SCIENTIFIC PROGRAMS AND ACTIVITIES
|May 18, 2013|
For better or worse, our physical world is constantly evolving in time. Many important physical phenomena depend fundamentally on time either deterministically or through dynamical system behavior. These leads to a multitude of challenges associated with identifying, quantifying, and controlling numerical errors in complex time dependent numerical simulations. For example, in turbulent flow simulations it is well known that the control of pointwise solution errors quickly becomes intractable as the flow Reynolds number increases but the control of errors occurring in statistics and space-time averaged quantities may still be tractable .
In this presentation, we consider the representation and control of numerical solution errors in space-time FE methods using standard duality techniques as succinctly described in [2,3]. For computed output quantities of interest that are mathematically described as functionals, this technique exploits the precise relationship between functional errors and weighted combinations of computable element residuals. The associated non-perturbative theory provides the exact form of these residual weights and elucidates via Galerkin orthogonality why these weights depend not on the dual solution but rather on the difference between the dual solution and any projection of the dual solution into the primal approximation space. This latter property often leads to a non-intuitive space-time dependence of the local solution error on the accuracy of computed output functionals. Finally, we briefly mention recent work in adapting established error control techniques for ODEs to a space-time FEM setting. In this approach, the normed-error in each time slab is adaptively made smaller than some prescribed tolerance before the next time slab interval is solved. If the time slab error tolerances are suitably chosen, then error control after N time steps is achieved. The presentation of numerical results will hopefully accentuate the challenges and difficulties in space-time error representation as well as stimulate fruitful discussions addressing the feasibility of genuine error control for time dependent fluid flow problems.
 J. Hoffman and C. Johnson,"Adaptive Finite Element Methods
 K. Ericksson, D. Estep, P. Hansbo and C. Johnson,
 R. Becker and R. Rannacher,"An Optimal Control Approach
The Cartesian grid embedded boundary approach has attracted much
interest in the last decade due to the ease of grid generation for
The analysis of high order schemes based on ENO and WENO methods will be considered. It will be shown that the use of high-order provably data bounded polynomials, based on extensions of the work of [Berzins SIAM Review December 07], provides a way to develop positivity preserving (W)ENO methods of potentially very high orders. A series of numerical experiments are used to determine optimal orders for typical solution profiles. A similar idea is then used to consider variable-order positivity preserving methods for time integration, based on the variable-step variable-order backward differentiation code DASSL.
It is well-known that stationary transonic solutions of the compressible Euler equations are hard to compute using the stationary form of the equations. Therefore, time marching methods with explicit or implicit time integration are normally employed. However, the computational complexity of the time marching approach is far from optimal, because convergence tends to be slow and tends to slow down even more as resolution increases. In this talk we explore the alternative of solving the stationary equations directly, which is a viable approach when the solution topology is known in advance. We first present a solution method for one-dimensional flows with critical points. The method is based on a dynamical systems formulation of the problem and uses adaptive integration combined with a two-by-two Newton shooting method. Example calculations show that the resulting method is fast and accurate. A sample application area for this method is the calculation of transonic hydrodynamic escape flows from extrasolar planets and the early Earth, and the method is also illustrated for quasi-one-dimensional nozzle flow and black hole accretion. The method can be extended easily to handle flows with shocks, using a Newton method applied to the Rankine-Hugoniot relations. Extension to flows with heat conduction is also discussed. The presentation will conclude with some thoughts on how the approach presented can be generalized to problems in higher dimensions.
The numerical solution of continuum and non-equilibrium flows by
using fully hyperbolic and realizable mathematical descriptions
which follow from a hierarchy of moment closures is described. A
somewhat novel hierarchy of both physically realizable and strictly
hyperbolic moment closures is considered which follows from a slight
modification to the more usual maximum-entropy closure hierarchies.
Hyperbolic equations with singular sources are found in many physical
and engineering applications. The solution of such
The Baer-Nunziato two phase flow model describes flame propagation
in gas-permeable reactive granular materials. We focus on the hydrodynamic
part of the system and neglect the terms due to combustion processes,
drag and heat transfer. The system is an averaged flow model, expressing
conservation of mass, and momentum and energy balance of the gas
and solid phases, plus an additional 'compaction' equation describing
the evolution of porosity. The system is only conditionally hyperbolic,
and may fail to have a complete set of eigenvectors. It is also
in non-conservation form due to momentum and energy exchange between
the phases. The presence of non-conservative terms has major consequences
both theoretically and computationally. If the porosities are piecewise
constant, the phases
We have formulated the Baer-Nunziato model in terms of the Riemann Invariants across the compaction wave, and propose a hybrid algorithm that uses the Riemann Invariants formulation across the compaction wave, and the conservative formulation away from the compaction wave. The talk will describe the hybrid scheme and present numerical results.
This is joint work with Gerardo Hernandez of the University of Michigan.
Abstract: Shocks (nonlinear discontinuities) separating a region of hyperbolic states from a non-hyperbolic region can occur in one of two ways in conservation laws. Steady transonic flow is a well-known phenomenon, and transonic shocks appear to be both physically and mathematically stable. Unsteady non-hyperbolic systems, on the other hand, would seem to form ill-posed, even catastrophically ill-posed problems. In this talk, I will review some work that Karen Ames and I did that suggested a sense in which shocks in these ill-posed systems may enjoy a type of stability. Later work with Milton Lopes has confirmed our initial, linear results. These results can be compared to more recent work with a number of co-authors on construction of transonic shocks.
