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1.
The convergence to steady state solutions of the Euler equations for high order weighted essentially non-oscillatory (WENO)
finite difference schemes with the Lax-Friedrichs flux splitting (Jiang and Shu, in J. Comput. Phys. 126:202–228, 1996) is investigated. Numerical evidence in Zhang and Shu (J. Sci. Comput. 31:273–305, 2007) indicates that there exist slight post-shock oscillations when we use high order WENO schemes to solve problems containing
shock waves. Even though these oscillations are small in their magnitude and do not affect the “essentially non-oscillatory”
property of the WENO schemes, they are indeed responsible for the numerical residue to hang at the truncation error level
of the scheme instead of settling down to machine zero. Differently from the strategy adopted in Zhang and Shu (J. Sci. Comput.
31:273–305, 2007), in which a new smoothness indicator was introduced to facilitate convergence to steady states, in this paper we study the
effect of the local characteristic decomposition on steady state convergence. Numerical tests indicate that the slight post-shock
oscillation has a close relationship with the local characteristic decomposition process. When this process is based on an
average Jacobian at the cell interface using the Roe average, as is the standard procedure for WENO schemes, such post-shock
oscillation appears. If we instead use upwind-biased interpolation to approximate the physical variables including the velocity
and enthalpy on the cell interface to compute the left and right eigenvectors of the Jacobian for the local characteristic
decomposition, the slight post-shock oscillation can be removed or reduced significantly and the numerical residue settles
down to lower values than other WENO schemes and can reach machine zero for many test cases. This new procedure is also effective
for higher order WENO schemes and for WENO schemes with different smoothness indicators. 相似文献
2.
This paper is devoted to the construction and analysis of a new smoothness index for WENO interpolation capable of dealing with corner discontinuities. The new smoothness index presented is initially developed for the point-value framework of Harten’s multiresolution. Even so, the ideas about how to extend the results to the cell-average framework are presented. The new smoothness index is inspired by the one proposed in Jiang and Shu (J Comput Phys 126(1):202–228, 1996). This index works very well for jump discontinuities as it was originally designed for the context of conservation laws in order to deal with problems that contain shocks and complicated fluid-structure interactions. Even so, it is easy to check that the mentioned index does not provide an appropriate performance for corner discontinuities. Our aim is to rise the order of accuracy of WENO interpolation near corner discontinuities. In order to do so, we will modify the original smoothness index proposed by Jiang and Shu such that the discontinuities in the first derivative of the function contribute effectively to the index. The modification proposed will produce a variation in the weights of WENO when dealing with a corner, that do not appear when using the smoothness indexes proposed by Jiang and Shu. The variation in the weights induced by the modification of the smoothness index will allow adaption to corner discontinuities, maintaining the adaption to jumps provided by the original smoothness index proposed by Jiang and Shu. The strategy proposed in Aràndiga et al. (SIAM J Numer Anal 49(2):893–915, 2011) can be adapted such that the accuracy is maintained near critical points at smooth zones. 相似文献
3.
Weighted essentially non-oscillatory (WENO) finite difference schemes, developed by Liu et al. (Comput Phys 115(1):200–212, 1994) and improved by Jiang and Shu (Comput Phys 126(1):202–228, 1996), are one of the most popular methods to approximate the solutions of hyperbolic equations. But these schemes fail to provide maximal order accuracy near smooth extrema, where the first derivative of the solution becomes zero. Some authors have addressed this problem with different weight designs. In this paper we focus on the weights proposed by Yamaleev and Carpenter (J Comput Phys 228:4248–4272, 2009). They propose new weights to provide faster weight convergence than those presented in Borges et al. (J Comput Phys 227:3191–3211, 2008) and deduce some constraints on the weights parameters to guarantee that the WENO scheme has maximal order for sufficiently smooth solutions with an arbitrary number of vanishing derivatives. We analyze the scheme with the weights proposed in Yamaleev and Carpenter (J Comput Phys 228:4248–4272, 2009) and prove that near discontinuities it achieves worse orders than classical WENO schemes. In order to solve these accuracy problems, we define new weights, based on those proposed in Yamaleev and Carpenter (J Comput Phys 228:4248–4272, 2009), and get some constraints on the weights parameters to guarantee maximal order accuracy for the resulting schemes. 相似文献
4.
