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1.
Arbitrarily High-Order (Weighted) Essentially Non-Oscillatory Finite Difference Schemes for Anelastic Flows on Staggered Meshes
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Siddhartha Mishra Carlos Paré s-Pulido & Kyle G. Pressel 《Communications In Computational Physics》2021,29(5):1299-1335
We propose a WENO finite difference scheme to approximate anelastic flows,
and scalars advected by them, on staggered grids. In contrast to existing WENO
schemes on staggered grids, the proposed scheme is designed to be arbitrarily high-order accurate as it judiciously combines ENO interpolations of velocities with WENO
reconstructions of spatial derivatives. A set of numerical experiments are presented
to demonstrate the increase in accuracy and robustness with the proposed scheme,
when compared to existing WENO schemes and state-of-the-art central finite difference schemes. 相似文献
2.
High Order Numerical Simulation of Detonation Wave Propagation Through Complex Obstacles with the Inverse Lax-Wendroff Treatment
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Cheng Wang Jianxu Ding Sirui Tan & Wenhu Han 《Communications In Computational Physics》2015,18(5):1264-1281
The high order inverse Lax-Wendroff (ILW) procedure is extended to boundary
treatment involving complex geometries on a Cartesian mesh. Our method ensures
that the numerical resolution at the vicinity of the boundary and the inner domain
keeps the fifth order accuracy for the system of the reactive Euler equations with the
two-step reaction model. Shock wave propagation in a tube with an array of rectangular
grooves is first numerically simulated by combining a fifth order weighted essentially
non-oscillatory (WENO) scheme and the ILW boundary treatment. Compared
with the experimental results, the ILW treatment accurately captures the evolution of
shock wave during the interactions of the shock waves with the complex obstacles.
Excellent agreement between our numerical results and the experimental ones further
demonstrates the reliability and accuracy of the ILW treatment. Compared with the
immersed boundary method (IBM), it is clear that the influence on pressure peaks in
the reflected zone is obviously bigger than that in the diffracted zone. Furthermore,
we also simulate the propagation process of detonation wave in a tube with three different
widths of wall-mounted rectangular obstacles located on the lower wall. It is
shown that the shock pressure along a horizontal line near the rectangular obstacles
gradually decreases, and the detonation cellular size becomes large and irregular with
the decrease of the obstacle width. 相似文献
3.
Comparison of Fifth-Order WENO Scheme and Finite Volume WENO-Gas-Kinetic Scheme for Inviscid and Viscous Flow Simulation
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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
4.
A Class of Hybrid DG/FV Methods for Conservation Laws III: Two-Dimensional Euler Equations
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Laiping Zhang Wei Liu Lixin He & Xiaogang Deng 《Communications In Computational Physics》2012,12(1):284-314
A concept of "static reconstruction" and "dynamic reconstruction" was introduced for higher-order (third-order or more) numerical methods in our previous
work. Based on this concept, a class of hybrid DG/FV methods had been developed
for one-dimensional conservation law using a "hybrid reconstruction" approach, and
extended to two-dimensional scalar equations on triangular and Cartesian/triangular
hybrid grids. In the hybrid DG/FV schemes, the lower-order derivatives of the piecewise polynomial are computed locally in a cell by the traditional DG method (called
as "dynamic reconstruction"), while the higher-order derivatives are reconstructed by
the "static reconstruction" of the FV method, using the known lower-order derivatives
in the cell itself and in its adjacent neighboring cells. In this paper, the hybrid DG/FV
schemes are extended to two-dimensional Euler equations on triangular and Cartesian/triangular hybrid grids. Some typical test cases are presented to demonstrate
the performance of the hybrid DG/FV methods, including the standard vortex evolution problem with exact solution, isentropic vortex/weak shock wave interaction,
subsonic flows past a circular cylinder and a three-element airfoil (30P30N), transonic
flow past a NACA0012 airfoil. The accuracy study shows that the hybrid DG/FV
method achieves the desired third-order accuracy, and the applications demonstrate
that they can capture the flow structure accurately, and can reduce the CPU time and
memory requirement greatly than the traditional DG method with the same order of
accuracy. 相似文献
5.
An Adaptive Cartesian Method for Prediction of the Unsteady Process of Aircraft Ice Accretion
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Chengxiang Zhu Huanyu Zhao Ning Zhao Chunling Zhu & Yu Liu 《Communications In Computational Physics》2021,30(2):515-535
An adaptive Cartesian method combined with the ghost cell method is proposed to solve problems of unsteady aircraft icing simulation in this paper. In the grid
generation module, Cartesian method is used to generate the background grids around
the clean and iced geometries. Boundary self-adaptive method is developed to update
the grids as the ice accumulates and the geometry is changing over time. Besides, local
encryption is carried out around the boundary grids in order to improve the prediction accuracy. An improved ray method is used to classify the grids into four types,
named fluid grids, solid grids, boundary grids and ghost grids. The flow field is obtained by solving Euler equations, and the ghost cell method is introduced to provide
the boundary conditions due to the non-body-fitted feature of Cartesian grids. Droplet
trajectories are calculated using Lagrangian method. And a new and efficient droplet
location judgment method is proposed to determine whether the droplet impinges on
the surface. Besides, the ice accretion behaviors are predicted using Messinger model.
