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1.
Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes 下载免费PDF全文
In this article we present a new class of high order accurate ArbitraryEulerian-Lagrangian (ALE) one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimensional unstructured triangular meshes. A WENO reconstruction algorithm is used to achieve high
order accuracy in space and a high order one-step time discretization is achieved by
using the local space-time Galerkin predictor proposed in [25]. For that purpose, a
new element-local weak formulation of the governing PDE is adopted on moving
space-time elements. The space-time basis and test functions are obtained considering Lagrange interpolation polynomials passing through a predefined set of nodes.
Moreover, a polynomial mapping defined by the same local space-time basis functions
as the weak solution of the PDE is used to map the moving physical space-time element onto a space-time reference element. To maintain algorithmic simplicity, the
final ALE one-step finite volume scheme uses moving triangular meshes with straight
edges. This is possible in the ALE framework, which allows a local mesh velocity that
is different from the local fluid velocity. We present numerical convergence rates for
the schemes presented in this paper up to sixth order of accuracy in space and time and
show some classical numerical test problems for the two-dimensional Euler equations
of compressible gas dynamics. 相似文献
2.
This paper is concerned with a new version of the Osher-Solomon Riemann
solver and is based on a numerical integration of the path-dependent dissipation matrix.
The resulting scheme is much simpler than the original one and is applicable to
general hyperbolic conservation laws, while retaining the attractive features of the original
solver: the method is entropy-satisfying, differentiable and complete in the sense
that it attributes a different numerical viscosity to each characteristic field, in particular
to the intermediate ones, since the full eigenstructure of the underlying hyperbolic system
is used. To illustrate the potential of the proposed scheme we show applications
to the following hyperbolic conservation laws: Euler equations of compressible gasdynamics
with ideal gas and real gas equation of state, classical and relativistic MHD
equations as well as the equations of nonlinear elasticity. To the knowledge of the authors,
apart from the Euler equations with ideal gas, an Osher-type scheme has never
been devised before for any of these complicated PDE systems. Since our new general
Riemann solver can be directly used as a building block of high order finite volume
and discontinuous Galerkin schemes we also show the extension to higher order of
accuracy and multiple space dimensions in the new framework of PNPM schemes on
unstructured meshes recently proposed in [9]. 相似文献
3.
A Multi-Domain Hybrid DG and WENO Method for Hyperbolic Conservation Laws on Hybrid Meshes 下载免费PDF全文
In [SIAM J. Sci. Comput., 35(2)(2013), A1049–A1072], a class of multi-domain
hybrid DG and WENO methods for conservation laws was introduced. Recent applications of this method showed that numerical instability may encounter if the DG flux
with Lagrangian interpolation is applied as the interface flux during the moment of
conservative coupling. In this continuation paper, we present a more robust approach
in the construction of DG flux at the coupling interface by using WENO procedures of
reconstruction. Based on this approach, such numerical instability is overcome very
well. In addition, the procedure of coupling a DG method with a WENO-FD scheme
on hybrid meshes is disclosed in detail. Typical testing cases are employed to demonstrate the accuracy of this approach and the stability under the flexibility of using either
WENO-FD flux or DG flux at the moment of requiring conservative coupling. 相似文献
4.
High Order Fixed-Point Sweeping WENO Methods for Steady State of Hyperbolic Conservation Laws and Its Convergence Study 下载免费PDF全文
Liang Wu Yong-Tao Zhang Shuhai Zhang & Chi-Wang Shu 《Communications In Computational Physics》2016,20(4):835-869
Fixed-point iterative sweeping methods were developed in the literature to
efficiently solve static Hamilton-Jacobi equations. This class of methods utilizes the
Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence
rate. They take advantage of the properties of hyperbolic partial differential equations
(PDEs) and try to cover a family of characteristics of the corresponding Hamilton-Jacobi
equation in a certain direction simultaneously in each sweeping order. Different
from other fast sweeping methods, fixed-point iterative sweeping methods have the
advantages such as that they have explicit forms and do not involve inverse operation
of nonlinear local systems. In principle, it can be applied to solving very general
equations using any monotone numerical fluxes and high order approximations easily.
