共查询到20条相似文献,搜索用时 15 毫秒
1.
Jun Zhu Xinghui Zhong Chi-Wang Shu & Jianxian Qiu 《Communications In Computational Physics》2016,19(4):944-969
In this paper, we propose a new type of weighted essentially non-oscillatory
(WENO) limiter, which belongs to the class of Hermite WENO (HWENO) limiters, for
the Runge-Kutta discontinuous Galerkin (RKDG) methods solving hyperbolic conservation
laws. This new HWENO limiter is a modification of the simple WENO limiter
proposed recently by Zhong and Shu [29]. Both limiters use information of the DG
solutions only from the target cell and its immediate neighboring cells, thus maintaining
the original compactness of the DG scheme. The goal of both limiters is to obtain
high order accuracy and non-oscillatory properties simultaneously. The main novelty
of the new HWENO limiter in this paper is to reconstruct the polynomial on the target
cell in a least square fashion [8] while the simple WENO limiter [29] is to use the entire
polynomial of the original DG solutions in the neighboring cells with an addition of
a constant for conservation. The modification in this paper improves the robustness
in the computation of problems with strong shocks or contact discontinuities, without
changing the compact stencil of the DG scheme. Numerical results for both one and
two dimensional equations including Euler equations of compressible gas dynamics
are provided to illustrate the viability of this modified limiter. 相似文献
2.
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
3.
High Order Finite Difference Hermite WENO Fixed-Point Fast Sweeping Method for Static Hamilton-Jacobi Equations 下载免费PDF全文
In this paper, we combine the nonlinear HWENO reconstruction in [43] andthe fixed-point iteration with Gauss-Seidel fast sweeping strategy, to solve the staticHamilton-Jacobi equations in a novel HWENO framework recently developed in [22].The proposed HWENO frameworks enjoys several advantages. First, compared withthe traditional HWENO framework, the proposed methods do not need to introduceadditional auxiliary equations to update the derivatives of the unknown function $phi$.They are now computed from the current value of $phi$ and the previous spatial derivatives of $phi$. This approach saves the computational storage and CPU time, which greatlyimproves the computational efficiency of the traditional HWENO scheme. In addition,compared with the traditional WENO method, reconstruction stencil of the HWENOmethods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller on the same mesh. Second, the fixed-point fast sweeping methodis used to update the numerical approximation. It is an explicit method and doesnot involve the inverse operation of nonlinear Hamiltonian, therefore any Hamilton-Jacobi equations with complex Hamiltonian can be solved easily. It also resolves someknown issues, including that the iterative number is very sensitive to the parameter $ε$ used in the nonlinear weights, as observed in previous studies. Finally, to furtherreduce the computational cost, a hybrid strategy is also presented. Extensive numerical experiments are performed on two-dimensional problems, which demonstrate thegood performance of the proposed fixed-point fast sweeping HWENO methods. 相似文献
4.
High-Order Local Discontinuous Galerkin Method with Multi-Resolution WENO Limiter for Navier-Stokes Equations on Triangular Meshes 下载免费PDF全文
Yizhou Lu Jun Zhu Shengzhu Cui Zhenming Wang Linlin Tian & Ning Zhao 《Communications In Computational Physics》2023,33(5):1217-1239
In this paper, a new multi-resolution weighted essentially non-oscillatory(MR-WENO) limiter for high-order local discontinuous Galerkin (LDG) method is designed for solving Navier-Stokes equations on triangular meshes. This MR-WENOlimiter is a new extension of the finite volume MR-WENO schemes. Such new limiteruses information of the LDG solution essentially only within the troubled cell itself, tobuild a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to thehighest degree of the LDG method. As an example, a third-order LDG method with associated same order MR-WENO limiter has been developed in this paper, which couldmaintain the original order of accuracy in smooth regions and could simultaneouslysuppress spurious oscillations near strong shocks or contact discontinuities. The linear weights of such new MR-WENO limiter can be any positive numbers on conditionthat their summation is one. This is the first time that a series of different degree polynomials within the troubled cell are applied in a WENO-type fashion to modify thefreedom of degrees of the LDG solutions in the troubled cell. This MR-WENO limiteris very simple to construct, and can be easily implemented to arbitrary high-order accuracy and in higher dimensions on unstructured meshes. Such spatial reconstructionmethodology improves the robustness in the numerical simulation on the same compact spatial stencil of the original LDG methods on triangular meshes. Some classicalviscous examples are given to show the good performance of this third-order LDGmethod with associated MR-WENO limiter. 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.
