9,249 research outputs found
Stability of Correction Procedure via Reconstruction With Summation-by-Parts Operators for Burgers' Equation Using a Polynomial Chaos Approach
In this paper, we consider Burgers' equation with uncertain boundary and
initial conditions. The polynomial chaos (PC) approach yields a hyperbolic
system of deterministic equations, which can be solved by several numerical
methods. Here, we apply the correction procedure via reconstruction (CPR) using
summation-by-parts operators. We focus especially on stability, which is proven
for CPR methods and the systems arising from the PC approach. Due to the usage
of split-forms, the major challenge is to construct entropy stable numerical
fluxes. For the first time, such numerical fluxes are constructed for all
systems resulting from the PC approach for Burgers' equation. In numerical
tests, we verify our results and show also the advantage of the given ansatz
using CPR methods. Moreover, one of the simulations, i.e. Burgers' equation
equipped with an initial shock, demonstrates quite fascinating observations.
The behaviour of the numerical solutions from several methods (finite volume,
finite difference, CPR) differ significantly from each other. Through careful
investigations, we conclude that the reason for this is the high sensitivity of
the system to varying dissipation. Furthermore, it should be stressed that the
system is not strictly hyperbolic with genuinely nonlinear or linearly
degenerate fields
Entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations
Non-linear entropy stability and a summation-by-parts framework are used to
derive entropy stable wall boundary conditions for the three-dimensional
compressible Navier--Stokes equations. A semi-discrete entropy estimate for the
entire domain is achieved when the new boundary conditions are coupled with an
entropy stable discrete interior operator. The data at the boundary are weakly
imposed using a penalty flux approach and a simultaneous-approximation-term
penalty technique. Although discontinuous spectral collocation operators on
unstructured grids are used herein for the purpose of demonstrating their
robustness and efficacy, the new boundary conditions are compatible with any
diagonal norm summation-by-parts spatial operator, including finite element,
finite difference, finite volume, discontinuous Galerkin, and flux
reconstruction/correction procedure via reconstruction schemes. The proposed
boundary treatment is tested for three-dimensional subsonic and supersonic
flows. The numerical computations corroborate the non-linear stability (entropy
stability) and accuracy of the boundary conditions.Comment: 43 page
SBP operators for CPR methods: Master's thesis
Summation-by-parts (SBP) operators have been used in the finite difference framework, providing means to prove conservation and discrete stability by the energy method, predominantly for linear (or linearised) equations. Recently, there have been some approaches to generalise the notion of SBP operators and to apply these ideas to other methods. The correction procedure via reconstruction (CPR), also known as flux reconstruction (FR) or lifting collocation penalty (LCP), is a unifying framework of high order methods for conservation laws, recovering some discontinuous Galerkin, spectral difference and spectral volume methods. Using a reformulation of CPR methods relying on SBP operators and simultaneous approximation terms (SATs), conservation and stability are investigated, recovering the linearly stable CPR schemes of Vincent et al. (2011, 2015). Extensions of SBP methods with diagonal-norm operators to Burgers’ equation are possible by a skew-symmetric form and the introduction of additional correction terms. An analytical setting allowing a generalised notion of SBP methods including modal bases is described and applied to Burgers’ equation, resulting in an extension of the previously mentioned skew-symmetric form. Finally, an extension of the results to multiple space dimensions is presented
Explicit reconstruction of the entanglement wedge
The problem of how the boundary encodes the bulk in AdS/CFT is still a
subject of study today. One of the major issues that needs more elucidation is
the problem of subregion duality; what information of the bulk a given boundary
subregion encodes. Although the proof given by Dong, Harlow, and Wall states
that the entanglement wedge of the bulk should be encoded in boundary
subregions, no explicit procedure for reconstructing the entanglement wedge was
given so far. In this paper, mode sum approach to obtaining smearing functions
for a single bulk scalar is generalised to include bulk reconstruction in the
entanglement wedge of boundary subregions. It is generally expectated that
solutions to the wave equation on a complicated coordinate patch are needed,
but this hard problem has been transferred to a less hard but tractable problem
of matrix inversion.Comment: version accepted by JHEP; added references and discussions on
covarianc
- …