Xiangxiong Zhang, Department of Mathematics, Purdue University Monotonicity in high order accurate schemes for diffusion operators with applications to compressible Navier-Stokes equations For gas dynamics equations such as compressible Euler and Navier-Stokes equations, preserving the positivity of density and pressure without losing conservation is crucial to stabilize the numerical computation. The L1-stability of mass and energy can be achieved by enforcing the positivity of density and pressure during the time evolution. However, high order schemes do not preserve the positivity. It is difficult to enforce the positivity without destroying the high order accuracy and the local conservation in an efficient manner for time-dependent gas dynamics equations. For explicit time discretizations, we show that any high order finite volume type scheme including discontinuous Galerkin method satisfies a weak monotonicity property, which can be used to enforce positivity. This allows us to obtain the first high order positivity-preserving schemes for compressible Navier-Stokes equations. For implicit time discretizations, it is a much harder problem which is related to the fact that second order centered difference and piecewise linear finite element method on triangular meshes... https://calendar.colorado.edu/event/computational_math_seminar_-_xiangxiong_zhang

Off

Challenges and insights from the application of spectral/hp methods to problems in computation medicine

Spectral/hp element methods for discretizing engineering-based PDE problems have been lauded for combining geometric (meshing) flexibility with superior convergence behavior.  Since their inception in the 1980s, there has been growing interest in their development and usage for solving real-world (i.e. not merely academic) engineering problems.  In spite of their relative maturity, each pivot towards trying to solve a new class of problems or towards exploiting newly available hardware opens up the possibility new research into the formulation, development and/or implementation of these methods.  In this talk, we will present our work on using spectral/hp element methods to solve problems related to computational medicine.  We will focus on three challenge areas:  solution positivity, linear system preconditioning, and hardware acceleration.  We will show that moving into applications areas not only benefits the area to which one applies spectral/hp element methods, but also provides return benefits on our understanding and implementations of these methods.

Off

Off

TBA

Ben Southworth

Applied Mathematics, 

Date and time: 

Tuesday, December 1, 2015 - 11:00am

Location: 

GRVW 105

Abstract: 

TBA

Off

Multilevel Monte Carlo via a Low Rank Control Variate

Hillary Fairbanks

Applied Mathematics, 

Date and time: 

Tuesday, November 17, 2015 - 11:00am

Location: 

GRVW 105

Abstract: 

Multilevel Monte Carlo (MLMC) has been shown to be a cost effective way to compute moments of desired quantities of interest in stochastic partial differential equations, when the uncertainty in the data is high-dimensional. As compared to standard Monte Carlo, the use of a series of nested grids in MLMC allows one to improve the convergence of the mean square error by allocating more computational work onto the coarse grids and less onto the costly, fine grid solves. In this talk, we present a variation of MLMC, called Multilevel Control Variates (MLCV), which relies on a low rank approximation of fine grid solutions from the samples of the coarse grid solutions to construct control variates for the estimation of expectations involved in MLMC. In addition, we present cost estimates as well as numerical examples demonstrating the advantage of this new MLCV approach.

Off

Fluidity based FOSLS Formulation of Nonlinear Stokes Flow for Glaciers

Jeffrey Allen

Applied Mathematics, 

Date and time: 

Tuesday, November 10, 2015 - 11:00am

Location: 

GRVW 105

Abstract: 

This talk is about modeling glaciers and ice sheets using a full nonlinear Stokes method. The first part will be a quick discussion about determine the basal topography of the Kennicott glacier using high resolution satellite imagery. The second part describes a First-order System Least Squares (FOSLS) formulation of a nonlinear Stokes flow model.

In Glen's law, the most commonly used constitutive equation for ice rheology, the ice viscosity becomes infinite as the velocity gradients (strain rates) approach zero, which typically occurs near the ice surface where deformation rates are low, or when the basal slip velocities are high.  The computational difficulties associated with the infinite viscosity are often overcome by an arbitrary modification of Glen's law that bounds the maximum viscosity.  The Stress-Vorticity-Fluidity formulation exploits the fact that only the product of the viscosity and strain rate appears in the nonlinear Stokes problem, a quantity that in fact approaches zero as the strain rate goes to zero.  This formulation is expressed in terms of a new set of variables and overcomes the problem of infinite viscosity.  The new formulation is well posed and $H^1$ elliptic away from spatial locations where the velocity gradients are zero.  A Nested Iteration (NI) Newton-FOSLS approach is used to solve the nonlinear Stokes problems, in which most of the iterations are performed on the coarsest grid. This fluidity based formulation demonstrate optimal finite element convergence and involves linear systems that are more amenable to solution by AMG.

Off

Plasma, Fusion, and Particle-in-Cell Simulation

Ben Sturdevant

Applied Mathematics, 

Date and time: 

Tuesday, November 3, 2015 - 11:00am

Location: 

GRVW 105

Abstract: 

In this talk some of the basic concepts of magnetically confined fusion will be introduced. In the temperature range for which fusion can occur, the fuel exists in a state of matter known as plasma. A plasma is a low density collection of free electrical charges (ions and electrons) which exhibits a wide variety of collective behaviors through long range electromagnetic forces. Essential to the development of a viable fusion reactor is an understanding of plasma dynamics and how collective behaviors can lead to particle and heat losses preventing sustained fusion reactions. The particle-in-cell method for plasma simulation will be presented along with a model for a plasma instability known as the ion temperature gradient (ITG) instability in a toroidal plasma.

Off

TBA

Alyson Fox

Applied Mathematics, 

Date and time: 

Tuesday, October 27, 2015 - 11:00am

Location: 

GRVW 105

Abstract: 

TBA

Off

A Comparison of Finite Element Spaces for H(div) Conforming First-Order System Least Squares

Steffen Münzenmaier

Applied Mathematics, 

Date and time: 

Tuesday, October 20, 2015 - 11:00am

Location: 

GRVW 105

Abstract: 

First-order system least squares (FOSLS) is a commonly used technique in a wide range of physical applications. FOSLS discretizations are straightforward to implement and offer many advantages over traditional Galerkin or saddle point formulations. Often these problems are formulated in H(div) spaces and H(div)-conforming elements are used. These elements have lesser regularity assumptions than the commonly used H1-conforming elements and are therefore believed to be more suited for singular problems arising in many applications. After introducing the least squares finite element method and the finite element spaces, this talk will compare the approximation properties of the H(div)-conforming Raviart-Thomas and Brezzi-Douglas-Marini elements to H1-conforming piecewise polynomials in a H(div)-setting. Furthermore a H1-formulation for these problems will be derived and compared to the H(div)-formulation. For the comparison typical Poisson/Stokes problems are examined and singular solutions will be addressed by  adaptive refinement strategies.

Off

Least squares method for a nonlinear Stokes problem in glaciology

, 

Date and time: 

Tuesday, October 13, 2015 - 11:00am

Location: 

GRVW 105

Abstract: 

In this paper, we analyze the nonlinear Stokes equations which is appeared in glaciology problems with least-squares finite element method. The modified Picard iteration is used to linearize the given nonlinear problem. We first establish minimization problem which finds minimizer of residual functional in corresponding Sobolev spaces and prove theoretical results. From numerical experiments, we find an approximation of weak solution and verify the error estimates.
 

Off