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
7.
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
8.
High Order Finite Difference WENO Methods with Unequal-Sized Sub-Stencils for the Degasperis-Procesi Type Equations 下载免费PDF全文
Jianfang Lin Yan Xu Huiwen Xue & Xinghui Zhong 《Communications In Computational Physics》2022,31(3):913-946
In this paper, we develop two finite difference weighted essentiallynon-oscillatory (WENO) schemes with unequal-sized sub-stencils for solving theDegasperis-Procesi (DP) and $mu$-Degasperis-Procesi ($mu$DP) equations, which containnonlinear high order derivatives, and possibly peakon solutions or shock waves. Byintroducing auxiliary variable(s), we rewrite the DP equation as a hyperbolic-ellipticsystem, and the $mu$DP equation as a first order system. Then we choose a linear finitedifference scheme with suitable order of accuracy for the auxiliary variable(s), andtwo finite difference WENO schemes with unequal-sized sub-stencils for the primalvariable. One WENO scheme uses one large stencil and several smaller stencils, andthe other WENO scheme is based on the multi-resolution framework which uses a series of unequal-sized hierarchical central stencils. Comparing with the classical WENOscheme which uses several small stencils of the same size to make up a big stencil, bothWENO schemes with unequal-sized sub-stencils are simple in the choice of the stenciland enjoy the freedom of arbitrary positive linear weights. Another advantage is thatthe final reconstructed polynomial on the target cell is a polynomial of the same degree as the polynomial over the big stencil, while the classical finite difference WENOreconstruction can only be obtained for specific points inside the target interval. Numerical tests are provided to demonstrate the high order accuracy and non-oscillatoryproperties of the proposed schemes. 相似文献
9.
A recent work of Li et al. [Numer. Math. Theor. Meth. Appl., 1(2008), pp.
92-112] proposed a finite volume solver to solve 2D steady Euler equations. Although
the Venkatakrishnan limiter is used to prevent the non-physical oscillations nearby the
shock region, the overshoot or undershoot phenomenon can still be observed. Moreover,
the numerical accuracy is degraded by using Venkatakrishnan limiter. To fix the
problems, in this paper the WENO type reconstruction is employed to gain both the
accurate approximations in smooth region and non-oscillatory sharp profiles near the
shock discontinuity. The numerical experiments will demonstrate the efficiency and
robustness of the proposed numerical strategy. 相似文献
10.
A Higher Order Interpolation Scheme of Finite Volume Method for Compressible Flow on Curvilinear Grids 下载免费PDF全文
Zhen-Hua Jiang Xi Deng Feng Xiao Chao Yan & Jian Yu 《Communications In Computational Physics》2020,28(4):1609-1638
A higher order interpolation scheme based on a multi-stage BVD (Boundary Variation Diminishing) algorithm is developed for the FV (Finite Volume) method
on non-uniform, curvilinear structured grids to simulate the compressible turbulent
flows. The designed scheme utilizes two types of candidate interpolants including
a higher order linear-weight polynomial as high as eleven and a THINC (Tangent of
Hyperbola for INterface Capturing) function with the adaptive steepness. We investigate not only the accuracy but also the efficiency of the methodology through the cost
efficiency analysis in comparison with well-designed mapped WENO (Weighted Essentially Non-Oscillatory) scheme. Numerical experimentation including benchmark
broadband turbulence problem as well as real-life wall-bounded turbulent flows has
been carried out to demonstrate the potential implementation of the present higher
order interpolation scheme especially in the ILES (Implicit Large Eddy Simulation) of
compressible turbulence. 相似文献
11.