Adaptive schemes provide computational advantages over uniform
mesh computations by allocating resources in regions where they
are most needed. The computational mesh can be refined ($h$-refinement)
or the order of the scheme can be modified ($p$-refinement) to better
resolve the solution. As a result, mesh sizes and local CFL numbers
can vary greatly throughout the computational domain. This can be
inefficient as a few small cells impose a restrictive time step
on the whole mesh. We will present an adaptive high-order time integration
scheme for solving partial differential equations with the method
of lines and explicit Runge-Kutta integrators. A time step on each
cell will be defined by the local CFL condition. That is, adaptivity
in time will consist of taking a time step $dt$ on cells of size
$h$, two $dt/2$ steps on cells of size $h/2$,and so on. The total
number of function evaluations will be, therefore, reduced. The
interface conditions will be imposed on interelemental boundaries
so that the order of the underlying Runge-Kutta scheme is preserved.
The novelty of this algorithm is its small stencil. This makes it
suitable for unstructured grids, where multi-layer reconstructions
are difficult. We will show how this
To model the dynamics of materials far from thermodynamic equilibrium, a kinetic model is often required, like the Boltzmann equation for dilute gases. Deterministic schemes for the solution of the Boltzmann equation are typically efficient and accurate but quite limited with respect to the collision operator that can be modeled. On the other hand, stochastic schemes, despite their slow convergence, are much simpler to implement and adapt for even the most complex collision kernels involving multiple complex species with long-range interactions. The key issue for stochastic schemes is to reduce the variance of the observed statistics while keeping the number of degrees of freedom under control.
Joint work with Kondo Assi
The talk is devoted to the reservoir technique coupled with finite
volume flux schemes for solving with low numerical diffusion nonlinear
hyperbolic conservation laws. I will present some convergence results,
simulations in 1d and 2d, and some results regarding the overall
algorithmic complexity of the method"
High-order finite-volume methods for unstructured mesh CFD are reasonably mature as research tools, with demonstrable accuracy and efficiency benefits compared with second-order methods. This maturity comes as a result of recent advances in limiting, convergence acceleration, and adaptive techniques as applied to high-order methods, which will be the focus of this talk.
Many applications require the solution of time-dependent partial differential equations (PDEs) on surfaces or more general manifolds. Methods for treating such problems include surface parameterization, methods on triangulated surfaces and embedding techniques. This talk describes an embedding approach based on the closest point representation of the surface and describes some of its advantages over other embedding methods. Noteworthy features of the method are its generality with respect to the underlying surface and its simplicity. In particular, the method requires only minimal changes to the corresponding three-dimensional codes to treat the evolution of PDEs on surfaces.
In this talk, we study the convergence and time evolution of the
error between the discontinuous Galerkin (DG) finite element solution
I will begin by describing the manner in which numerical methods designed for hyperbolic systems improve the simulations of internal waves in an incompressible, density stratified fluid, especially in the high Reynolds number limit. I will pay particular attention to situations in which density overturns occur for which limiting has been termed "implicit Large Eddy Simulation (LES)". The relationship between methods can be turned around, and results for fully nonlinear internal waves can be used to improve the numerical simulation of classical, weakly nonlinear corrections to hyperbolic waves, such as the Korteweg de Vries equation. Finally I will consider porous media acoustics for which the corrections to hyperbolic systems are non-conservative and consist of absorbing terms. I will survey some of the techniques available for the numerical solution of such systems and demonstrate that classical, or Biot, frequency dependent absorption is non-causal.
Numerical solutions of weak shock reflection problems for the unsteady transonic small disturbance equations, the nonlinear wave system, and the compressible Euler equations at a set of parameter values for which regular reflection is impossible contain a complex structure. Instead of a mathematically inadmissible Mach reflection, as is apparently observed in experiments, the solutions contain a cascade of triple points and tiny supersonic patches behind a leading triple point. A centered expansion wave originates at each triple point. We call this structure of repeating supersonic patches and triple points ``Guderley Mach reflection,'' or GMR.
At the upstream side of each patch in GMR, a sonic line separates the patch from a region of subsonic flow. This sonic line can be considered a free boundary in the formulation of a free boundary problem, with the states on either side coupled through the boundary. As a step towards the goal of formulating this free boundary problem, we present a problem which retains its main features, but which is simpler. We choose the simplest model of weak shock reflection, the unsteady transonic small disturbance equations (UTSDE). We choose initial data which results in an expansion wave that reflects off a sonic line, similar to the reflection of a centered rarefaction off the sonic line in a single supersonic patch in GMR. In both GMR and our simpler problem for the UTSDE, the states on either side of the free boundary are coupled through the boundary, and the solutions in both the supersonic and subsonic regions are a priori unknown. These features make these problems analogous to two-phase Stefan problems for the heat equation. At the moment, we have linearized the simplified problem and solved it exactly. We have not yet formulated the free boundary problem, but we have solved the full nonlinear simplified problem numerically. We will present and describe our solutions and our solution method.
A range of important features relating to the practical application
of discontinuous Galerkin (DG) method for wave propagation will
Given the suitability of DG for solving Maxwell's equations and
their ability to propagate waves over long distance, it is natural
There is an Achilles heel to high order discontinuous Galerkin
methods when applied to conservation laws. The methods are typically
Finally, I will discuss the use of GPU hardware to accelerate computation for time-domain electromagnetics simulations.