In this paper, based on the recently developed fifth order WENO schemes which improve
the convergence of the classical WENO schemes by removing slight post-shock
oscillations, we design fifth order fixed-point sweeping WENO methods for efficient
computation of steady state solution of hyperbolic conservation laws. Especially, we
show that although the methods do not have linear computational complexity, they
converge to steady state solutions much faster than regular time-marching approach
by stability improvement for high order schemes with a forward Euler time-marching. 相似文献
5.
In this paper, we present a new type of Hermite weighted essentially non-oscillatory
(HWENO) schemes for solving the Hamilton-Jacobi equations on the finite
volume framework. The cell averages of the function and its first one (in one dimension)
or two (in two dimensions) derivative values are together evolved via time
approaching and used in the reconstructions. And the major advantages of the new
HWENO schemes are their compactness in the spacial field, purely on the finite volume
framework and only one set of small stencils is used for different type of the
polynomial reconstructions. Extensive numerical tests are performed to illustrate the
capability of the methodologies. 相似文献
6.
In this paper, we introduce a new type of troubled-cell indicator to improvehybrid weighted essentially non-oscillatory (WENO) schemes for solving the hyperbolic conservation laws. The hybrid WENO schemes selectively adopt the high-orderlinear upwind scheme or the WENO scheme to avoid the local characteristic decompositions and calculations of the nonlinear weights in smooth regions. Therefore,they can reduce computational cost while maintaining non-oscillatory properties innon-smooth regions. Reliable troubled-cell indicators are essential for efficient hybridWENO methods. Most of troubled-cell indicators require proper parameters to detectdiscontinuities precisely, but it is very difficult to determine the parameters automatically. We develop a new troubled-cell indicator derived from the mean value theoremthat does not require any variable parameters. Additionally, we investigate the characteristics of indicator variable; one of the conserved properties or the entropy is considered as indicator variable. Detailed numerical tests for 1D and 2D Euler equations areconducted to demonstrate the performance of the proposed indicator. The results withthe proposed troubled-cell indicator are in good agreement with pure WENO schemes.Also the new indicator has advantages in the computational cost compared with theother indicators. 相似文献
7.
In this paper, a high-order moment-based multi-resolution Hermiteweighted essentially non-oscillatory (HWENO) scheme is designed for hyperbolic conservation laws. The main idea of this scheme is derived from our previous work [J.Comput. Phys., 446 (2021) 110653], in which the integral averages of the function andits first order derivative are used to reconstruct both the function and its first orderderivative values at the boundaries. However, in this paper, only the function values atthe Gauss-Lobatto points in the one or two dimensional case need to be reconstructedby using the information of the zeroth and first order moments. In addition, an extramodification procedure is used to modify those first order moments in the troubled-cells, which leads to an improvement of stability and an enhancement of resolutionnear discontinuities. To obtain the same order of accuracy, the size of the stencil required by this moment-based multi-resolution HWENO scheme is still the same as thegeneral HWENO scheme and is more compact than the general WENO scheme. Moreover, the linear weights are not unique and are independent of the node position, andthe CFL number can still be 0.6 whether for the one or two dimensional case, which hasto be 0.2 in the two dimensional case for other HWENO schemes. Extensive numericalexamples are given to demonstrate the stability and resolution of such moment-basedmulti-resolution HWENO scheme. 相似文献
8.
A New Approach of High Order Well-Balanced Finite Volume WENO Schemes and Discontinuous Galerkin Methods for a Class of Hyperbolic Systems with Source Terms 下载免费PDF全文
Hyperbolic balance laws have steady state solutions in which the flux gradients are
nonzero but are exactly balanced by the source terms. In our earlier work [31–33], we designed
high order well-balanced schemes to a class of hyperbolic systems with separable source terms.