Adaptive Order WENO Reconstructions for the Semi-Lagrangian Finite Difference Scheme for Advection Problem 下载免费PDF全文
Jiajie Chen Xiaofeng Cai Jianxian Qiu & Jing-Mei Qiu 《Communications In Computational Physics》2021,30(1):67-96
We present a new conservative semi-Lagrangian finite difference weighted
essentially non-oscillatory scheme with adaptive order. This is an extension of the
conservative semi-Lagrangian (SL) finite difference WENO scheme in [Qiu and Shu,
JCP, 230 (4) (2011), pp. 863-889], in which linear weights in SL WENO framework
were shown not to exist for variable coefficient problems. Hence, the order of accuracy is not optimal from reconstruction stencils. In this paper, we incorporate a recent
WENO adaptive order (AO) technique [Balsara et al., JCP, 326 (2016), pp. 780-804]
to the SL WENO framework. The new scheme can achieve an optimal high order of
accuracy, while maintaining the properties of mass conservation and non-oscillatory
capture of solutions from the original SL WENO. The positivity-preserving limiter is
further applied to ensure the positivity of solutions. Finally, the scheme is applied to
high dimensional problems by a fourth-order dimensional splitting. We demonstrate
the effectiveness of the new scheme by extensive numerical tests on linear advection
equations, the Vlasov-Poisson system, the guiding center Vlasov model as well as the
incompressible Euler equations. 相似文献
12.
Divergence-Free WENO Reconstruction-Based Finite Volume Scheme for Solving Ideal MHD Equations on Triangular Meshes 下载免费PDF全文
Zhiliang Xu Dinshaw S. Balsara & Huijing Du 《Communications In Computational Physics》2016,19(4):841-880
In this paper, we introduce a high-order accurate constrained transport
type finite volume method to solve ideal magnetohydrodynamic equations on two-dimensional
triangular meshes. A new divergence-free WENO-based reconstruction
method is developed to maintain exactly divergence-free evolution of the numerical
magnetic field. In this formulation, the normal component of the magnetic field at each
face of a triangle is reconstructed uniquely and with the desired order of accuracy. Additionally,
a new weighted flux interpolation approach is also developed to compute
the z-component of the electric field at vertices of grid cells. We also present numerical
examples to demonstrate the accuracy and robustness of the proposed scheme. 相似文献
13.
A NURBS-Enhanced Finite Volume Method for Steady Euler Equations with Goal-Oriented $h$-Adaptivity 下载免费PDF全文
Xucheng Meng & Guanghui Hu 《Communications In Computational Physics》2022,32(2):490-523
In [A NURBS-enhanced finite volume solver for steady Euler equations, X. C.Meng, G. H. Hu, J. Comput. Phys., Vol. 359, pp. 77–92], a NURBS-enhanced finite volumemethod was developed to solve the steady Euler equations, in which the desired highorder numerical accuracy was obtained for the equations imposed in the domain witha curved boundary. In this paper, the method is significantly improved in the following ways: (i) a simple and efficient point inversion technique is designed to computethe parameter values of points lying on a NURBS curve, (ii) with this new point inversion technique, the $h$-adaptive NURBS-enhanced finite volume method is introducedfor the steady Euler equations in a complex domain, and (iii) a goal-oriented a posteriorierror indicator is designed to further improve the efficiency of the algorithm towardsaccurately calculating a given quantity of interest. Numerical results obtained from avariety of numerical experiments with different flow configurations successfully showthe effectiveness and robustness of the proposed method. 相似文献
14.
Ruo Li & Wei Zhong 《Communications In Computational Physics》2021,30(5):1545-1588
We propose a new family of mapped WENO schemes by using several adaptive control functions and a smoothing approximation of the signum function. Theproposed schemes admit an extensive permitted range of the parameters in the mapping functions. Consequently, they have the capacity to achieve optimal convergencerates, even near critical points. Particularly, the new schemes with fine-tuned parameters illustrates a significant advantage when solving problems with discontinuities. Itproduces numerical solutions with high resolution without generating spurious oscillations, especially for long output times. 相似文献
15.
