We develop an efficient, adaptive locally weighted projection regression (ALWPR) framework for uncertainty quantification (UQ) of systems governed by ordinary and partial differential equations. The algorithm adaptively selects the new input points with the largest predictive variance and decides when and where to add new local models. It effectively learns the local features and accurately quantifies the uncertainty in the prediction of the statistics. The developed methodology provides predictions and confidence intervals at any query input and can deal with multi-output cases. Numerical examples are presented to show the accuracy and efficiency of the ALWPR framework including problems with non-smooth local features such as discontinuities in the stochastic space.

An existing phase-field model of two immiscible fluids with a single soluble surfactant present is discussed in detail. We analyze the well-posedness of the model and provide strong evidence that it is mathematically ill-posed for a large set of physically relevant parameters. As a consequence, critical modifications to the model are suggested that substantially increase the domain of validity. Carefully designed numerical simulations offer informative demonstrations as to the sharpness of our theoretical results and the qualities of the physical model. A fully coupled hydrodynamic test-case demonstrates the potential to also capture non-trivial effects on the overall flow.

A fully discrete discontinuous Galerkin method is introduced for solving time-dependent Maxwell's equations. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), in our scheme, discontinuous Galerkin methods are used to discretize not only the spatial domain but also the temporal domain. The proposed numerical scheme is proved to be unconditionally stable, and a convergent rate $\mathcal{O}((∆t)^{r+1}+h^{k+1/2})$ is established under the $L^2$ -norm when polynomials of degree at most $r$ and $k$ are used for temporal and spatial approximation, respectively. Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction. An ultra-convergence of order $(∆t)^{2r+1}$ in time step is observed numerically for the numerical fluxes w.r.t. temporal variable at the grid points.

In this paper, we propose a space-time rescaling scheme for computing the long time evolution of multiple precipitates in an elastically stressed medium. The algorithm is second order accurate in time, spectrally accurate in space and enables one to simulate the evolution of precipitates in a fraction of the time normally used by fixed-frame algorithms. In particular, we extend the algorithm recently developed for single particle by Li et al. (Li, Lowengrub and Leo, J. Comput. Phys., 335 (2007), 554) to the multiple particle case, which involves key differences in the method. Our results show that without elasticity there are successive tip splitting phenomena accompanied by the formation of narrow channels between the precipitates. In presence of applied elastic field, the precipitates form dendrite-like structures with the primary arms aligned in the principal directions of the elastic field. We demonstrate that when the far-field flux decreases with the effective radius of the system, tip-splitting and dendrite formation can be suppressed, as in the one particle case. Depending on the initial position of the precipitates, we further observe that some precipitates grow while others may shrink, even when a positive far field flux is applied.

This paper presents a gas kinetic study and analytical results on high speed rarefied gas flows from a planar exit. The beginning of this paper reviews the results for planar free jet expanding into a vacuum, followed by an investigation of jet impingement on normally set plates with either a diffuse or a specular surface. Presented results include exact solutions for flowfield and surface properties. Numerical simulations with the direct simulation Monte Carlo method were performed to validate these analytical results, and good agreement with this is obtained for flows at high Knudsen numbers. These highly rarefied jet and jet impingement results can provide references for real jet and jet impingement flows.

In this paper, we present the Multiscale Finite Element Method (MsFEM) for problems on rough heterogeneous surfaces. We consider the diffusion equation on oscillatory surfaces. Our objective is to represent small-scale features of the solution via multiscale basis functions described on a coarse grid. This problem arises in many applications where processes occur on surfaces or thin layers. We present a unified multiscale finite element framework that entails the use of transformations that map the reference surface to the deformed surface. The main ingredients of MsFEM are (1) the construction of multiscale basis functions and (2) a global coupling of these basis functions. For the construction of multiscale basis functions, our approach uses the transformation of the reference surface to a deformed surface. On the deformed surface, multiscale basis functions are defined where reduced (1D) problems are solved along the edges of coarse-grid blocks to calculate nodal multiscale basis functions. Furthermore, these basis functions are transformed back to the reference configuration. We discuss the use of appropriate transformation operators that improve the accuracy of the method. The method has an optimal convergence if the transformed surface is smooth and the image of the coarse partition in the reference configuration forms a quasiuniform partition. In this paper, we consider such transformations based on harmonic coordinates (following H. Owhadi and L. Zhang [Comm. Pure and Applied Math., LX(2007), pp. 675–723]) and discuss gridding issues in the reference configuration. Numerical results are presented where we compare the MsFEM when two types of deformations are used for multiscale basis construction. The first deformation employs local information and the second deformation employs a global information. Our numerical results show that one can improve the accuracy of the simulations when a global information is used.