In this paper, we present a different approach to the same purpose: designing high order
well-balanced finite volume weighted essentially non-oscillatory (WENO) schemes and RungeKutta
discontinuous Galerkin (RKDG) finite element methods. We make the observation that
the traditional RKDG methods are capable of maintaining certain steady states exactly, if a
small modification on either the initial condition or the flux is provided. The computational
cost to obtain such a well balanced RKDG method is basically the same as the traditional
RKDG method. The same idea can be applied to the finite volume WENO schemes. We
will first describe the algorithms and prove the well balanced property for the shallow water
equations, and then show that the result can be generalized to a class of other balance laws.
We perform extensive one and two dimensional simulations to verify the properties of these
schemes 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. 相似文献
9.
A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws 下载免费PDF全文
Rapha& euml l Loub& egrave re Michael Dumbser & Steven Diot 《Communications In Computational Physics》2014,16(3):718-763
In this paper, we investigate the coupling of the Multi-dimensional Optimal
Order Detection (MOOD) method and the Arbitrary high order DERivatives (ADER)
approach in order to design a new high order accurate, robust and computationally
efficient Finite Volume (FV) scheme dedicated to solving nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and
three space dimensions, respectively. The Multi-dimensional Optimal Order Detection
(MOOD) method for 2D and 3D geometries has been introduced in a recent series of
papers for mixed unstructured meshes. It is an arbitrary high-order accurate Finite
Volume scheme in space, using polynomial reconstructions with a posteriori detection
and polynomial degree decrementing processes to deal with shock waves and other
discontinuities. In the following work, the time discretization is performed with an
elegant and efficient one-step ADER procedure. Doing so, we retain the good properties of the MOOD scheme, that is to say, the optimal high-order of accuracy is reached
on smooth solutions, while spurious oscillations near singularities are prevented. The
ADER technique not only reduces the cost of the overall scheme as shown
on a set of numerical tests in 2D and 3D, but also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO
schemes and the new ADER-MOOD approach has been carried out for high-order
schemes in space and time in terms of cost, robustness, accuracy and efficiency. The
main finding of this paper is that the combination of ADER with MOOD generally
outperforms the one of ADER and WENO either because at given accuracy MOOD isless expensive (memory and/or CPU time), or because it is more accurate for a given
grid resolution. A large suite of classical numerical test problems has been solved
on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations
of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations
(RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic
partial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements. 相似文献
10.
A Space-Time Conservative Method for Hyperbolic Systems with Stiff and Non-Stiff Source Terms 下载免费PDF全文
In this article we propose a higher-order space-time conservative method
for hyperbolic systems with stiff and non-stiff source terms as well as relaxation systems. We call the scheme a slope propagation (SP) method. It is an extension of our
scheme derived for homogeneous hyperbolic systems [1]. In the present inhomogeneous
systems the relaxation time may vary from order of one to a very small value. These
small values make the relaxation term stronger and highly stiff. In such situations
underresolved numerical schemes may produce spurious numerical results. However,
our present scheme has the capability to correctly capture the behavior of the physical
phenomena with high order accuracy even if the initial layer and the small relaxation
time are not numerically resolved. The scheme treats the space and time in a unified
manner. The flow variables and their slopes are the basic unknowns in the scheme. The
source term is treated by its volumetric integration over the space-time control volume
and is a direct part of the overall space-time flux balance. We use two approaches
for the slope calculations of the flow variables, the first one results directly from the
flux balance over the control volumes, while in the second one we use a finite difference approach. The main features of the scheme are its simplicity, its Jacobian-free
and Riemann solver-free recipe, as well as its efficiency and high order accuracy. In
particular we show that the scheme has a discrete analog of the continuous asymptotic limit. We have implemented our scheme for various test models available in the
literature such as the Broadwell model, the extended thermodynamics equations, the
shallow water equations, traffic flow and the Euler equations with heat transfer. The
numerical results validate the accuracy, versatility and robustness of the present scheme. 相似文献
11.
