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1.
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 essentially
non-oscillatory (WENO) schemes with unequal-sized sub-stencils for solving the
Degasperis-Procesi (DP) and $\mu$-Degasperis-Procesi ($\mu$DP) equations, which contain
nonlinear high order derivatives, and possibly peakon solutions or shock waves. By
introducing auxiliary variable(s), we rewrite the DP equation as a hyperbolic-elliptic
system, and the $\mu$DP equation as a first order system. Then we choose a linear finite
difference scheme with suitable order of accuracy for the auxiliary variable(s), and
two finite difference WENO schemes with unequal-sized sub-stencils for the primal
variable. One WENO scheme uses one large stencil and several smaller stencils, and
the other WENO scheme is based on the multi-resolution framework which uses a series of unequal-sized hierarchical central stencils. Comparing with the classical WENO
scheme which uses several small stencils of the same size to make up a big stencil, both
WENO schemes with unequal-sized sub-stencils are simple in the choice of the stencil
and enjoy the freedom of arbitrary positive linear weights. Another advantage is that
the 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 WENO
reconstruction can only be obtained for specific points inside the target interval. Numerical tests are provided to demonstrate the high order accuracy and non-oscillatory
properties of the proposed schemes. 相似文献
2.
Yifei Wan & Yinhua Xia 《Communications In Computational Physics》2023,33(5):1270-1331
For steady Euler equations in complex boundary domains, high-order shockcapturing schemes usually suffer not only from the difficulty of steady-state convergence but also from the problem of dealing with physical boundaries on Cartesian
grids to achieve uniform high-order accuracy. In this paper, we utilize a fifth-order
finite difference hybrid WENO scheme to simulate steady Euler equations, and the
same fifth-order WENO extrapolation methods are developed to handle the curved
boundary. The values of the ghost points outside the physical boundary can be obtained by applying WENO extrapolation near the boundary, involving normal derivatives acquired by the simplified inverse Lax-Wendroff procedure. Both equivalent expressions involving curvature and numerical differentiation are utilized to transform
the tangential derivatives along the curved solid wall boundary. This hybrid WENO
scheme is robust for steady-state convergence and maintains high-order accuracy in
the smooth region even with the solid wall boundary condition. Besides, the essentially non-oscillation property is achieved. The numerical spectral analysis also shows
that this hybrid WENO scheme has low dispersion and dissipation errors. Numerical
examples are presented to validate the high-order accuracy and robust performance of
the hybrid scheme for steady Euler equations in curved domains with Cartesian grids. 相似文献
3.
A Sufficient and Necessary Condition of the Existence of WENO-Like Linear Combination for Finite Difference Schemes 下载免费PDF全文
Jian Kang & Xinliang Li 《Communications In Computational Physics》2021,29(2):534-570
In the finite difference WENO (weighted essentially non-oscillatory) method, the final scheme on the whole stencil was constructed by linear combinations of
highest order accurate schemes on sub-stencils, all of which share the same total count
of grid points. The linear combination method which the original WENO applied was
generalized to arbitrary positive-integer-order derivative on an arbitrary (uniform or
non-uniform) mesh, still applying finite difference method. The possibility of expressing the final scheme on the whole stencil as a linear combination of highest order accurate schemes on WENO-like sub-stencils was investigated. The main results include:
(a) the highest order of accuracy a finite difference scheme can achieve and (b) a sufficient and necessary condition that the linear combination exists. This is a sufficient
and necessary condition for all finite difference schemes in a set (rather than a specific
finite difference scheme) to have WENO-like linear combinations. After the proofs
of the results, some remarks on the WENO schemes and TENO (targeted essentially
non-oscillatory) schemes were given. 相似文献
4.
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
5.
We propose an a-posteriori error/smoothness indicator for standard semi-discrete
finite volume schemes for systems of conservation laws, based on the numerical
production of entropy. This idea extends previous work by the first author limited
to central finite volume schemes on staggered grids. We prove that the indicator converges
to zero with the same rate of the error of the underlying numerical scheme on
smooth flows under grid refinement. We construct and test an adaptive scheme for
systems of equations in which the mesh is driven by the entropy indicator. The adaptive
scheme uses a single nonuniform grid with a variable timestep. We show how
to implement a second order scheme on such a space-time non uniform grid, preserving
accuracy and conservation properties. We also give an example of a p-adaptive
strategy. 相似文献
6.
A Simple Solver for the Two-Fluid Plasma Model Based on PseudoSpectral Time-Domain Algorithm 下载免费PDF全文
Benoit Morel Remo Giust Kazem Ardaneh & Francois Courvoisier 《Communications In Computational Physics》2021,29(3):955-978
We present a solver of 3D two-fluid plasma model for the simulation of
short-pulse laser interactions with plasma. This solver resolves the equations of the
two-fluid plasma model with ideal gas closure. We also include the Bhatnagar-Gross-Krook collision model. Our solver is based on PseudoSpectral Time-Domain (PSTD)
method to solve Maxwell's curl equations. We use a Strang splitting to integrate Euler
equations with source term: while Euler equations are solved with a composite scheme
mixing Lax-Friedrichs and Lax-Wendroff schemes, the source term is integrated with
a fourth-order Runge-Kutta scheme. This two-fluid plasma model solver is simple to
implement because it only relies on finite difference schemes and Fast Fourier Transforms. It does not require spatially staggered grids. The solver was tested against
several well-known problems of plasma physics. Numerical simulations gave results
in excellent agreement with analytical solutions or with previous results from the literature. 相似文献
7.