Comparison of Fifth-Order WENO Scheme and Finite Volume WENO-Gas-Kinetic Scheme for Inviscid and Viscous Flow Simulation 下载免费PDF全文
The development of high-order schemes has been mostly concentrated on
the limiters and high-order reconstruction techniques. In this paper, the effect of the
flux functions on the performance of high-order schemes will be studied. Based on the
same WENO reconstruction, two schemes with different flux functions, i.e., the fifth-order WENO method and the WENO-Gas-Kinetic scheme (WENO-GKS), will be compared. The fifth-order finite difference WENO-SW scheme is a characteristic variable
reconstruction based method which uses the Steger-Warming flux splitting for inviscid terms, the sixth-order central difference for viscous terms, and three stages Runge-Kutta time stepping for the time integration. On the other hand, the finite volume
WENO-GKS is a conservative variable reconstruction based method with the same
WENO reconstruction. But it evaluates a time dependent gas distribution function
along a cell interface, and updates the flow variables inside each control volume by
integrating the flux function along the boundary of the control volume in both space
and time. In order to validate the robustness and accuracy of the schemes, both methods are tested under a wide range of flow conditions: vortex propagation, Mach 3
step problem, and the cavity flow at Reynolds number 3200. Our study shows that
both WENO-SW and WENO-GKS yield quantitatively similar results and agree with
each other very well provided a sufficient grid resolution is used. With the reduction of mesh points, the WENO-GKS behaves to have less numerical dissipation and
present more accurate solutions than those from the WENO-SW in all test cases. For
the Navier-Stokes equations, since the WENO-GKS couples inviscid and viscous terms
in a single flux evaluation, and the WENO-SW uses an operator splitting technique, it
appears that the WENO-SW is more sensitive to the WENO reconstruction and boundary treatment. In terms of efficiency, the finite volume WENO-GKS is about 4 times
slower than the finite difference WENO-SW in two dimensional simulations. The current study clearly shows that besides high-order reconstruction, an accurate gas evolution model or flux function in a high-order scheme is also important in the capturing of physical solutions. In a physical flow, the transport, stress deformation, heat conduction, and viscous heating are all coupled in a single gas evolution process. Therefore,
it is preferred to develop such a scheme with multi-dimensionality, and unified treatment of inviscid and dissipative terms. A high-order scheme does prefer a high-order
gas evolution model. Even with the rapid advances of high-order reconstruction techniques, the first-order dynamics of the Riemann solution becomes the bottleneck for
the further development of high-order schemes. In order to avoid the weakness of the
low order flux function, the development of high-order schemes relies heavily on the
weak solution of the original governing equations for the update of additional degree
of freedom, such as the non-conservative gradients of flow variables, which cannot be
physically valid in discontinuous regions. 相似文献
16.
A Novel Iterative Penalty Method to Enforce Boundary Conditions in Finite Volume POD-Galerkin Reduced Order Models for Fluid Dynamics Problems 下载免费PDF全文
S. Kelbij Star Giovanni Stabile Francesco Belloni Gianluigi Rozza & Joris Degroote 《Communications In Computational Physics》2021,30(1):34-66
A Finite-Volume based POD-Galerkin reduced order model is developed for
fluid dynamics problems where the (time-dependent) boundary conditions are controlled using two different boundary control strategies: the lifting function method,
whose aim is to obtain homogeneous basis functions for the reduced basis space and
the penalty method where the boundary conditions are enforced in the reduced order
model using a penalty factor. The penalty method is improved by using an iterative
solver for the determination of the penalty factor rather than tuning the factor with a
sensitivity analysis or numerical experimentation.The boundary control methods are compared and tested for two cases: the classical lid driven cavity benchmark problem and a Y-junction flow case with two inlet
channels and one outlet channel. The results show that the boundaries of the reduced
order model can be controlled with the boundary control methods and the same order
of accuracy is achieved for the velocity and pressure fields. Finally, the reduced order
models are 270-308 times faster than the full order models for the lid driven cavity test
case and 13-24 times for the Y-junction test case. 相似文献
17.
A Hermite WENO Method with Modified Ghost Fluid Method for Compressible Two-Medium Flow Problems 下载免费PDF全文
Zhuang Zhao Yong-Tao Zhang Yibing Chen & Jianxian Qiu 《Communications In Computational Physics》2021,30(3):851-873
In this paper, we develop a novel approach by combining a new robust finite difference Hermite weighted essentially non-oscillatory (HWENO) method [51]
with the modified ghost fluid method (MGFM) [25] to simulate the compressible two-medium flow problems. The main idea is that we first use the technique of the MGFM
to transform a two-medium flow problem to two single-medium cases by defining the
ghost fluids status based on the predicted interface status. Then the efficient and robust
HWENO finite difference method is applied for solving the single-medium flow cases.