Jun Zhu & Jianxian Qiu 《Communications In Computational Physics》2020,27(3):897-920
In this paper, a new type of third-order and fourth-order weighted essentially non-oscillatory (WENO) schemes is designed for simulating the Hamilton-Jacobi equations on triangular meshes. We design such schemes with the use of the nodal information defined on five unequal-sized spatial stencils, the application of monotone Hamiltonians as a building block, the artificial set of positive linear weights to make up high-order approximations in smooth regions simultaneously avoiding spurious oscillations nearby discontinuities of the derivatives of the solutions. The spatial reconstructions are convex combinations of the derivatives of a modified cubic/quartic polynomial defined on a big spatial stencil and four quadratic polynomials defined on small spatial stencils, and a third-order TVD Runge-Kutta method is used for the time discretization. The main advantages of these WENO schemes are their efficiency, simplicity, and can be easily implemented to higher dimensional unstructured meshes. Extensive numerical tests are performed to illustrate the good performance of such new WENO schemes. 相似文献
12.
On the Order of Accuracy and Numerical Performance of Two Classes of Finite Volume WENO Schemes 下载免费PDF全文
Rui Zhang Mengping Zhang & Chi-Wang Shu 《Communications In Computational Physics》2011,9(3):807-827
In this paper we consider two commonly used classes of finite volume
weighted essentially non-oscillatory (WENO) schemes in two dimensional Cartesian
meshes. We compare them in terms of accuracy, performance for smooth and shocked
solutions, and efficiency in CPU timing. For linear systems both schemes are high
order accurate, however for nonlinear systems, analysis and numerical simulation results
verify that one of them (Class A) is only second order accurate, while the other
(Class B) is high order accurate. The WENO scheme in Class A is easier to implement
and costs less than that in Class B. Numerical experiments indicate that the resolution
for shocked problems is often comparable for schemes in both classes for the same
building blocks and meshes, despite of the difference in their formal order of accuracy.
The results in this paper may give some guidance in the application of high order finite
volume schemes for simulating shocked flows. 相似文献
13.
Adaptive Bayesian Inference for Discontinuous Inverse Problems,Application to Hyperbolic Conservation Laws 下载免费PDF全文
Alexandre Birolleau Gaë l Poë tte & Didier Lucor 《Communications In Computational Physics》2014,16(1):1-34
Various works from the literature aimed at accelerating Bayesian inference
in inverse problems. Stochastic spectral methods have been recently proposed as surrogate approximations of the forward uncertainty propagation model over the support
of the prior distribution. These representations are efficient because they allow affordable simulation of a large number of samples from the posterior distribution. Unfortunately, they do not perform well when the forward model exhibits strong nonlinear
behavior with respect to its input.In this work, we first relate the fast (exponential) $L^2$-convergence of the forward
approximation to the fast (exponential) convergence (in terms of Kullback-Leibler divergence) of the approximate posterior. In particular, we prove that in case the prior
distribution is uniform, the posterior is at least twice as fast as the convergence rate of
the forward model in those norms. The Bayesian inference strategy is developed in the
framework of a stochastic spectral projection method. The predicted convergence rates
are then demonstrated for simple nonlinear inverse problems of varying smoothness.We then propose an efficient numerical approach for the Bayesian solution of inverse problems presenting strongly nonlinear or discontinuous system responses. This
comes with the improvement of the forward model that is adaptively approximated by
an iterative generalized Polynomial Chaos-based representation. The numerical approximations and predicted convergence rates of the former approach are compared
to the new iterative numerical method for nonlinear time-dependent test cases of varying dimension and complexity, which are relevant regarding our hydrodynamics motivations and therefore regarding hyperbolic conservation laws and the apparition of
discontinuities in finite time. 相似文献
14.
关节镜下松解治疗膝关节僵直 总被引:9,自引:3,他引:6
目的探讨关节镜下松解治疗膝关节纤维僵直的方法和效果.方法21例膝关节僵直行关节镜下松解,切除关节内粘连束带和疤痕组织、松解挛缩关节囊、分离股四头肌与股骨间粘连.结果术后2例出现关节血肿.随访4~49个月,平均19个月,膝关节活动度从术前平均41°增至115°,平均提高74°.结论关节镜下松解治疗膝关节僵直创伤小,康复快,效果好. 相似文献
15.