With the method proposed in this paper, extensive numerical tests in various icing
temperatures are simulated. And, the computational results are compared with test
results. It can be seen that the proposed new method can predict the unsteady process
of aircraft ice accretion and the compared results also show better agreements for both
glaze and rime ice. 相似文献
6.
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
7.
Angelo L. Scandaliato & Meng-Sing Liou 《Communications In Computational Physics》2012,12(4):1096-1120
In this paper we demonstrate the accuracy and robustness of combining the
advection upwind splitting method (AUSM), specifically AUSM+-UP [9], with high-order upwind-biased interpolation procedures, the weighted essentially non-oscillatory
(WENO-JS) scheme [8] and its variations [2, 7], and the monotonicity preserving (MP)
scheme [16], for solving the Euler equations. MP is found to be more effective than the
three WENO variations studied. AUSM+-UP is also shown to be free of the so-called "carbuncle" phenomenon with the high-order interpolation. The characteristic variables are preferred for interpolation after comparing the results using primitive and
conservative variables, even though they require additional matrix-vector operations.
Results using the Roe flux with an entropy fix and the Lax-Friedrichs approximate Riemann solvers are also included for comparison. In addition, four reflective boundary
condition implementations are compared for their effects on residual convergence and
solution accuracy. Finally, a measure for quantifying the efficiency of obtaining high
order solutions is proposed; the measure reveals that a maximum return is reached
after which no improvement in accuracy is possible for a given grid size. 相似文献
8.
A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws
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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
9.
High-Order Gas-Kinetic Scheme in Curvilinear Coordinates for the Euler and Navier-Stokes Solutions
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Liang Pan & Kun Xu 《Communications In Computational Physics》2020,28(4):1321-1351
The high-order gas-kinetic scheme (HGKS) has achieved success in simulating compressible flows with Cartesian meshes. To study the flow problems in general
geometries, such as the flow over a wing-body, the development of HGKS in general
curvilinear coordinates becomes necessary. In this paper, a two-stage fourth-order gas-kinetic scheme is developed for the Euler and Navier-Stokes solutions in the curvilinear coordinates from one-dimensional to three-dimensional computations. Based on
the coordinate transformation, the kinetic equation is transformed first to the computational space, and the flux function in the gas-kinetic scheme is obtained there and
is transformed back to the physical domain for the update of flow variables inside
each control volume. To achieve the expected order of accuracy, the dimension-by-dimension reconstruction based on the WENO scheme is adopted in the computational domain, where the reconstructed variables are the cell averaged Jacobian and the
Jacobian-weighted conservative variables. In the two-stage fourth-order gas-kinetic
scheme, the point values as well as the spatial derivatives of conservative variables at
Gaussian quadrature points have to be used in the evaluation of the time dependent
flux function. The point-wise conservative variables are obtained by ratio of the above
reconstructed data, and the spatial derivatives are reconstructed through orthogonalization in physical space and chain rule. A variety of numerical examples from the
accuracy tests to the solutions with strong discontinuities are presented to validate the
accuracy and robustness of the current scheme for both inviscid and viscous flows.
The precise satisfaction of the geometrical conservation law in non-orthogonal mesh is
also demonstrated through the numerical example. 相似文献
10.
A Lattice Boltzmann and Immersed Boundary Scheme for Model Blood Flow in Constricted Pipes: Part 1 – Steady Flow
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S. C. Fu W. W. F. Leung & R. M. C. So 《Communications In Computational Physics》2013,14(1):126-152
Hemodynamics is a complex problem with several distinct characteristics;
fluid is non-Newtonian, flow is pulsatile in nature, flow is three-dimensional due to
cholesterol/plague built up, and blood vessel wall is elastic. In order to simulate this
type of flows accurately, any proposed numerical scheme has to be able to replicate
these characteristics correctly, efficiently, as well as individually and collectively. Since
the equations of the finite difference lattice Boltzmann method (FDLBM) are hyperbolic, and can be solved using Cartesian grids locally, explicitly and efficiently on parallel computers, a program of study to develop a viable FDLBM numerical scheme
that can mimic these characteristics individually in any model blood flow problem
was initiated. The present objective is to first develop a steady FDLBM with an immersed boundary (IB) method to model blood flow in stenoic artery over a range of
Reynolds numbers. The resulting equations in the FDLBM/IB numerical scheme can
still be solved using Cartesian grids; thus, changing complex artery geometry can be
treated without resorting to grid generation. The FDLBM/IB numerical scheme is validated against known data and is then used to study Newtonian and non-Newtonian
fluid flow through constricted tubes. The investigation aims to gain insight into the
constricted flow behavior and the non-Newtonian fluid effect on this behavior. 相似文献
11.
