110 research outputs found
Summation-By-Parts Operators and High-Order Quadrature
Summation-by-parts (SBP) operators are finite-difference operators that mimic
integration by parts. This property can be useful in constructing energy-stable
discretizations of partial differential vequations. SBP operators are defined
by a weight matrix and a difference operator, with the latter designed to
approximate to a specified order of accuracy. The accuracy of the weight
matrix as a quadrature rule is not explicitly part of the SBP definition. We
show that SBP weight matrices are related to trapezoid rules with end
corrections whose accuracy matches the corresponding difference operator at
internal nodes. The accuracy of SBP quadrature extends to curvilinear domains
provided the Jacobian is approximated with the same SBP operator used for the
quadrature. This quadrature has significant implications for SBP-based
discretizations; for example, the discrete norm accurately approximates the
norm for functions, and multi-dimensional SBP discretizations
accurately mimic the divergence theorem.Comment: 18 pages, 3 figure
Review of Summation-by-parts schemes for initial-boundary-value problems
High-order finite difference methods are efficient, easy to program, scales
well in multiple dimensions and can be modified locally for various reasons
(such as shock treatment for example). The main drawback have been the
complicated and sometimes even mysterious stability treatment at boundaries and
interfaces required for a stable scheme. The research on summation-by-parts
operators and weak boundary conditions during the last 20 years have removed
this drawback and now reached a mature state. It is now possible to construct
stable and high order accurate multi-block finite difference schemes in a
systematic building-block-like manner. In this paper we will review this
development, point out the main contributions and speculate about the next
lines of research in this area
Output error estimation strategies for discontinuous Galerkin discretizations of unsteady convectionâdominated flows
We study practical strategies for estimating numerical errors in scalar outputs calculated from unsteady simulations of convectionâdominated flows, including those governed by the compressible NavierâStokes equations. The discretization is a discontinuous Galerkin finite element method in space and time on static spatial meshes. Timeâintegral quantities are considered for scalar outputs and these are shown to superconverge with temporal refinement. Output error estimates are calculated using the adjointâweighted residual method, where the unsteady adjoint solution is obtained using a discrete approach with an iterative solver. We investigate the accuracy versus computational cost tradeâoff for various approximations of the fineâspace adjoint and find that exact adjoint solutions are accurate but expensive. To reduce the cost, we propose a local temporal reconstruction that takes advantage of superconvergence properties at Radau points, and a spatial reconstruction based on nearestâneighbor elements. This inexact adjoint yields output error estimates at a computational cost of less than 2.5 times that of the forward problem for the cases tested. The calculated error estimates account for numerical error arising from both the spatial and temporal discretizations, and we present a method for identifying the percentage contributions of each discretization to the output error. Copyright © 2011 John Wiley & Sons, Ltd.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/88080/1/3224_ftp.pd
Entropy-split multidimensional summation-by-parts discretization of the Euler and compressible Navier-Stokes equations
High-order Hadamard-form entropy stable multidimensional summation-by-parts
discretizations of the Euler and compressible Navier-Stokes equations are
considerably more expensive than the standard divergence-form discretization.
In search of a more efficient entropy stable scheme, we extend the
entropy-split method for implementation on unstructured grids and investigate
its properties. The main ingredients of the scheme are Harten's entropy
functions, diagonal- summation-by-parts operators with diagonal
norm matrix, and entropy conservative simultaneous approximation terms (SATs).
We show that the scheme is high-order accurate and entropy conservative on
periodic curvilinear unstructured grids for the Euler equations. An entropy
stable matrix-type interface dissipation operator is constructed, which can be
added to the SATs to obtain an entropy stable semi-discretization.
Fully-discrete entropy conservation is achieved using a relaxation Runge-Kutta
method. Entropy stable viscous SATs, applicable to both the Hadamard-form and
entropy-split schemes, are developed for the compressible Navier-Stokes
equations. In the absence of heat fluxes, the entropy-split scheme is entropy
stable for the compressible Navier-Stokes equations. Local conservation in the
vicinity of discontinuities is enforced using an entropy stable hybrid scheme.
Several numerical problems involving both smooth and discontinuous solutions
are investigated to support the theoretical results. Computational cost
comparison studies suggest that the entropy-split scheme offers substantial
efficiency benefits relative to Hadamard-form multidimensional SBP-SAT
discretizations.Comment: 34 pages, 8 figure
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