In [J. Comput. Phys. 193:115–135, 2004] and [Comput. Fluids 34:642–663, 2005], Qiu and Shu developed a class of high order weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials,
termed HWENO (Hermite WENO) schemes, for solving nonlinear hyperbolic conservation law systems, and applied them as limiters
for the Runge-Kutta discontinuous Galerkin (RKDG) methods on structured meshes. In this continuation paper, we extend the
method to solve two dimensional problems on unstructured meshes. The emphasis is again on the application of such HWENO finite
volume methodology as limiters for RKDG methods to maintain compactness of RKDG methods. Numerical experiments for two dimensional
Burgers’ equation and Euler equations of compressible gas dynamics are presented to show the effectiveness of these methods.
The research was partially supported by the European project ADIGMA on the development of innovative solution algorithms for
aerodynamic simulations, NSFC grant 10671091 and JSNSF BK2006511. 相似文献
5.
A New Smoothness Indicator for the WENO Schemes and Its Effect on the Convergence to Steady State Solutions 总被引:1,自引:0,他引:1
The convergence to steady state solutions of the Euler equations for the fifth-order weighted essentially non-oscillatory
(WENO) finite difference scheme with the Lax–Friedrichs flux splitting [7, (1996) J. Comput. Phys. 126, 202–228.] is studied through systematic numerical tests. Numerical evidence indicates that this type of WENO scheme suffers
from slight post-shock oscillations. Even though these oscillations are small in magnitude and do not affect the “essentially
non-oscillatory” property of WENO schemes, they are indeed responsible for the numerical residue to hang at the truncation
error level of the scheme instead of settling down to machine zero. We propose a new smoothness indicator for the WENO schemes
in steady state calculations, which performs better near the steady shock region than the original smoothness indicator in
[7, (1996) J. Comput. Phys. 126, 202–228.]. With our new smoothness indicator, the slight post-shock oscillations are either removed or significantly reduced
and convergence is improved significantly. Numerical experiments show that the residue for the WENO scheme with this new smoothness
indicator can converge to machine zero for one and two dimensional (2D) steady problems with strong shock waves when there
are no shocks passing through the domain boundaries.
Dedicated to the memory of Professor Xu-Dong Liu. 相似文献
6.
In Zhang and Shu (J. Comput. Phys. 229:3091–3120, 2010), two of the authors constructed uniformly high order accurate finite volume and discontinuous Galerkin (DG) schemes satisfying
a strict maximum principle for scalar conservation laws on rectangular meshes. The technique is generalized to positivity
preserving (of density and pressure) high order DG or finite volume schemes for compressible Euler equations in Zhang and
Shu (J. Comput. Phys. 229:8918–8934, 2010). The extension of these schemes to triangular meshes is conceptually plausible but highly nontrivial. In this paper, we
first introduce a special quadrature rule which is exact for two-variable polynomials over a triangle of a given degree and
satisfy a few other conditions, by which we can construct high order maximum principle satisfying finite volume schemes (e.g.
essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or DG method solving two dimensional scalar conservation
laws on triangular meshes. The same method can preserve the maximum principle for DG or finite volume schemes solving two-dimensional
incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible
velocity field. We also obtain positivity preserving (for density and pressure) high order DG or finite volume schemes solving
compressible Euler equations on triangular meshes. Numerical tests for the third order Runge-Kutta DG (RKDG) method on unstructured
meshes are reported. 相似文献
7.
In (J. Comput. Phys. 229: 8105–8129, 2010), Li and Qiu investigated the hybrid weighted essentially non-oscillatory (WENO) schemes with different indicators for Euler
equations of gas dynamics. In this continuation paper, we extend the method to solve the one- and two-dimensional shallow
water equations with source term due to the non-flat bottom topography, with a goal of obtaining the same advantages of the
schemes for the Euler equations, such as the saving computational cost, essentially non-oscillatory property for general solution
with discontinuities, and the sharp shock transition. Extensive simulations in one- and two-dimensions are provided to illustrate
the behavior of this procedure. 相似文献
8.