Guoliang Zhang & Tao Xiong 《Communications In Computational Physics》2022,32(1):126-155
We propose a high order finite difference linear scheme combined with a
high order bound preserving maximum-principle-preserving (MPP) flux limiter to
solve the incompressible flow system. For such problem with highly oscillatory structure but not strong shocks, our approach seems to be less dissipative and much less
costly than a WENO type scheme, and has high resolution due to a Hermite reconstruction. Spurious numerical oscillations can be controlled by the weak MPP flux
limiter. Numerical tests are performed for the Vlasov-Poisson system, the 2D guiding-center model and the incompressible Euler system. The comparison between the linear
and WENO type schemes, with and without the MPP flux limiter, will demonstrate the
good performance of our proposed approach. 相似文献
8.
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
9.
An Explicit MUSCL Scheme on Staggered Grids with Kinetic-Like Fluxes for the Barotropic and Full Euler System 下载免费PDF全文
Thierry Goudon Julie Llobell & Sebastian Minjeaud 《Communications In Computational Physics》2020,27(3):672-724
We present a second order scheme for the barotropic and full Euler equations. The scheme works on staggered grids, with numerical unknowns stored at dual locations, while the numerical fluxes are derived in the spirit of kinetic schemes. We identify stability conditions ensuring the positivity of the discrete density and energy. We illustrate the ability of the scheme to capture the structure of complex flows with 1D and 2D simulations on MAC grids. 相似文献
10.
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
11.
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
12.
This paper presents a new and better suited formulation to implement the
limiting projection to high-order schemes that make use of high-order local reconstructions
for hyperbolic conservation laws. The scheme, so-called MCV-WENO4 (multi-moment
Constrained finite Volume with WENO limiter of 4th order) method, is an
extension of the MCV method of Ii & Xiao (2009) by adding the 1st order derivative
(gradient or slope) at the cell center as an additional constraint for the cell-wise local
reconstruction. The gradient is computed from a limiting projection using the WENO
(weighted essentially non-oscillatory) reconstruction that is built from the nodal values
at 5 solution points within 3 neighboring cells. Different from other existing methods
where only the cell-average value is used in the WENO reconstruction, the present
method takes account of the solution structure within each mesh cell, and thus minimizes
the stencil for reconstruction. The resulting scheme has 4th-order accuracy and
is of significant advantage in algorithmic simplicity and computational efficiency. Numerical
results of one and two dimensional benchmark tests for scalar and Euler conservation
laws are shown to verify the accuracy and oscillation-less property of the
scheme. 相似文献
13.
In this paper, we introduce a new type of troubled-cell indicator to improve
hybrid weighted essentially non-oscillatory (WENO) schemes for solving the hyperbolic conservation laws. The hybrid WENO schemes selectively adopt the high-order
linear 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 in
non-smooth regions. Reliable troubled-cell indicators are essential for efficient hybrid
WENO methods. Most of troubled-cell indicators require proper parameters to detect
discontinuities precisely, but it is very difficult to determine the parameters automatically. We develop a new troubled-cell indicator derived from the mean value theorem
that 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 are
conducted to demonstrate the performance of the proposed indicator. The results with
the proposed troubled-cell indicator are in good agreement with pure WENO schemes.
Also the new indicator has advantages in the computational cost compared with the
other indicators. 相似文献
14.
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
15.
Comparative Study of Three High Order Schemes for LES of Temporally Evolving Mixing Layers 下载免费PDF全文
Helen C. Yee Bjorn Sjö green & Abdellah Hadjadj 《Communications In Computational Physics》2012,12(5):1603-1622
Three high order shock-capturing schemes are compared for large eddy
simulations (LES) of temporally evolving mixing layers for different convective Mach
numbers ranging from the quasi-incompressible regime to highly compressible supersonic regime. The considered high order schemes are fifth-order WENO (WENO5),
seventh-order WENO (WENO7) and the associated eighth-order central spatial base
scheme with the dissipative portion of WENO7 as a nonlinear post-processing filter
step (WENO7fi). This high order nonlinear filter method of Yee & Sjogreen is designed for accurate and efficient simulations of shock-free compressible turbulence,
turbulence with shocklets and turbulence with strong shocks with minimum tuning
of scheme parameters. The LES results by WENO7fi using the same scheme parameter agree well with experimental results compiled by Barone et al., and published
direct numerical simulations (DNS) work of Rogers & Moser and Pantano & Sarkar,
whereas results by WENO5 and WENO7 compare poorly with experimental data and
DNS computations. 相似文献
16.
