We consider the initial boundary value problem for the three dimensional Navier-Stokes equations with Navier-type slip boundary conditions. After having properly formulated the problem, we prove that weak solutions constructed by approximating the time-derivative by backward finite differences (with Euler schemes) are suitable. The main novelty is the proof of the local energy inequality in the case of a weak solution constructed by time discretization. Moreover, the problem is analyzed with boundary conditions which are of particular interest in view of applications to turbulent flows.

A statistical turbulence model is proposed for ensemble calculations with two fluids coupled across a flat interface, motivated by atmosphere-ocean interaction. For applications, like climate research, the response of an equilibrium climate state to variations in forcings is important to interrogate predictive capabilities of simulations. The method proposed here focuses on the computation of the ensemble mean-flow fluid velocities. In particular, a closure model is used for the Reynolds stresses that accounts for the fluid behavior at the interface. The model is shown to converge at long times to statistical equilibrium and an analogous, discrete result is shown for two numerical methods. Some matrix assembly costs are reduced with this approach. Computations are performed with monolithic (implicit) and partitioned coupling of the fluid velocities; the former being too expensive for practical computing, but providing a point of comparison to see the effect of partitioning on the ensemble statistics. It is observed that the partitioned methods reproduce the mean-flow behavior well, but may introduce some long-time statistical bias.

The third stage of Fourier analysis is considered herein. A generalized Fourier series is considered with real valued, locally integrable functions.

This paper presents two algorithms for calculating an ensemble of solutions to laminar natural convection problems. The ensemble average is the most likely temperature distribution and its variance gives an estimate of prediction reliability. Solutions are calculated by solving two coupled linear systems, each involving a shared coefficient matrix, for multiple right-hand sides at each timestep. Storage requirements and computational costs to solve the system are thereby reduced. Stability and convergence of the method are proven under a timestep condition involving fluctuations. A series of numerical tests, including predictability horizons, are provided which confirm the theoretical analyses and illustrate uses of ensemble simulations.

We consider a problem of recovering the time-dependent diffusion coefficient in a parabolic system. To ensure uniqueness the system is constrained by the integral of the solution at all times. This problem has applications in geology where the parabolic equation models the accumulation and diffusion of argon in micas. Argon is generated by the decay of potassium and the diffusion is thermally activated. We introduce a time discretization, on which we base an application of Rothe’s method to prove existence of solutions. The numerical scheme corresponding to the semi-discretization exhibits convergence that is consistent with that in Euler’s method.

This expository paper is presented in celebration of Professor Layton’s 60th birthday. In mathematics genealogy, he is the gg-grandson of Göttingen’s Ludwig Prandtl (1875-1953), who contributed enormously to both fluid and solid mechanics. The Prandtl-Meyer fan is a simple tool to visualize solutions of hyperbolic PDEs in the context of gasdynamics. In later years, Prandtl adapted the fan to visualize solutions of hyperbolic PDEs in the context of plasticity in metals. By 1960, the fan had been used by Sokolovskii in the context of granular plasticity. Because the last introduces logarithms into plasticity’s equivalent of the Riemann invariant, granular materials exhibit unexpected behaviors that are being ignored by the construction community with costly consequences. Interestingly, one of the consequences for soil is analogous to the Rankine-Hugoniot relation in gasdynamics.

We consider a fluid-structure interaction (FSI) problem that consists of a viscoelastic fluid flow and a linear elastic structure. The system is formulated as (i) a monolithic problem, where the matching conditions at the moving interface are satisfied implicitly, and (ii) a partitioned problem, where the fluid and structure subproblems are coupled by Robin-type boundary conditions along the interface. Numerical algorithms are designed based on the Arbitrary Lagrangian-Eulerian (ALE) formulation for the time-dependent fluid domain. We perform numerical experiments to compare monolithic and partitioned schemes and study the effect of stress boundary conditions on the inflow portion of moving interface.

