2,364 research outputs found
A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws
In this article we consider one-dimensional random systems of hyperbolic
conservation laws. We first establish existence and uniqueness of random
entropy admissible solutions for initial value problems of conservation laws
which involve random initial data and random flux functions. Based on these
results we present an a posteriori error analysis for a numerical approximation
of the random entropy admissible solution. For the stochastic discretization,
we consider a non-intrusive approach, the Stochastic Collocation method. The
spatio-temporal discretization relies on the Runge--Kutta Discontinuous
Galerkin method. We derive the a posteriori estimator using continuous
reconstructions of the discrete solution. Combined with the relative entropy
stability framework this yields computable error bounds for the entire
space-stochastic discretization error. The estimator admits a splitting into a
stochastic and a deterministic (space-time) part, allowing for a novel
residual-based space-stochastic adaptive mesh refinement algorithm. We conclude
with various numerical examples investigating the scaling properties of the
residuals and illustrating the efficiency of the proposed adaptive algorithm
Adaptive-quadrature fluctuation-splitting schemes for the Euler equations
In this paper, we present fluctuation-splitting schemes that can capture an isolated shock over a suitably oriented single triangular element and also recognize a rarefaction. A particular focus is on the evaluation of the fluctuation (or the cell residual): a one-parameter-family quadrature rule is employed to evaluate the fluctuation, which endows the fluctuation with a wave recognition capability. The parameter value is chosen based on the nature of the nonlinear wave passing through the element, and then the resulting fluctuation is distributed to the nodes. This strategy, combined with various distribution schemes, defines a family of adaptive-quadrature fluctuation-splitting schemes. The results demonstrate the superior ability of the new schemes in handling nonlinear waves compared with standard fluctuation-splitting schemes that cannot capture shocks over a single element and also admits nonphysical shocks unless some kind of entropy fix is incorporated. Copyright © 2007 John Wiley & Sons, Ltd.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/58563/1/1609_ftp.pd
Comparison of data-driven uncertainty quantification methods for a carbon dioxide storage benchmark scenario
A variety of methods is available to quantify uncertainties arising with\-in
the modeling of flow and transport in carbon dioxide storage, but there is a
lack of thorough comparisons. Usually, raw data from such storage sites can
hardly be described by theoretical statistical distributions since only very
limited data is available. Hence, exact information on distribution shapes for
all uncertain parameters is very rare in realistic applications. We discuss and
compare four different methods tested for data-driven uncertainty
quantification based on a benchmark scenario of carbon dioxide storage. In the
benchmark, for which we provide data and code, carbon dioxide is injected into
a saline aquifer modeled by the nonlinear capillarity-free fractional flow
formulation for two incompressible fluid phases, namely carbon dioxide and
brine. To cover different aspects of uncertainty quantification, we incorporate
various sources of uncertainty such as uncertainty of boundary conditions, of
conceptual model definitions and of material properties. We consider recent
versions of the following non-intrusive and intrusive uncertainty
quantification methods: arbitary polynomial chaos, spatially adaptive sparse
grids, kernel-based greedy interpolation and hybrid stochastic Galerkin. The
performance of each approach is demonstrated assessing expectation value and
standard deviation of the carbon dioxide saturation against a reference
statistic based on Monte Carlo sampling. We compare the convergence of all
methods reporting on accuracy with respect to the number of model runs and
resolution. Finally we offer suggestions about the methods' advantages and
disadvantages that can guide the modeler for uncertainty quantification in
carbon dioxide storage and beyond
Differential formulation of discontinuous Galerkin and related methods for compressible Euler and Navier-Stokes equations
A new approach to high-order accuracy for the numerical solution of conservation laws introduced by Huynh and extended to simplexes by the current work is renamed CPR (correction procedure or collocation penalty via reconstruction). The CPR approach employs the differential form of the equation and accounts for the jumps in flux values at the cell boundaries by a correction procedure. In addition to being simple and economical, it unifies several existing methods including discontinuous Galerkin (DG), staggered grid, spectral volume (SV), and spectral difference (SD).
The approach is then extended to diffusion equation and Navier-Stokes equations. In the discretization of the diffusion terms, the BR2 (Bassi and Rebay), interior penalty, compact DG (CDG), and I-continuous approaches are used. The first three of these approaches, originally derived using the integral formulation, were recast here in the CPR framework, whereas the I-continuous scheme, originally derived for a quadrilateral mesh, was extended to a triangular mesh.
The current work also includes a study of high-order curve boundaries representations. A new boundary representation based on the Bezier curve is then developed and analyzed, which is shown to have several advantages for complicated geometries.
To further enhance the efficiency, the capability of h/p mesh adaptation is developed for the CPR solver. The adaptation is driven by an efficient multi-p a posteriori error estimator. P-adaptation is applied to smooth regions of the flow field while h-adaptation targets the non-smooth regions, identified by accuracy-preserving TVD marker. Several numerical tests are presented to demonstrate the capability of the technique
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