Simulation of Two-Dimensional Scramjet Combustor Reacting Flow Field Using Reynolds Averaged Navier-Stokes WENO Solver
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Juan-Chen Huang Yu-Hsuan Lai Jeng-Shan Guo & Jaw-Yen Yang 《Communications In Computational Physics》2015,18(4):1181-1210
The non-equilibrium chemical reacting combustion flows of a proposed long
slender scramjet system were numerically studied by solving the turbulent Reynolds
averaged Navier-Stokes (RANS) equations. The Spalart-Allmaras one equation turbulence
model is used which produces better results for near wall and boundary layer
flow field problems. The lower-upper symmetric Gauss-Seidel implicit scheme, which
enables results converge efficiently under steady state condition, is combined with the
weighted essentially non-oscillatory (WENO) scheme to yield an accurate simulation
tool for scramjet combustion flow field analysis. Using the WENO schemes high-order
accuracy and its non-oscillatory solution at flow discontinuities, better resolution of
the hypersonic flow problems involving complex shock-shock/shock-boundary layer
interactions inside the flow path, can be achieved. Two types of scramjet combustor
with cavity-based and strut-based fuel injector were considered as the testing models.
The flow characteristics with and without combustion reactions of the two types of combustor models were studied with a transient hydrogen/oxygen combustion model.
The detailed results of aerodynamic data are obtained and discussed, moreover, the
combustion properties of varying the equivalent ratio of hydrogen, including the concentration
of reacting species, hydrogen and oxygen, and the reacting products, water,
are demonstrated to study the combustion process and performance of the combustor.
The comparisons of flow field structures, pressure on wall and velocity profiles
between the experimental data and the solutions of the present algorithms, showed
qualitatively as well as the quantitatively in good agreement, and validated the adequacy
of the present simulation tool for hypersonic scramjet reacting flow analysis. 相似文献
12.
A Strong Stability-Preserving Predictor-Corrector Method for the Simulation of Elastic Wave Propagation in Anisotropic Media
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In this paper, we propose a strong stability-preserving predictor-corrector
(SSPC) method based on an implicit Runge-Kutta method to solve the acoustic- and
elastic-wave equations. We first transform the wave equations into a system of ordinary differential equations (ODEs) and apply the local extrapolation method to discretize the spatial high-order derivatives, resulting in a system of semi-discrete ODEs.
Then we use the SSPC method based on an implicit Runge-Kutta method to solve
the semi-discrete ODEs and introduce a weighting parameter into the SSPC method.
On top of such a structure, we develop a robust numerical algorithm to effectively
suppress the numerical dispersion, which is usually caused by the discretization of
wave equations when coarse grids are used or geological models have large velocity
contrasts between adjacent layers. Meanwhile, we investigate the performance of the
SSPC method including numerical errors and convergence rate, numerical dispersion,
and stability criteria with different choices of the weighting parameter to solve 1-D
and 2-D acoustic- and elastic-wave equations. When the SSPC is applied to seismic
simulations, the computational efficiency is also investigated by comparing the SSPC,
the fourth-order Lax-Wendroff correction (LWC) method, and the staggered-grid (SG)
finite difference method. Comparisons of synthetic waveforms computed by the SSPC
and analytic solutions for acoustic and elastic models are given to illustrate the accuracy and the validity of the SSPC method. Furthermore, several numerical experiments
are conducted for the geological models including a 2-D homogeneous transversely
isotropic (TI) medium, a two-layer elastic model, and the 2-D SEG/EAGE salt model.
The results show that the SSPC can be used as a practical tool for large-scale seismic
simulation because of its effectiveness in suppressing numerical dispersion even in the
situations such as coarse grids, strong interfaces, or high frequencies. 相似文献
13.
A NURBS-Enhanced Finite Volume Method for Steady Euler Equations with Goal-Oriented $h$-Adaptivity
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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 volume
method was developed to solve the steady Euler equations, in which the desired high
order numerical accuracy was obtained for the equations imposed in the domain with
a 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 compute
the 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 introduced
for the steady Euler equations in a complex domain, and (iii) a goal-oriented a posteriori
error indicator is designed to further improve the efficiency of the algorithm towards
accurately calculating a given quantity of interest. Numerical results obtained from a
variety of numerical experiments with different flow configurations successfully show
the effectiveness and robustness of the proposed method. 相似文献
14.