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
17.
Ruo Li & Wei Zhong 《Communications In Computational Physics》2022,31(2):548-592
Existing mapped WENO schemes can hardly prevent spurious oscillations
while preserving high resolutions at long output times. We reveal in this paper the essential reason of such phenomena. It is actually caused by that the mapping function
in these schemes can not preserve the order of the nonlinear weights of the stencils.
The nonlinear weights may be increased for non-smooth stencils and be decreased for
smooth stencils. It is then indicated to require the set of mapping functions to be order-preserving in mapped WENO schemes. Therefore, we propose a new mapped WENO
scheme with a set of mapping functions to be order-preserving which exhibits a remarkable advantage over the mapped WENO schemes in references. For long output
time simulations of the one-dimensional linear advection equation, the new scheme
has the capacity to attain high resolutions and avoid spurious oscillations near discontinuities meanwhile. In addition, for the two-dimensional Euler problems with strong
shock waves, the new scheme can significantly reduce the numerical oscillations. 相似文献
18.
A Simple 3D Immersed Interface Method for Stokes Flow with Singular Forces on Staggered Grids 下载免费PDF全文
Weiyi Wang & Zhijun Tan 《Communications In Computational Physics》2021,30(1):227-254
In this paper, a fairly simple 3D immersed interface method based on the
CG-Uzawa type method and the level set representation of the interface is employed
for solving three-dimensional Stokes flow with singular forces along the interface. The
method is to apply the Taylor's expansions only along the normal direction and incorporate the jump conditions up to the second normal derivatives into the finite difference schemes. A second order geometric iteration algorithm is employed for computing orthogonal projections on the surface with third-order accuracy. The Stokes
equations are discretized involving the correction terms on staggered grids and then
solved by the conjugate gradient Uzawa type method. The major advantages of the
present method are the special simplicity, the ability in handling the Dirichlet boundary conditions, and no need of the pressure boundary condition. The method can
also preserve the volume conservation and the discrete divergence free condition very
well. The numerical results show that the proposed method is second order accurate
and efficient. 相似文献
19.
Piecewise Polynomial Mapping Method and Corresponding WENO Scheme with Improved Resolution 下载免费PDF全文
The method of mapping function was first proposed by Henrick et al. [J.
Comput. Phys. 207:542-547 (2005)] to adjust nonlinear weights in [0,1] for the fifth-order
WENO scheme, and through which the requirement of convergence order is
satisfied and the performance of the scheme is improved. Different from Henrick's
method, a concept of piecewise polynomial function is proposed in this study and
corresponding WENO schemes are obtained. The advantage of the new method is
that the function can have a gentle profile at the location of the linear weight (or the
mapped nonlinear weight can be close to its linear counterpart), and therefore is favorable
for the resolution enhancement. Besides, the function also has the flexibility
of quick convergence to identity mapping near two endpoints of [0,1], which is favorable
for improved numerical stability. The fourth-, fifth- and sixth-order polynomial
functions are constructed correspondingly with different emphasis on aforementioned
flatness and convergence. Among them, the fifth-order version has the flattest profile.
To check the performance of the methods, the 1-D Shu-Osher problem, the 2-D
Riemann problem and the double Mach reflection are tested with the comparison of
WENO-M, WENO-Z and WENO-NS. The proposed new methods show the best resolution
for describing shear-layer instability of the Riemann problem, and they also
indicate high resolution in computations of double Mach reflection, where only these
proposed schemes successfully resolved the vortex-pairing phenomenon. Other investigations
have shown that the single polynomial mapping function has no advantage
over the proposed piecewise one, and it is of no evident benefit to use the proposed
method for the symmetric fifth-order WENO. Overall, the fifth-order piecewise polynomial
and corresponding WENO scheme are suggested for resolution improvement. 相似文献
20.
Xiaogang Deng Meiliang Mao Guohua Tu Hanxin Zhang & Yifeng Zhang 《Communications In Computational Physics》2012,11(4):1081-1102
The purpose of this article is to summarize our recent progress in high-order
and high accurate CFD methods for flow problems with complex grids as well as
to discuss the engineering prospects in using these methods. Despite the rapid development of high-order algorithms in CFD, the applications of high-order and high
accurate methods on complex configurations are still limited. One of the main reasons which hinder the widely applications of these methods is the complexity of grids.
Many aspects which can be neglected for low-order schemes must be treated carefully
for high-order ones when the configurations are complex. In order to implement high-order finite difference schemes on complex multi-block grids, the geometric conservation law and block-interface conditions are discussed. A conservative metric method is
applied to calculate the grid derivatives, and a characteristic-based interface condition
is employed to fulfil high-order multi-block computing. The fifth-order WCNS-E-5
proposed by Deng [9, 10] is applied to simulate flows with complex grids, including
a double-delta wing, a transonic airplane configuration, and a hypersonic X-38 configuration. The results in this paper and the references show pleasant prospects in
engineering-oriented applications of high-order schemes. 相似文献