## 4th Annual JohnFest / SIAM Student Conference (2012)

### Enhanced Physics Schemes for the 2D NS-alpha Model of Incompressible Flow

We study algorithms for the 2D NS-alpha model of incompressible flow that conserve both discrete energy and discrete enstrophy in the absence of viscous and external forces, and otherwise admit exact balances for them analogous to those of true fluid flow. This model belongs to a very small group that conserves both of these quantities in the continuous case (i.e. before discretizing). We develop finite element algorithms for this model that will preserve numerical energy and enstrophy in the computed solutions. For 2D flows, energy and enstrophy are believed fundamental to the organization of large structures, and thus predicting these conserved quantities correctly is paramount for many problems.

### Finite Element Solution to the Pure Stream Function Form of the Static Quasi-Geostophic Flow Equations

With the continuous increase in computational power, complex mathematical models are becoming more and more popular in the numerical simulation of oceanic and atmospheric flows. For some geophysical flows in which computational efficiency is of paramount importance, however, simplified mathematical models are central. For example, in climate modeling the quasi-geostrophic barotropic potential vorticity equations (QGE), are often used. I will present a new finite element discretization of the streamfunction formulation of the stationary quasi-geostrophic equations. Rigorous error estimates for this finite element discretization are presented and Numerical results for the Argyris finite element are used to confirm the theoretical error estimates.

### Approximation of Viscoelastic Fluid Flows with Defective Boundary Conditions

We investigate numerical algorithms for viscoelastic fluid flows with defective boundary conditions, where only flow rates or mean pressures are prescribed on parts of the boundary. The defective boundary condition problem is formulated as a minimization problem, where we seek boundary conditions of the flow equations which yield an optimal functional value. Two different approaches are considered in developing computational algorithms for the constrained optimization problem, and results of numerical experiments are presented to compare performance of the algorithms.

### Reduced Order Model Study of Burgers Equation using Proper Orthogonal Decomposition

We demonstrate a numerical study of the 1D viscous Burgers equation and several Reduced Order Models (ROMs) over a range of parameter values. This study is motivated by the need for robust reduced order models that can be used both during a design phase to allow for selection of optimal parameter values in a trade space and throughout a life cycle to identify impacts in changes of parameter values that occur during development, production and sustainment of the designs. To facilitate this study we apply the Finite Element Method (FEM) and where applicable the Group Finite Element Method (GFE) due its proven stability and reduced complexity over the standard FEM. We also utilize Proper Orthogonal Decomposition (POD) as a model reduction technique and modifications of POD that include Global POD, and the sensitivity based modifications, Extrapolated POD and Expanded POD. We then use a single baseline parameter in the parameter range to develop a ROM basis for each method above and investigate the error of each ROM method against a full order 'truth' solution for the full parameter range.

### A POD Manual for the Coupled Burgers Equation

Understanding the thermal-flow in rooms is crucial for the design of energy efficient buildings. The coupled Burgers equation is motivated by the Boussinesq equations that are often used to model the thermal-fluid dynamics of air in buildings. In many applications in dynamical systems, one wants to balance accuracy and computational efficiency of numerical solutions. Proper Orthogonal Decomposition (POD) has been used frequently and with considerable success as a tool for model reduction in the CFD community. In our work, the data to generate the POD basis is gathered from Finite Element simulations. Especially for the linear part, the POD model can be obtained by fast matrix-vector multiplications from the FE model. We present an outline of the conversion from FE to POD model and provide numerical results with regards to efficiency and accuracy of the reduced order model.

### Uncoupling Evolutionary Groundwater-Surface Water Flows Using the Crank-Nicolson Leapfrog Method

Consider an incompressible fluid in a region $\Omega_f$ flowing both ways across an interface, $I$, into a porous media domain $\Omega_p$ saturated with the same fluid. The physical processes in each domain have been well studied and are described by the Stokes equations in the fluid region and the Darcy equations in the porous media region. Taking the interfacial conditions into account produces a system with an exactly skew symmetric coupling. Spatial discretization by finite element method and time discretization by Crank-Nicolson LeapFrog gives a second order partitioned method requiring only one Stokes and one Darcy sub-physics and sub-domain solver per time step for the fully evolutionary Stokes-Darcy problem. Analysis of this method leads to a time-step condition sufficient for stability and convergence. Numerical tests verify predicted rates of convergence, however stability tests reveal the problem of growth of numerical noise in unstable modes in some cases. In such instances, the addition of time filters adds stability.

