4,346 research outputs found
Hybrid Spectral Difference/Embedded Finite Volume Method for Conservation Laws
A novel hybrid spectral difference/embedded finite volume method is
introduced in order to apply a discontinuous high-order method for large scale
engineering applications involving discontinuities in the flows with complex
geometries. In the proposed hybrid approach, the finite volume (FV) element,
consisting of structured FV subcells, is embedded in the base hexahedral
element containing discontinuity, and an FV based high-order shock-capturing
scheme is employed to overcome the Gibbs phenomena. Thus, a discontinuity is
captured at the resolution of FV subcells within an embedded FV element. In the
smooth flow region, the SD element is used in the base hexahedral element.
Then, the governing equations are solved by the SD method. The SD method is
chosen for its low numerical dissipation and computational efficiency
preserving high-order accurate solutions. The coupling between the SD element
and the FV element is achieved by the globally conserved mortar method. In this
paper, the 5th-order WENO scheme with the characteristic decomposition is
employed as the shock-capturing scheme in the embedded FV element, and the
5th-order SD method is used in the smooth flow field.
The order of accuracy study and various 1D and 2D test cases are carried out,
which involve the discontinuities and vortex flows. Overall, it is shown that
the proposed hybrid method results in comparable or better simulation results
compared with the standalone WENO scheme when the same number of solution DOF
is considered in both SD and FV elements.Comment: 27 pages, 17 figures, 2 tables, Accepted for publication in the
Journal of Computational Physics, April 201
Non-Linear Shallow Water Equations numerical integration on curvilinear boundary-conforming grids
An Upwind Weighted Essentially Non-Oscillatory scheme for the solution of the Shallow Water Equations on generalized curvilinear coordinate systems is proposed. The Shallow Water Equations are expressed in a contravariant formulation in which Christoffel symbols are avoided. The equations are solved by using a high-resolution finite-volume method incorporated with an exact Riemann Solver. A procedure developed in order to correct errors related to the difficulties of numerically satisfying the metric identities on generalized boundary-conforming grids is presented; this procedure allows the numerical scheme to satisfy the freestream preservation property on highly-distorted grids. The capacity of the proposed model is verified against test cases present in literature. The results obtained are compared with analytical solutions and alternative numerical solutions
High-order conservative reconstruction schemes for finite volume methods in cylindrical and spherical coordinates
High-order reconstruction schemes for the solution of hyperbolic conservation
laws in orthogonal curvilinear coordinates are revised in the finite volume
approach. The formulation employs a piecewise polynomial approximation to the
zone-average values to reconstruct left and right interface states from within
a computational zone to arbitrary order of accuracy by inverting a
Vandermonde-like linear system of equations with spatially varying
coefficients. The approach is general and can be used on uniform and
non-uniform meshes although explicit expressions are derived for polynomials
from second to fifth degree in cylindrical and spherical geometries with
uniform grid spacing. It is shown that, in regions of large curvature, the
resulting expressions differ considerably from their Cartesian counterparts and
that the lack of such corrections can severely degrade the accuracy of the
solution close to the coordinate origin. Limiting techniques and monotonicity
constraints are revised for conventional reconstruction schemes, namely, the
piecewise linear method (PLM), third-order weighted essentially non-oscillatory
(WENO) scheme and the piecewise parabolic method (PPM).
The performance of the improved reconstruction schemes is investigated in a
number of selected numerical benchmarks involving the solution of both scalar
and systems of nonlinear equations (such as the equations of gas dynamics and
magnetohydrodynamics) in cylindrical and spherical geometries in one and two
dimensions. Results confirm that the proposed approach yields considerably
smaller errors, higher convergence rates and it avoid spurious numerical
effects at a symmetry axis.Comment: 37 pages, 12 Figures. Accepted for publication in Journal of
Compuational Physic
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
Multi-Dimensional Astrophysical Structural and Dynamical Analysis I. Development of a Nonlinear Finite Element Approach
A new field of numerical astrophysics is introduced which addresses the
solution of large, multidimensional structural or slowly-evolving problems
(rotating stars, interacting binaries, thick advective accretion disks, four
dimensional spacetimes, etc.). The technique employed is the Finite Element
Method (FEM), commonly used to solve engineering structural problems. The
approach developed herein has the following key features:
1. The computational mesh can extend into the time dimension, as well as
space, perhaps only a few cells, or throughout spacetime.
2. Virtually all equations describing the astrophysics of continuous media,
including the field equations, can be written in a compact form similar to that
routinely solved by most engineering finite element codes.
3. The transformations that occur naturally in the four-dimensional FEM
possess both coordinate and boost features, such that
(a) although the computational mesh may have a complex, non-analytic,
curvilinear structure, the physical equations still can be written in a simple
coordinate system independent of the mesh geometry.
(b) if the mesh has a complex flow velocity with respect to coordinate space,
the transformations will form the proper arbitrary Lagrangian- Eulerian
advective derivatives automatically.
4. The complex difference equations on the arbitrary curvilinear grid are
generated automatically from encoded differential equations.
This first paper concentrates on developing a robust and widely-applicable
set of techniques using the nonlinear FEM and presents some examples.Comment: 28 pages, 9 figures; added integral boundary conditions, allowing
very rapidly-rotating stars; accepted for publication in Ap.
Implicit High-Order Flux Reconstruction Solver for High-Speed Compressible Flows
The present paper addresses the development and implementation of the first
high-order Flux Reconstruction (FR) solver for high-speed flows within the
open-source COOLFluiD (Computational Object-Oriented Libraries for Fluid
Dynamics) platform. The resulting solver is fully implicit and able to simulate
compressible flow problems governed by either the Euler or the Navier-Stokes
equations in two and three dimensions. Furthermore, it can run in parallel on
multiple CPU-cores and is designed to handle unstructured grids consisting of
both straight and curved edged quadrilateral or hexahedral elements. While most
of the implementation relies on state-of-the-art FR algorithms, an improved and
more case-independent shock capturing scheme has been developed in order to
tackle the first viscous hypersonic simulations using the FR method. Extensive
verification of the FR solver has been performed through the use of
reproducible benchmark test cases with flow speeds ranging from subsonic to
hypersonic, up to Mach 17.6. The obtained results have been favorably compared
to those available in literature. Furthermore, so-called super-accuracy is
retrieved for certain cases when solving the Euler equations. The strengths of
the FR solver in terms of computational accuracy per degree of freedom are also
illustrated. Finally, the influence of the characterizing parameters of the FR
method as well as the the influence of the novel shock capturing scheme on the
accuracy of the developed solver is discussed
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