In this paper, a sixth-order finite difference weighted essentially non-oscillatory (WENO) scheme is developed to approximate the viscosity solution of the Hamilton–Jacobi equations. This new WENO scheme has the same spatial nodes as the classical fifth-order WENO scheme proposed by Jiang and Peng [Weighted ENO schemes for Hamilton–Jacobi equations, SIAM. J. Sci. Comput. 21 (2000), pp. 2126–2143] but can be as high as sixth-order accurate in smooth region while keeping sharp discontinuous transitions with no spurious oscillations near discontinuities. Extensive numerical experiments in one- and two-dimensional cases are carried out to illustrate the capability of the proposed scheme. 相似文献
9.
In this paper we continue the study, which was initiated in (Ben-Artzi et al. in Math. Model. Numer. Anal. 35(2):313–303,
2001; Fishelov et al. in Lecture Notes in Computer Science, vol. 2667, pp. 809–817, 2003; Ben-Artzi et al. in J. Comput. Phys. 205(2):640–664, 2005 and SIAM J. Numer. Anal. 44(5):1997–2024, 2006) of the numerical resolution of the pure streamfunction formulation of the time-dependent two-dimensional Navier-Stokes equation.
Here we focus on enhancing our second-order scheme, introduced in the last three afore-mentioned articles, to fourth order
accuracy. We construct fourth order approximations for the Laplacian, the biharmonic and the nonlinear convective operators.
The scheme is compact (nine-point stencil) for the Laplacian and the biharmonic operators, which are both treated implicitly
in the time-stepping scheme. The approximation of the convective term is compact in the no-leak boundary conditions case and
is nearly compact (thirteen points stencil) in the case of general boundary conditions. However, we stress that in any case
no unphysical boundary condition was applied to our scheme. Numerical results demonstrate that the fourth order accuracy is
actually obtained for several test-cases. 相似文献
10.
In this paper, we generalize the high order well-balanced finite difference weighted essentially non-oscillatory (WENO) scheme, designed earlier by us in Xing and Shu (2005, J. Comput. phys. 208, 206–227) for the shallow water equations, to solve a wider class of hyperbolic systems with separable source terms including the elastic wave equation, the hyperbolic model for a chemosensitive movement, the nozzle flow and a two phase flow model. Properties of the scheme for the shallow water equations (Xing and Shu 2005, J. Comput. phys. 208, 206–227), such as the exact preservation of the balance laws for certain steady state solutions, the non-oscillatory property for general solutions with discontinuities, and the genuine high order accuracy in smooth regions, are maintained for the scheme when applied to this general class of hyperbolic systems 相似文献
11.
N. V. Banichuk F. Ragnedda M. Serra 《Structural and Multidisciplinary Optimization》2009,38(5):491-498
The problem of maximization of the depth of penetration of rigid impactor into semi-infinite solid media (concrete shield)
is investigated analytically and numerically using two-stage model and experimental data of Forrestal and Tzou (Int J Solids
Struct 34(31–32):4127–4146, 1997). The shape of the axisymmetric rigid impactor has been taken as an unknown design variable. To solve the formulated optimization
problem for nonadditive functional, we expressed the depth of penetration (DOP) under some isoperimetric constraints. We apply
approaches based on analytical and qualitative variational methods and numerical optimization algorithm of global search.
Basic attention for considered optimization problem was given to constraints on the mass of penetrated bodies, expressed by
the volume in the case of penetrated solid body and by the surface area in the case of penetrated thin-walled rigid shell.
As a result of performed investigation, based on two-term and three-term two stage models proposed by Forrestal et al. (Int
J Impact Eng 15(4):396–405, 1994), Forrestal and Tzou (Int J Solids Struct 34(31–32):4127–4146, 1997) and effectively developed by Ben-Dor et al. (Comp Struct 56:243–248, 2002, Comput Struct 81(1):9–14, 2003a, Int J Solids Struct 40(17):4487–4500, 2003b, Mech Des Struct Mach 34(2): 139–156, 2006), we found analytical and numerical solutions and analyzed singularities of optimal forms. 相似文献
12.