In this paper, we study the numerical stability of reduced order models for convection-dominated stochastic systems in a relatively simple setting: a stochastic Burgers equation with linear multiplicative noise. Our preliminary results suggest that, in a convection-dominated regime, standard reduced order models yield inaccurate results in the form of spurious numerical oscillations. To alleviate these oscillations, we use the Leray reduced order model, which increases the numerical stability of the standard model by smoothing (regularizing) the convective term with an explicit spatial filter. The Leray reduced order model yields significantly better results than the standard reduced order model and is more robust with respect to changes in the strength of the noise.

Partial differential equations (PDE) often involve parameters, such as viscosity or density. An analysis of the PDE may involve considering a large range of parameter values, as occurs in uncertainty quantification, control and optimization, inference, and several statistical techniques. The solution for even a single case may be quite expensive; whereas parallel computing may be applied, this reduces the total elapsed time but not the total computational effort. In the case of flows governed by the Navier-Stokes equations, a method has been devised for computing an ensemble of solutions. Recently, a reduced-order model derived from a proper orthogonal decomposition (POD) approach was incorporated into a first-order accurate in time version of the ensemble algorithm. In this work, we expand on that work by incorporating the POD reduced order model into a second-order accurate ensemble algorithm. Stability and convergence results for this method are updated to account for the POD/ROM approach. Numerical experiments illustrate the accuracy and efficiency of the new approach.

The goal of this work is to solve nonlocal diffusion and anomalous diffusion problems by approximating the nonlocal integral appearing in the integro-differential equation by novel quadrature rules. These quadrature rules are derived so that they are exact for a nonlocal integral evaluated at translations of a given radial basis function (RBF). We first illustrate how to derive RBF-generated quadrature rules in one dimension and demonstrate their accuracy for approximating a nonlocal integral. Once the quadrature rules are derived as a preprocessing step, we apply them to approximate the nonlocal integral in a nonlocal diffusion problem and when the temporal derivative is approximated by a standard difference approximation a system of difference equations are obtained. This approach is extended to two dimensions where both a circular and rectangular nonlocal neighborhood are considered. Numerical results are provided and we compare our results to published results solving nonlocal problems using standard finite element methods.

We study and compare fully discrete numerical approximations for the Cahn-Hilliard-Navier-Stokes (CHNS) system of equations that enforce the divergence constraint in different ways, one method via penalization in a projection-type splitting scheme, and the other via strongly divergence-free elements in a fully coupled scheme. We prove a connection between these two approaches, and test the methods against standard ones with several numerical experiments. The tests reveal that CHNS system solutions can be efficiently and accurately computed with penalty-projection methods.

A mantle convection model consisting of the stationary Stokes equations and a time-dependent convection-diffusion equation for the temperature is studied. The Stokes problem is discretized with a conforming inf-sup stable pair of finite element spaces and the temperature equation is stabilized with the SUPG method. Finite element error estimates are derived which show the dependency of the error of the solution of one problem on the error of the solution of the other equation. The dependency of the error bounds on the coefficients of the problem is monitored.

In this article, we propose and develop a time relaxation implementation of the modular nonlinear filter model of [21]. A complete numerical analysis of the scheme, that includes the computability of its numerical solutions, its stability, and velocity error estimates, is given. This is followed by 2D and 3D numerical experiments that show the advantage of the proposed scheme.

The existence of solutions to the Boussinesq system driven by random exterior forcing terms both in the velocity field and the temperature is proven using a semigroup approach. We also obtain the existence and uniqueness of an invariant measure via coupling methods.

We construct a pair of conforming and inf–sup stable finite element spaces for the two–dimensional Stokes problem yielding divergence–free approximations on general convex quadrilateral partitions. The velocity and pressure spaces consist of piecewise quadratic and piecewise constant polynomials, respectively. We show that the discrete velocity and a locally post–processed pressure solution are second–order convergent.