High Order Fixed-Point Sweeping WENO Methods for Steady State of Hyperbolic Conservation Laws and Its Convergence Study
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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
15.
Adaptive Order WENO Reconstructions for the Semi-Lagrangian Finite Difference Scheme for Advection Problem
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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
16.
Piecewise Polynomial Mapping Method and Corresponding WENO Scheme with Improved Resolution
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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
17.
An Implicit Algorithm for High-Order DG/FV Schemes for Compressible Flows on 2D Arbitrary Grids
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Laiping Zhang Ming Li Wei Liu & Xin He 《Communications In Computational Physics》2015,17(1):287-316
A Newton/LU-SGS (lower-upper symmetric Gauss-Seidel) iteration implicit
method was developed to solve two-dimensional Euler and Navier-Stokes equations
by the DG/FV hybrid schemes on arbitrary grids. The Newton iteration was employed
to solve the nonlinear system, while the linear system was solved with LU-SGS iteration.
The effect of several parameters in the implicit scheme, such as the CFL number,
the Newton sub-iteration steps, and the update frequency of Jacobian matrix, was investigated
to evaluate the performance of convergence history. Several typical test
cases were simulated, and compared with the traditional explicit Runge-Kutta (RK)
scheme. Firstly the Couette flow was tested to validate the order of accuracy of the
present DG/FV hybrid schemes. Then a subsonic inviscid flow over a bump in a channel
was simulated and the effect of parameters was alsoinvestigated. Finally, the implicit
algorithm was applied to simulate a subsonic inviscid flow over a circular cylinder
and the viscous flow in a square cavity. The numerical results demonstrated that
the present implicit scheme can accelerate the convergence history efficiently. Choosing
proper parameters would improve the efficiency of the implicit scheme. Moreover,
in the same framework, the DG/FV hybrid schemes are more efficient than the same
order DG schemes. 相似文献
18.
A Compact Third-Order Gas-Kinetic Scheme for Compressible Euler and Navier-Stokes Equations
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In this paper, a compact third-order gas-kinetic scheme is proposed for the
compressible Euler and Navier-Stokes equations. The main reason for the feasibility
to develop such a high-order scheme with compact stencil, which involves only
neighboring cells, is due to the use of a high-order gas evolution model. Besides the
evaluation of the time-dependent flux function across a cell interface, the high-order
gas evolution model also provides an accurate time-dependent solution of the flow
variables at a cell interface. Therefore, the current scheme not only updates the cell
averaged conservative flow variables inside each control volume, but also tracks the
flow variables at the cell interface at the next time level. As a result, with both cell averaged
and cell interface values, the high-order reconstruction in the current scheme
can be done compactly. Different from using a weak formulation for high-order accuracy
in the Discontinuous Galerkin method, the current scheme is based on the strong
solution, where the flow evolution starting from a piecewise discontinuous high-order
initial data is precisely followed. The cell interface time-dependent flow variables can
be used for the initial data reconstruction at the beginning of next time step. Even with
compact stencil, the current scheme has third-order accuracy in the smooth flow regions,
and has favorable shock capturing property in the discontinuous regions. It can
be faithfully used from the incompressible limit to the hypersonic flow computations,
and many test cases are used to validate the current scheme. In comparison with many
other high-order schemes, the current method avoids the use of Gaussian points for
the flux evaluation along the cell interface and the multi-stage Runge-Kutta time stepping
technique. Due to its multidimensional property of including both derivatives of
flow variables in the normal and tangential directions of a cell interface, the viscous
flow solution, especially those with vortex structure, can be accurately captured. With
the same stencil of a second order scheme, numerical tests demonstrate that the current
scheme is as robust as well-developed second-order shock capturing schemes, but
provides more accurate numerical solutions than the second order counterparts. 相似文献
19.
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
20.
Chaoqun Liu Ping Lu Maria Oliveira & Peng Xie 《Communications In Computational Physics》2012,11(3):1022-1042
Standard compact scheme and upwinding compact scheme have high order accuracy and high resolution, but cannot capture the shock which is a discontinuity. This work developed a modified upwinding compact scheme which uses an effective shock detector to block compact scheme to cross the shock and a control function to mix the flux with WENO scheme near the shock. The new scheme makes the original compact scheme able to capture the shock sharply and, more importantly, keep high order accuracy and high resolution in the smooth area which is particularly important for shock boundary layer and shock acoustic interactions. Numerical results show the scheme is successful for 2-D Euler and 2-D Navier-Stokes solvers. The examples include 2-D incident shock, 2-D incident shock and boundary layer interaction. The scheme is robust, which does not involve case related parameters. 相似文献