In this paper we describe a new pseudo-spectral method to solve numerically two- and three-dimensional nonlinear diffusion equations over unbounded domains, taking Hermite functions, sinc functions, and rational Chebyshev polynomials as basis functions. The idea is to discretize the equations by means of differentiation matrices and to relate them to Sylvester-type equations by means of a fourth-order implicit-explicit scheme, being of particular interest the treatment of three-dimensional Sylvester equations that we make. The resulting method is easy to understand and express, and can be implemented in a transparent way by means of a few lines of code. We test numerically the three choices of basis functions, showing the convenience of this new approach, especially when rational Chebyshev polynomials are considered.

In this paper a second-order two-scale (SOTS) analysis method is developed for a static heat conductive problem in a periodical porous domain with radiation boundary condition on the surfaces of cavities. By using asymptotic expansion for the temperature field and a proper regularity assumption on the macroscopic scale, the cell problem, effective material coefficients, homogenization problem, first-order correctors and second-order correctors are obtained successively. The characteristics of the asymptotic model is the coupling of the cell problems with the homogenization temperature field due to the nonlinearity and nonlocality of the radiation boundary condition. The error estimation is also obtained for the original solution and the SOTS's approximation solution. Finally the corresponding finite element algorithms are developed and a simple numerical example is presented.

In this paper the pressure distribution of the gaseous flow in a microchannel is studied via a lattice Boltzmann equation (LBE) method. With effective relaxation times and a generalized second order slip boundary condition, the LBE can be used to simulate rarefied gas flows from slip to transition regimes. The Knudsen minimum phenomena of mass flow rate in the pressure driven flow is also investigated. The effects of Knudsen number (rarefaction effect), pressure ratio and aspect ratio (compression effect) on the pressure distribution are analyzed. It is found the rarefaction effect tends to the curvature of the nonlinear pressure distribution, while the compression effect tends to enhance its nonlinearity. The combined effects lead to a local minimum of the pressure deviation. Furthermore, it is also found that the relationship between the pressure deviation and the aspect ratio follows a pow-law.

Boundary integral methods are naturally suited for the computation of harmonic functions on a region having inclusions or cells with different material properties. However, accuracy deteriorates when the cell boundaries are close to each other. We present a boundary integral method in two dimensions which is specially designed to maintain second order accuracy even if boundaries are arbitrarily close. The method uses a regularization of the integral kernel which admits analytically determined corrections to maintain accuracy. For boundaries with many components we use the fast multipole method for efficient summation. We compute electric potentials on a domain with cells whose conductivity differs from that of the surrounding medium. We first solve an integral equation for a source term on the cell interfaces and then find values of the potential near the interfaces via integrals. Finally we use a Poisson solver to extend the potential to a regular grid covering the entire region. A number of examples are presented. We demonstrate that increased refinement is not needed to maintain accuracy as interfaces become very close.

The Manley-Rowe constants of motion (MRC) are conservation laws written out for a dynamical system describing the time evolution of the amplitudes in resonant triad. In this paper we extend the concept of MRC to resonance clusters of any form yielding generalized Manley-Rowe constants (gMRC) and give a constructive method how to compute them. We also give details of a $Mathematica$ implementation of this method. While MRC provide integrability of the underlying dynamical system, gMRC generally do not but may be used for qualitative and numerical study of dynamical systems describing generic resonance clusters.

Factorization of the incompressible Stokes operator linking pressure and velocity is revisited. The main purpose is to use the inverse of the Stokes operator with a large time step as a preconditioner for Newton and Arnoldi iterations applied to computation of steady three-dimensional flows and study of their stability. It is shown that the Stokes operator can be inversed within an acceptable computational effort. This inverse includes fast direct inverses of several Helmholtz operators and iterative inverse of the pressure matrix. It is shown, additionally, that fast direct solvers can be attractive for the inverse of the Helmholtz and Laplace operators on fine grids and at large Reynolds numbers, as well as for other problems where convergence of iterative methods slows down. Implementation of the Stokes operator inverse to time-stepping-based formulation of the Newton and Arnoldi iterations is discussed.

Linear/nonlinear and Stokes based-stabilizations for the filter equations for damping out primitive variable (PV) solutions corrupted by uniformly distributed random noises are numerically studied through the natural convection (NC) as well as the mixed convection (MC) environment. The most recognizable filter-scheme is based on a combination of the negative Laplace equation multiplied with the selection of the spatial scale and a linear function in order to preserve the uniqueness of the filtered solution. A more complicated filter-scheme, based on a Stokes problem which couples a filtered velocity and a filtered (artificial) pressure (or Lagrange multiplier) in order to enforce the incompressibility constraint, is also studied. Linear and Stokes based-filters via nested iterative (NI) filters and the consistent splitting scheme (CSS) are proposed for the NC/MC problems. Inspired by the total-variation (TV) model of image diffusion, well preserved feature flow patterns from the corrupted NC/MC environment are obtained by TV-Stokes based-filters together with the CSS. Our experimental results show that our proposed algorithms are effective and efficient in eliminating the unwanted spurious oscillations and preserving the accuracy of thermal convective fluid flows.