902 research outputs found
Unsplit Schemes for Hyperbolic Conservation Laws with Source Terms in One Space Dimension
The present work is concerned with the extension of the theory of characteristics to conservation laws with source terms in one space dimension, such as the
Euler equations for reacting flows. New spacetime curves are introduced on which the equations decouple to the characteristic set of O.D.E's for the corresponding
homogeneous laws, thus allowing the introduction of functions analogous to the Riemann Invariants. The geometry of these curves depends on the spatial gradients
for the solution. This particular decomposition can be used in the design of efficient unsplit algorithms for the numerical integration of the equations. As a first step,
these ideas are implemented for the case of a scalar conservation law with a nonlinear
source term. The resulting algorithm belongs to the class of MUSCL-type, shock-capturing schemes. Its accuracy and robustness are checked through a series
of tests. The aspect of the stiffness of the source term is also studied. Then, the algorithm is generalized for a system of hyperbolic equations, namely the Euler
equations for reacting flows. An extensive numerical study of unstable detonations is performed
Investigation of finite-volume methods to capture shocks and turbulence spectra in compressible flows
The aim of the present paper is to provide a comparison between several
finite-volume methods of different numerical accuracy: second-order Godunov
method with PPM interpolation and high-order finite-volume WENO method. The
results show that while on a smooth problem the high-order method perform
better than the second-order one, when the solution contains a shock all the
methods collapse to first-order accuracy. In the context of the decay of
compressible homogeneous isotropic turbulence with shocklets, the actual
overall order of accuracy of the methods reduces to second-order, despite the
use of fifth-order reconstruction schemes at cell interfaces. Most important,
results in terms of turbulent spectra are similar regardless of the numerical
methods employed, except that the PPM method fails to provide an accurate
representation in the high-frequency range of the spectra. It is found that
this specific issue comes from the slope-limiting procedure and a novel hybrid
PPM/WENO method is developed that has the ability to capture the turbulent
spectra with the accuracy of a high-order method, but at the cost of the
second-order Godunov method. Overall, it is shown that virtually the same
physical solution can be obtained much faster by refining a simulation with the
second-order method and carefully chosen numerical procedures, rather than
running a coarse high-order simulation. Our results demonstrate the importance
of evaluating the accuracy of a numerical method in terms of its actual
spectral dissipation and dispersion properties on mixed smooth/shock cases,
rather than by the theoretical formal order of convergence rate.Comment: This paper was previously composed of 2 parts, and this submission
was part 1. It is now replaced by the combined pape
Stabilizing Discontinuous Galerkin Methods Using Dafermos' Entropy Rate Criterion: II -- Systems of Conservation Laws and Entropy Inequality Predictors
A novel approach for the stabilization of the Discontinuous Galerkin method
based on the Dafermos entropy rate crition is presented. First, estimates for
the maximal possible entropy dissipation rate of a weak solution are derived.
Second, families of conservative Hilbert-Schmidt operators are identified to
dissipate entropy. Steering these operators using the bounds on the entropy
dissipation results in high-order accurate shock-capturing DG schemes for the
Euler equations, satisfying the entropy rate criterion and an entropy
inequality
Unstructured mesh algorithms for aerodynamic calculations
The use of unstructured mesh techniques for solving complex aerodynamic flows is discussed. The principle advantages of unstructured mesh strategies, as they relate to complex geometries, adaptive meshing capabilities, and parallel processing are emphasized. The various aspects required for the efficient and accurate solution of aerodynamic flows are addressed. These include mesh generation, mesh adaptivity, solution algorithms, convergence acceleration, and turbulence modeling. Computations of viscous turbulent two-dimensional flows and inviscid three-dimensional flows about complex configurations are demonstrated. Remaining obstacles and directions for future research are also outlined
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