are useful, and thus frame selection is necessary and highly beneficial. In this work, we propose a variational model for joint subsampling of frames and extraction of a clear image. A fine image and a suitable choice of subsample are simultaneously obtained by iteratively reducing an energy functional. The energy consists of a fidelity term measuring the discrepancy between the extracted image and the subsampled frames, as well as regularization terms on the extracted image and the subsample. Different choices of fidelity and regularization terms are explored. By carefully selecting suitable frames and extracting the image, the quality of the reconstructed image can be significantly improved. Extensive experiments have been carried out, which demonstrate the efficacy of our proposed model. In addition, the extracted subsamples and images can be put in existing algorithms to produce improved results.

We propose an alternative approach to the direct polynomial chaos expansion in order to approximate one-dimensional uncertain field exhibiting steep fronts. The principle of our non-intrusive approach is to decompose the level points of the quantity of interest in order to avoid the spurious oscillations encountered in the direct approach. This method is more accurate and less expensive than the direct approach since the regularity of the level points with respect to the input parameters allows achieving the convergence with low-order polynomial series. The additional computational cost induced in the post-processing phase is largely offset by the use of low-level sparse grids that require a weak number of direct model evaluations in comparison with high-level sparse grids. We apply the method to subsurface flows problem with uncertain hydraulic conductivity. Infiltration test cases having different levels of complexity are presented.

Computational simulation of plasma diagnostics via microwave absorption has been successfully accomplished. This simulation capability is developed from solutions to a combination of the three-dimensional Maxwell equations and the generalized Ohm’s law in the time domain. As the simulation procedure developed, numerical results were obtained for a range of plasma transport properties including electrical conductivity, permittivity, and plasma frequency. The present results reveal the wave reflection at the media interface and substantial distortion of the electromagnetic field within a thin plasma sheet from a guided microwave. The present numerical simulation also accurately predicts the microwave blackout phenomenon as the wave propagates through a thick plasma sheet. The diffractions and refractions occurring at antenna apertures and passing through a plasma column are captured numerically. Finally, the numerical simulation has successfully duplicated a plasma diagnostic experiment in a hypersonic magneto-hydrodynamic channel.

The viscosity solution of static Hamilton-Jacobi equations with a point-source condition has an upwind singularity at the source, which makes all formally high-order finite-difference scheme exhibit first-order convergence and relatively large errors. To obtain designed high-order accuracy, one needs to treat this source singularity during computation. In this paper, we apply the factorization idea to numerically compute viscosity solutions of anisotropic eikonal equations with a point-source condition. The idea is to factor the unknown traveltime function into two functions, either additively or multiplicatively. One of these two functions is specified to capture the source singularity so that the other function is differentiable in a neighborhood of the source. Then we design monotone fast sweeping schemes to solve the resulting factored anisotropic eikonal equation. Numerical examples show that the resulting monotone schemes indeed yield clean first-order convergence rather than polluted first-order convergence and both factorizations are able to treat the source singularity successfully.