Nonlinear partial differential equations modeling turbulentv fluid flow and similar processes present special challanges in numerical analysis. Regions of stability of implicit-explicit methods are reviewed, and an energy norm based on Dahlquist's concept of G-stability is developed. Using this norm, a time-stepping Crank-Nicolson Adams-Bashforth 2 implicit-explicit method for solving spatially-discretized convection-diffusion equations of this type is analyzed and shown to be unconditionally stable.

In this paper, we investigate the superconvergence properties and a posteriori error estimates of a local discontinuous Galerkin (LDG) method for solving the one-dimensional linear fourth-order initial-boundary value problems arising in study of transverse vibrations of beams. We present a local error analysis to show that the leading terms of the local spatial discretization errors for the k-degree LDG solution and its spatial derivatives are proportional to (k+1)-degree Radau polynomials. Thus, the k-degree LDG solution and its derivatives are O(h^{k+2}) superconvergent at the roots of (k+1)-degree Radau polynomials. Computational results indicate that global superconvergence holds for LDG solutions. We discuss how to apply our superconvergence results to construct efficient and asymptotically exact a posteriori error estimates in regions where solutions are smooth. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori error estimates under mesh refinement. Our results are valid for arbitrary regular meshes and for P^k polynomials with k ≥ 1, and for various types of boundary conditions.

This paper puts forth several closure models for the proper orthogonal decomposition (POD) reduced order modeling of fluid flows. These new closure models, together with other standard closure models, are investigated in the numerical simulation of the Burgers equation. This simplified setting represents just the first step in the investigation of the new closure models. It allows a thorough assessment of the performance of the new models, including a parameter sensitivity study. Two challenging test problems displaying moving shock waves are chosen in the numerical investigation. The closure models and a standard Galerkin POD reduced order model are benchmarked against the fine resolution numerical simulation. Both numerical accuracy and computational effciency are used to assess the performance of the models.

Highly imbalanced datasets occur in domains like fraud detection, fraud prediction, and clinical diagnosis of rare diseases, among others. These datasets are characterized by the existence of a prevalent class (e.g. legitimate sellers) while the other is relatively rare (e.g. fraudsters). Although small in proportion, the observations belonging to the minority class can be of a crucial importance. In this work we extend an unsupervised learning technique-Self-Organizing Maps-to use labeled data for binary classification under a constraint on the proportion of false positives. The resulting technique was applied to two highly imbalanced real datasets, achieving good results while being easier to interpret.

Developing a variational model that is capable of restoring both smooth (no edges) and non-smooth (with edges) images is still a valid challenge in the image processing. In this paper, we present two methods for image denoising problems based on the use of the LLT model (see [14]) and iterated total variation refinement. The idea of our methods is, first make use of the LLT model to get a smooth primal sketch, and then get some meaningful signal by iterated total variation refinement from the removed noise image. Numerical experiments show that our method is able to maintain some important information such as small details in the image, and at the same time to get a better visualization.

The performance of horizontal wells producing at constant wellbore pressure is a critical problem in petroleum engineering. But few articles on the well performance under constant wellbore pressure can be found in the literature due to the difficulty of hypersingular integral equations, which are needed for this problem. This article proposes and studies a new model using a hypersingular integral equation for the productivity of horizontal wells producing at constant wellbore pressure. An efficient numerical method is developed for this hypersingular integral equation based on a new expansion with respect to the singularity at arbitrary points. And numerical examples are provided to illustrate the convergence of the numerical methods. By using fluid potential superposition principle, productivity equations for a line sink model are derived from the point sink solution to the diffusivity equation. By solving the hypersingular integral equation, the authors obtain the productivity formulae of a horizontal well producing at constant wellbore pressure, which provide fast analytical tools to evaluate production performance of horizontal wells. Numerical examples are provided to illustrate the features of the model and the numerical method.

The implicit-explicit combination of Crank-Nicolson and Leap-Frog methods is widely used for atmosphere, ocean and climate simulations. Its stability under a CFL condition in the autonomous case was proven by Fourier methods in 1962 and by energy methods for autonomous systems in 2012. We provide an energy estimate showing that solution energy can grow with time in the nonautonomous case, with worst case rate proportional to time step size. We present two constructions showing that this worst case growth rate is attained for a sequence of timesteps Δt → 0. The construction exhibiting this growth for leapfrog is for a problem with a periodic coefficient.