关节镜下松解治疗膝关节僵直 总被引:5,自引:0,他引:5
目的:探讨关节镜下松解治疗膝关节纤维僵直的方法和效果。方法:21例膝关节僵直行关节镜下松解,切除关节内粘连束带和疤痕组织、松解挛缩关节囊、分离股四头肌与股骨间粘连。结果:术后2例出现关节血肿。随访4~49个月,平均19个月,膝关节活动度从术前平均41°增至115°,平均提高74°。结论:关节镜下松解治疗膝关节僵直创伤小,康复快,效果好。 相似文献
16.
Franç ois Dubois & Pierre Lallemand 《Communications In Computational Physics》2013,13(3):649-670
We propose to extend the d'Humières version of the lattice Boltzmann scheme to triangular meshes. We use Bravais lattices or more general lattices with the property that the degree of each internal vertex is supposed to be constant. On such meshes,
it is possible to define the lattice Boltzmann scheme as a discrete particle method, without need of finite volume formulation or Delaunay-Voronoi hypothesis for the lattice.
We test this idea for the heat equation and perform an asymptotic analysis with the
Taylor expansion method for two schemes named D2T4 and D2T7. The results show a
convergence up to second order accuracy and set new questions concerning a possible
super-convergence. 相似文献
17.
A Finite Volume Upwind-Biased Centred Scheme for Hyperbolic Systems of Conservation Laws: Application to Shallow Water Equations 下载免费PDF全文
Guglielmo Stecca Annunziato Siviglia & Eleuterio F. Toro 《Communications In Computational Physics》2012,12(4):1183-1214
We construct a new first-order central-upwind numerical method for solving systems of hyperbolic equations in conservative form. It applies in multidimensional structured and unstructured meshes. The proposed method is an extension of
the UFORCE method developed by Stecca, Siviglia and Toro [25], in which the upwind
bias for the modification of the staggered mesh is evaluated taking into account the
smallest and largest wave of the entire Riemann fan. The proposed first-order method
is shown to be identical to the Godunov upwind method in applications to a 2×2 linear
hyperbolic system. The method is then extended to non-linear systems and its performance is assessed by solving the two-dimensional inviscid shallow water equations.
Extension to second-order accuracy is carried out using an ADER-WENO approach in
the finite volume framework on unstructured meshes. Finally, numerical comparison
with current competing numerical methods enables us to identify the salient features
of the proposed method. 相似文献
18.
A Review of Residual Distribution Schemes for Hyperbolic and Parabolic Problems: The July 2010 State of the Art 下载免费PDF全文
Remi Abgrall 《Communications In Computational Physics》2012,11(4):1043-1080
We describe and review non oscillatory residual distribution schemes that are rather natural extension of high order finite volume schemes when a special emphasis is put on the structure of the computational stencil. We provide their connections with standard stabilized finite element and discontinuous Galerkin schemes, show that their are really non oscillatory. We also discuss the extension to these methods to parabolic problems. We also draw some research perspectives. 相似文献
19.
Numerical Methods for Balance Laws with Space Dependent Flux: Application to Radiotherapy Dose Calculation 下载免费PDF全文
Christophe Berthon Martin Frank Cé line Sarazin & Rodolphe Turpault 《Communications In Computational Physics》2011,10(5):1184-1210
The present work is concerned with the derivation of numerical methods
to approximate the radiation dose in external beam radiotherapy. To address this issue,
we consider a moment approximation of radiative transfer, closed by an entropy
minimization principle. The model under consideration is governed by a system of
hyperbolic equations in conservation form supplemented by source terms. The main
difficulty coming from the numerical approximation of this system is an explicit space
dependence in the flux function. Indeed, this dependence will be seen to be stiff and
specific numerical strategies must be derived in order to obtain the needed accuracy. A
first approach is developed considering the 1D case, where a judicious change of variables
allows to eliminate the space dependence in the flux function. This is not possible
in multi-D. We therefore reinterpret the 1D scheme as a scheme on two meshes, and
generalize this to 2D by alternating transformations between separate meshes. We call
this procedure projection method. Several numerical experiments, coming from medical
physics, illustrate the potential applicability of the developed method. 相似文献
20.