By using immediate neighbor information to deal with both the solution and its derivatives, the fifth order finite difference HWENO scheme adopted in this paper is more
compact and has higher resolution than the classical fifth order finite difference WENO
scheme of Jiang and Shu [14]. Furthermore, by combining the HWENO scheme with
the MGFM to simulate the two-medium flow problems, less ghost point information
is needed than that in using the classical WENO scheme in order to obtain the same
numerical accuracy. Various one-dimensional and two-dimensional two-medium flow
problems are solved to illustrate the good performances of the proposed method. 相似文献
18.
A Bilinear Immersed Finite Volume Element Method for the Diffusion Equation with Discontinuous Coefficient 下载免费PDF全文
This paper is to present a finite volume element (FVE) method based on the
bilinear immersed finite element (IFE) for solving the boundary value problems of the
diffusion equation with a discontinuous coefficient (interface problem). This method
possesses the usual FVE method's local conservation property and can use a structured
mesh or even the Cartesian mesh to solve a boundary value problem whose coefficient
has discontinuity along piecewise smooth nontrivial curves. Numerical examples are
provided to demonstrate features of this method. In particular, this method can produce
a numerical solution to an interface problem with the usual O(h2) (in L2 norm)
and O(h) (in H1 norm) convergence rates. 相似文献
19.
A Geometry-Preserving Finite Volume Method for Compressible Fluids on Schwarzschild Spacetime 下载免费PDF全文
We consider the relativistic Euler equations governing spherically symmetric, perfect fluid flows on the outer domain of communication of Schwarzschild spacetime, and we introduce a version of the finite volume method which is formulated
from the geometric formulation (and thus takes the geometry into account at the discretization level) and is well-balanced, in the sense that it preserves steady solutions to
the Euler equations on the curved geometry under consideration. In order to formulate our method, we first derive a closed formula describing all steady and spherically
symmetric solutions to the Euler equations posed on Schwarzschild spacetime. Second, we describe a geometry-preserving, finite volume method which is based on the family of steady solutions to the Euler system. Our scheme is second-order accurate and, as required, preserves the family of steady solutions at the discrete level.
Numerical experiments are presented which demonstrate the efficiency and robustness of the proposed method even for solutions containing shock waves and nonlinear
interacting wave patterns. As an application, we investigate the late-time asymptotics
of perturbed steady solutions and demonstrate its convergence for late time toward
another steady solution, taking the overall effect of the perturbation into account. 相似文献
20.
Conservative Semi-Lagrangian Finite Difference WENO Formulations with Applications to the Vlasov Equation 下载免费PDF全文
In this paper, we propose a new conservative semi-Lagrangian (SL) finite
difference (FD) WENO scheme for linear advection equations, which can serve as a
base scheme for the Vlasov equation by Strang splitting [4]. The reconstruction procedure
in the proposed SL FD scheme is the same as the one used in the SL finite volume
(FV) WENO scheme [3]. However, instead of inputting cell averages and approximate
the integral form of the equation in a FV scheme, we input point values and approximate
the differential form of equation in a FD spirit, yet retaining very high order
(fifth order in our experiment) spatial accuracy. The advantage of using point values,
rather than cell averages, is to avoid the second order spatial error, due to the shearing
in velocity (v) and electrical field (E) over a cell when performing the Strang splitting
to the Vlasov equation. As a result, the proposed scheme has very high spatial accuracy,
compared with second order spatial accuracy for Strang split SL FV scheme for
solving the Vlasov-Poisson (VP) system. We perform numerical experiments on linear
advection, rigid body rotation problem; and on the Landau damping and two-stream
instabilities by solving the VP system. For comparison, we also apply (1) the conservative
SL FD WENO scheme, proposed in [22] for incompressible advection problem, (2)
the conservative SL FD WENO scheme proposed in [21] and (3) the non-conservative
version of the SL FD WENO scheme in [3] to the same test problems. The performances
of different schemes are compared by the error table, solution resolution of sharp interface,
and by tracking the conservation of physical norms, energies and entropies,
which should be physically preserved. 相似文献