In this paper we provide the full spectral decomposition of the Multi-Class Lighthill Whitham Richards (MCLWR) traffic models
described in (Wong et al. in Transp. Res. Part A 36:827–841, 2002; Benzoni-Gavage and Colombo in Eur. J. Appl. Math. 14:587–612, 2003). Even though the eigenvalues of these models can only be found numerically, the knowledge of the spectral structure allows
the use of characteristic-based High Resolution Shock Capturing (HRSC) schemes. We compare the characteristic-based approach
to the component-wise schemes used in (Zhang et al. in J. Comput. Phys. 191:639–659, 2003), and propose two strategies to minimize the oscillatory behavior that can be observed when using the component-wise approach. 相似文献
13.
Mitsuhiro T. Nakao Yoshitaka Watanabe Nobito Yamamoto Takaaki Nishida Myoung-Nyoung Kim 《Journal of scientific computing》2010,43(3):388-401
In previous works (Nakao et al., Reliab. Comput., 9(5):359–372, 2003; Watanabe et al., J. Math. Fluid Mech., 6(1):1–20, 2004), the authors considered the numerical verification method of solutions for two-dimensional heat convection problems known
as Rayleigh-Bénard problem. In the present paper, to make the arguments self-contained, we first summarize these results including
the basic formulation of the problem with numerical examples. Next, we will give a method to verify the bifurcation point
itself, which should be an important information to clarify the global bifurcation structure, and show a numerical example.
Finally, an extension to the three dimensional case will be described. 相似文献
14.
This work deliberately introduces collective-rotation noise into quantum states to prevent an intercept-resend attack on Zhang’s
quantum secret sharing scheme over a collective-noise quantum channel (Zhang in Phys A 361:233–238, 2006). The noise recovering capability of the scheme remains intact. With this design, the quantum bit efficiency of the protocol
is doubled when compared to Sun et al.’s improvement on Zhang’s scheme (Sun et al. in Opt Commun 283:181–183, 2010). 相似文献
15.
Computing the duplication history of a tandem repeated region is an important problem in computational biology (Fitch in Genetics
86:623–644, 1977; Jaitly et al. in J. Comput. Syst. Sci. 65:494–507, 2002; Tang et al. in J. Comput. Biol. 9:429–446, 2002). In this paper, we design a polynomial-time approximation scheme (PTAS) for the case where the size of the duplication block
is 1. Our PTAS is faster than the previously best PTAS in Jaitly et al. (J. Comput. Syst. Sci. 65:494–507, 2002). For example, to achieve a ratio of 1.5, our PTAS takes O(n
5) time while the PTAS in Jaitly et al. (J. Comput. Syst. Sci. 65:494–507, 2002) takes O(n
11) time. We also design a ratio-6 polynomial-time approximation algorithm for the case where the size of each duplication block
is at most 2. This is the first polynomial-time approximation algorithm with a guaranteed ratio for this case.
Part of work was done during a Z.-Z. Chen visit at City University of Hong Kong. 相似文献
16.
We present an improved technique for data hiding in polygonal meshes, which is based on the work of Bogomjakov et al. (Comput.
Graph. Forum 27(2):637–642, 2008). Like their method, we use an arrangement on primitives relative to a reference ordering to embed a message. But instead
of directly interpreting the index of a primitive in the reference ordering as the encoded/decoded bits, our method slightly
modifies the mapping so that our modification doubles the chance of encoding an additional bit compared to Bogomjakov et al.’s
(Comput. Graph. Forum 27(2):637–642, 2008). We illustrate the inefficiency in the original mapping of Bogomjakov et al. (Comput. Graph. Forum 27(2):637–642, 2008) with an intuitive representation using a binary tree. 相似文献
17.
Comparison theorems between the spectral radii of different matrices are a useful tool for judging the efficiency of preconditioners.
For single splittings of different monotone matrices, Elsner et al. (Linear Algebra Appl. 363:65–80, 2003) gave out comparison theorems for spectral radii. For double splittings, some convergence and comparison theorems of a monotone
matrix are presented by Shen et al. (Comput. Math. Appl. 51:1751–1760, 2006). In this note we give the comparison theorem for the spectral radii of matrices arising from double splittings of different
monotone matrices. 相似文献
18.
