90 research outputs found
An asymptotic preserving method for linear systems of balance laws based on Galerkin's method
We apply the concept of Asymptotic Preserving (AP) schemes to the linearized
p-system and discretize the resulting elliptic equation using standard
continuous Finite Elements instead of Finite Differences. The fully discrete
method is analyzed with respect to consistency, and we compare it numerically
with more traditional methods such as Implicit Euler's method. Numerical
results indicate that the AP method is indeed superior to more traditional
methods.Comment: Journal of Scientific Computing, 201
Flux Splitting for stiff equations: A notion on stability
For low Mach number flows, there is a strong recent interest in the
development and analysis of IMEX (implicit/explicit) schemes, which rely on a
splitting of the convective flux into stiff and nonstiff parts. A key
ingredient of the analysis is the so-called Asymptotic Preserving (AP)
property, which guarantees uniform consistency and stability as the Mach number
goes to zero. While many authors have focussed on asymptotic consistency, we
study asymptotic stability in this paper: does an IMEX scheme allow for a CFL
number which is independent of the Mach number? We derive a stability criterion
for a general linear hyperbolic system. In the decisive eigenvalue analysis,
the advective term, the upwind diffusion and a quadratic term stemming from the
truncation in time all interact in a subtle way. As an application, we show
that a new class of splittings based on characteristic decomposition, for which
the commutator vanishes, avoids the deterioration of the time step which has
sometimes been observed in the literature
A temporally adaptive hybridized discontinuous Galerkin method for time-dependent compressible flows
The potential of the hybridized discontinuous Galerkin (HDG) method has been
recognized for the computation of stationary flows. Extending the method to
time-dependent problems can, e.g., be done by backward difference formulae
(BDF) or diagonally implicit Runge-Kutta (DIRK) methods. In this work, we
investigate the use of embedded DIRK methods in an HDG solver, including the
use of adaptive time-step control. Numerical results demonstrate the
performance of the method for both linear and nonlinear (systems of)
time-dependent convection-diffusion equations
Adjoint-Based Error Estimation and Mesh Adaptation for Hybridized Discontinuous Galerkin Methods
We present a robust and efficient target-based mesh adaptation methodology,
building on hybridized discontinuous Galerkin schemes for (nonlinear)
convection-diffusion problems, including the compressible Euler and
Navier-Stokes equations. Hybridization of finite element discretizations has
the main advantage, that the resulting set of algebraic equations has globally
coupled degrees of freedom only on the skeleton of the computational mesh.
Consequently, solving for these degrees of freedom involves the solution of a
potentially much smaller system. This not only reduces storage requirements,
but also allows for a faster solution with iterative solvers. The mesh
adaptation is driven by an error estimate obtained via a discrete adjoint
approach. Furthermore, the computed target functional can be corrected with
this error estimate to obtain an even more accurate value. The aim of this
paper is twofold: Firstly, to show the superiority of adjoint-based mesh
adaptation over uniform and residual-based mesh refinement, and secondly to
investigate the efficiency of the global error estimate
A Comparison of Hybridized and Standard DG Methods for Target-Based hp-Adaptive Simulation of Compressible Flow
We present a comparison between hybridized and non-hybridized discontinuous
Galerkin methods in the context of target-based hp-adaptation for compressible
flow problems. The aim is to provide a critical assessment of the computational
efficiency of hybridized DG methods. Hybridization of finite element
discretizations has the main advantage, that the resulting set of algebraic
equations has globally coupled degrees of freedom only on the skeleton of the
computational mesh. Consequently, solving for these degrees of freedom involves
the solution of a potentially much smaller system. This not only reduces
storage requirements, but also allows for a faster solution with iterative
solvers. Using a discrete-adjoint approach, sensitivities with respect to
output functionals are computed to drive the adaptation. From the error
distribution given by the adjoint-based error estimator, h- or p-refinement is
chosen based on the smoothness of the solution which can be quantified by
properly-chosen smoothness indicators. Numerical results are shown for
subsonic, transonic, and supersonic flow around the NACA0012 airfoil.
hp-adaptation proves to be superior to pure h-adaptation if discontinuous or
singular flow features are involved. In all cases, a higher polynomial degree
turns out to be beneficial. We show that for polynomial degree of approximation
p=2 and higher, and for a broad range of test cases, HDG performs better than
DG in terms of runtime and memory requirements
A note on adjoint error estimation for one-dimensional stationary balance laws with shocks
We consider one-dimensional steady-state balance laws with discontinuous
solutions. Giles and Pierce realized that a shock leads to a new term in the
adjoint error representation for target functionals.This term disappears if and
only if the adjoint solution satisfies an internal boundary condition.
Curiously, most computer codes implementing adjoint error estimation ignore the
new term in the functional, as well as the internal adjoint boundary condition.
The purpose of this note is to justify this omission as follows: if one
represents the exact forward and adjoint solutions as vanishing viscosity
limits of the corresponding viscous problems, then the internal boundary
condition is naturally satisfied in the limit
Functional-preserving predictor-corrector multiderivative schemes
In this work, we develop a class of high-order multiderivative time
integration methods that is able to preserve certain functionals discretely.
Important ingredients are the recently developed
Hermite-Birkhoff-Predictor-Corrector methods and the technique of relaxation
for numerical methods of ODEs. We explain the algorithm in detail and show
numerical results for two- and three-derivative methods, comparing relaxed and
unrelaxed methods. The numerical results demonstrate that, at the slight cost
of the relaxation, an improved scheme is obtained.Comment: Submitted to the Proceedings in Applied Mathematics and Mechanic
(GAMM Annual Meeting 2023
Effects of hydrothermal aging on co and no oxidation activity over monometallic and bimetallic ptâpd catalysts
By combining scanning transmission electron microscopy, CO chemisorption, and energy dispersive X-ray spectroscopy with CO and NO oxidation light-off measurements we investigated deactivation phenomena of Pt/AlO, Pd/AlO, and Pt-Pd/AlO model diesel oxidation catalysts during stepwise hydrothermal aging. Aging induces significant particle sintering that results in a decline of the catalytic activity for all catalyst formulations. While the initial aging step caused the most pronounced deactivation and sintering due to Ostwald ripening, the deactivation rates decline during further aging and the catalyst stabilizes at a low level of activity. Most importantly, we observed pronounced morphological changes for the bimetallic catalyst sample: hydrothermal aging at 750 °C causes a stepwise transformation of the Pt-Pd alloy via core-shell structures into inhomogeneous agglomerates of palladium and platinum. Our study shines a light on the aging behavior of noble metal catalysts under industrially relevant conditions and particularly underscores the highly complex transformation of bimetallic Pt-Pd diesel oxidation catalysts during hydrothermal treatmen
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