### On the Quasistatic approximation in the Stokes-Darcy Model of Groundwater-Surface Water Flows

We study the validity of the quasistatic approximation in the fully evolutionary Stokes-Darcy problem for the coupling of groundwater and surface water flows, as well as dependence of the problem upon the specific storage parameter. In the coupled equations that describe the groundwater and surface water flows for an incompressible fluid, the specific storage, $S_0$, represents the volume of water that a fully saturated porous medium will expel (or absorb) per unit volume per unit change in hydraulic head. In confined aquifers, $S_0$ takes values ranging from $10^{-6}$ to $10^{-2}$. In this work we analyze the validity of the previously studied quasistatic approximation (setting $S_0 =0$ in the Stokes-Darcy equations) by proving that the weak solution of the Stokes-Darcy problem approaches the weak solution of the quasistatic problem as $S_0 \rightarrow 0$. We also estimate the rate of convergence.

### Frechet Sensitivity Analysis for the Convection-Diffusion Equation

In this work, we consider Fréchet derivatives of solutions to the convection-diffusion equation with respect to distributed parameters and demonstrate their applicability in parameter estimation. Under certain conditions, the Fréchet derivative operator is Hilbert-Schmidt and thus there are parametric variations that have more impact on the solution than others. These variations can be used to identify locations for new measurements or data samples. We also analyze an algorithm for approximating these parametric variations.

### Explicitly uncoupled VMS stabilization of fluid flow

Variational multiscale methods have proven to be an accurate and systematic approach to the simulation of turbulent flows. Many turbulent flows are solved by legacy codes or by ones written by a team of programmers and of great complexity so implementing a new approach to turbulence in such cases can be daunting. We propose a new approach to inducing a VMS treatment of turbulence in such cases. The method adds a separate, uncoupled and modular postprocessing step to each time step. Adding this step requires the ability to solve a Stokes problem either on the same mesh or to solve and interpolate between the postprocessing step's mesh and the code's mesh. We prove stability and convergence for the combination and quantify the VMS dissipation induced. Numerical experiments confirming the theory are given. In particular, the performance of the two step, modular VMS method is comparable to a monolithic (fully coupled) VMS method for the benchmark problem of decaying homogeneous turbulence.

### A Variational Approach for the Identification of Uncertain Distributed Parameters in Elliptic Systems

The formulation of stochastic inverse problems within a least squares framework affords us the use of techniques from optimization theory, with which to prove existence and necessary optimality conditions, and from regularization theory, with which to connect the infinite dimensional problem with it's finite noise discretization. We use approximations in the space of functions with bounded mixed derivatives to estimate the random diffusion coefficient in an elliptic PDE and present related numerical results.

### A Finite Element Discretization of the Streamfunction-Vorticity Formulation of the Quasi-Geostrophic Equations

In this talk, we will discuss the finite element discretization of the Quasi-Geostrophic equations in the streamfuction-vorticity formulation, which is widely used in the geophysics community to describe the large scale motion of the ocean currents. The method is tested numerically on an ocean basin flow driven by a symmetric double-gyre wind force, which displays a four-gyre mean circulation pattern.

### A Two-Level Finite Element Discretization of the Stream Function Form of the Steady-State Quasi-Geostrophic Equations

The Quasigeostrophic Equations are commonly used in the numerical simulation of large scale wind-driven ocean flows. As the geometry of the ocean is complicated, the finite element method is appropriate for accurate depictions of boundaries. We investigate the use of two-level techniques that linearize the resulting system of equations around a solution generated on a coarser grid. This approach decreases the computational time without compromising the numerical accuracy.

### Numerical Approximations of the Voigt Regularization of Incompressible Navier-Stokes and Magnetohydrodynamics Equations

We study numerical approximations to the Voigt regularization models of the incompressible Navier-Stokes (NSE) and magnetohydrodynamics (MHD) equations. We present finite element numerical schemes, which are linearized and enforce solenodial constraints exactly, for the two models. The methods are shown to be unconditionally stable and optimally accurate. We then verify the effectiveness of the numerical schemes by testing on benchmark problems.

### Modular Nonlinear Filter Stabilization and Error Estimates of a Projection Method for Navier-Stokes Equations

In this talk, a nonlinearly filtered projection method (or fractional step method) is given in the semi-discretized form for the Navier-Stokes equations in a two or three-dimensional bounded domain. Stability analysis and error estimates for the velocity of the projection scheme are established via the energy method.

### Numerical study of a Fluid-Structure Interaction Problem

We consider a fluid-structure problem where the fluid is viscoelastic and the structure is represented by the one-dimensional string model. The coupled problem is decomposed into a fluid subproblem and a structure subproblem. In the fluid problem the Arbitrary Lagrangian Eulerian (ALE) formulation is used to develop a numerical algorithm in the setting of finite element method, while the geometric conservation law (GCL) is considered in time discretization. We will discuss a numerical algorithm for a time stepping scheme to carry out a fluid-structure simulation.

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