The potential flow equations which govern the free-surface motion of an ideal fluid (the water wave problem) are notoriously
difficult to solve for a number of reasons. First, they are a classical free-boundary problem where the domain shape is one
of the unknowns to be found. Additionally, they are strongly nonlinear (with derivatives appearing in the nonlinearity) without
a natural dissipation mechanism so that spurious high-frequency modes are not damped. In this contribution we address the
latter of these difficulties using a surface formulation (which addresses the former complication) supplemented with physically-motivated
viscous effects recently derived by Dias et al. (Phys. Lett. A 372:1297–1302, 2008). The novelty of our approach is to derive a weakly nonlinear model from the surface formulation of Zakharov (J. Appl. Mech.
Tech. Phys. 9:190–194, 1968) and Craig and Sulem (J. Comput. Phys. 108:73–83, 1993), complemented with the viscous effects mentioned above. Our new model is simple to implement while being both faithful to
the physics of the problem and extremely stable numerically. 相似文献
19.
Thomas Franke Ronald H. W. Hoppe Christopher Linsenmann Lothar Schmid Carina Willbold Achim Wixforth 《Computing and Visualization in Science》2011,14(4):167-180
We study the mathematical modeling and numerical simulation of the motion of red blood cells (RBC) and vesicles subject to
an external incompressible flow in a microchannel. RBC and vesicles are viscoelastic bodies consisting of a deformable elastic
membrane enclosing an incompressible fluid. We provide an extension of the finite element immersed boundary method by Boffi
and Gastaldi (Comput Struct 81:491–501, 2003), Boffi et al. (Math Mod Meth Appl Sci 17:1479–1505, 2007), Boffi et al. (Comput Struct 85:775–783, 2007) based on a model for the membrane that additionally accounts for bending energy and also consider inflow/outflow conditions
for the external fluid flow. The stability analysis requires both the approximation of the membrane by cubic splines (instead
of linear splines without bending energy) and an upper bound on the inflow velocity. In the fully discrete case, the resulting
CFL-type condition on the time step size is also more restrictive. We perform numerical simulations for various scenarios
including the tank treading motion of vesicles in microchannels, the behavior of ‘healthy’ and ‘sick’ RBC which differ by
their stiffness, and the motion of RBC through thin capillaries. The simulation results are in very good agreement with experimentally
available data. 相似文献
20.
Jake Porway Qiongchen Wang Song Chun Zhu 《International Journal of Computer Vision》2010,88(2):254-283
In this paper we present a hierarchical and contextual model for aerial image understanding. Our model organizes objects (cars,
roofs, roads, trees, parking lots) in aerial scenes into hierarchical groups whose appearances and configurations are determined
by statistical constraints (e.g. relative position, relative scale, etc.). Our hierarchy is a non-recursive grammar for objects
in aerial images comprised of layers of nodes that can each decompose into a number of different configurations. This allows
us to generate and recognize a vast number of scenes with relatively few rules. We present a minimax entropy framework for
learning the statistical constraints between objects and show that this learned context allows us to rule out unlikely scene
configurations and hallucinate undetected objects during inference. A similar algorithm was proposed for texture synthesis
(Zhu et al. in Int. J. Comput. Vis. 2:107–126, 1998) but didn’t incorporate hierarchical information. We use a range of different bottom-up detectors (AdaBoost, TextonBoost,
Compositional Boosting (Freund and Schapire in J. Comput. Syst. Sci. 55, 1997; Shotton et al. in Proceedings of the European Conference on Computer Vision, pp. 1–15, 2006; Wu et al. in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–8, 2007)) to propose locations of objects in new aerial images and employ a cluster sampling algorithm (C4 (Porway and Zhu, 2009)) to choose the subset of detections that best explains the image according to our learned prior model. The C4 algorithm
can quickly and efficiently switch between alternate competing sub-solutions, for example whether an image patch is better
explained by a parking lot with cars or by a building with vents. We also show that our model can predict the locations of
objects our detectors missed. We conclude by presenting parsed aerial images and experimental results showing that our cluster
sampling and top-down prediction algorithms use the learned contextual cues from our model to improve detection results over
traditional bottom-up detectors alone. 相似文献