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1.
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
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.
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
4.
This paper develops a high-order accurate gas-kinetic scheme in the framework
of the finite volume method for the one- and two-dimensional flow simulations,
which is an extension of the third-order accurate gas-kinetic scheme [Q.B. Li, K. Xu, and
S. Fu, J. Comput. Phys., 229(2010), 6715-6731] and the second-order accurate gas-kinetic
scheme [K. Xu, J. Comput. Phys., 171(2001), 289-335]. It is formed by two parts: quartic
polynomial reconstruction of the macroscopic variables and fourth-order accurate flux
evolution. The first part reconstructs a piecewise cell-center based quartic polynomial
and a cell-vertex based quartic polynomial according to the "initial" cell average approximation
of macroscopic variables to recover locally the non-equilibrium and equilibrium
single particle velocity distribution functions around the cell interface. It is in
view of the fact that all macroscopic variables become moments of a single particle velocity
distribution function in the gas-kinetic theory. The generalized moment limiter
is employed there to suppress the possible numerical oscillation. In the second part,
the macroscopic flux at the cell interface is evolved in fourth-order accuracy by means
of the simple particle transport mechanism in the microscopic level, i.e. free transport
and the Bhatnagar-Gross-Krook (BGK) collisions. In other words, the fourth-order
flux evolution is based on the solution (i.e. the particle velocity distribution function)
of the BGK model for the Boltzmann equation. Several 1D and 2D test problems are
numerically solved by using the proposed high-order accurate gas-kinetic scheme. By
comparing with the exact solutions or the numerical solutions obtained the second-order
or third-order accurate gas-kinetic scheme, the computations demonstrate that
our scheme is effective and accurate for simulating invisid and viscous fluid flows,
and the accuracy of the high-order GKS depends on the choice of the (numerical) collision
time. 相似文献
5.
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
6.
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
7.
Performance Enhancement for High-Order Gas-Kinetic Scheme Based on WENO-Adaptive-Order Reconstruction
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Xing Ji & Kun Xu 《Communications In Computational Physics》2020,28(2):539-590
High-order gas-kinetic scheme (HGKS) has been well-developed in the past
years. Abundant numerical tests including hypersonic flow, turbulence, and aeroacoustic problems, have been used to validate its accuracy, efficiency, and robustness.
However, there is still room for its further improvement. Firstly, the reconstruction
in the previous scheme mainly achieves a fifth-order accuracy for the point-wise values at a cell interface due to the use of standard WENO reconstruction, and the slopes
of the initial non-equilibrium states have to be reconstructed from the cell interface
values and cell averages again. The same order of accuracy for slopes as the original
WENO scheme cannot be achieved. At the same time, the equilibrium state in space
and time in HGKS has to be reconstructed separately. Secondly, it is complicated to get
reconstructed data at Gaussian points from the WENO-type method in high dimensions. For HGKS, besides the point-wise values at the Gaussian points it also requires
the slopes in both normal and tangential directions of a cell interface. Thirdly, there exists visible spurious overshoot/undershoot at weak discontinuities from the previous
HGKS with the standard WENO reconstruction. In order to overcome these difficulties, in this paper we use an improved reconstruction for HGKS. The WENO with
adaptive order (WENO-AO) [2] method is implemented for reconstruction. Equipped
with WENO-AO reconstruction, the performance enhancement of HGKS is fully explored. WENO-AO not only provides the interface values, but also the slopes. In other
words, a whole polynomial inside each cell is provided by the WENO-AO reconstruction. The available polynomial may not benefit to the high-order schemes based on the
Riemann solver, where only points-wise values at the cell interface are needed. But,
it can be fully utilized in the HGKS. As a result, the HGKS becomes simpler than the
previous one with the direct implementation of cell interface values and their slopes
from WENO-AO. The additional reconstruction of equilibrium state at the beginning
of each time step can be avoided as well by dynamically merging the reconstructed non-equilibrium slopes. The new HGKS essentially releases or totally removes the
above existing problems in the previous HGKS. The accuracy of the scheme from 1D
to 3D from the new HGKS can recover the theoretical order of accuracy of the WENO
reconstruction. In the two- and three-dimensional simulations, the new HGKS shows
better robustness and efficiency than the previous scheme in all test cases. 相似文献
8.
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
9.
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
10.
High-Order Conservative Asymptotic-Preserving Schemes for Modeling Rarefied Gas Dynamical Flows with Boltzmann-BGK Equation
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Manuel A. Diaz Min-Hung Chen & Jaw-Yen Yang 《Communications In Computational Physics》2015,18(4):1012-1049
High-order and conservative phase space direct solvers that preserve the
Euler asymptotic limit of the Boltzmann-BGK equation for modelling rarefied gas
flows are explored and studied. The approach is based on the conservative discrete ordinate
method for velocity space by using Gauss Hermite or Simpsons quadrature rule
and conservation of macroscopic properties are enforced on the BGK collision operator.
High-order asymptotic-preserving time integration is adopted and the spatial evolution
is performed by high-order schemes including a finite difference weighted essentially
non-oscillatory method and correction procedure via reconstruction schemes.
An artificial viscosity dissipative model is introduced into the Boltzmann-BGK equation
when the correction procedure via reconstruction scheme is used. The effects of
the discrete velocity conservative property and accuracy of high-order formulations of
kinetic schemes based on BGK model methods are provided. Extensive comparative
tests with one-dimensional and two-dimensional problems in rarefied gas flows have
been carried out to validate and illustrate the schemes presented. Potentially advantageous
schemes in terms of stable large time step allowed and higher-order of accuracy
are suggested. 相似文献
11.
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
12.
Conservative Semi-Lagrangian Finite Difference WENO Formulations with Applications to the Vlasov Equation
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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
13.
Divergence-Free WENO Reconstruction-Based Finite Volume Scheme for Solving Ideal MHD Equations on Triangular Meshes
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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
14.
Fifth-Order A-WENO Path-Conservative Central-Upwind Scheme for Behavioral Non-Equilibrium Traffic Models
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Shaoshuai Chu Alexander Kurganov Saeed Mohammadian & Zuduo Zheng 《Communications In Computational Physics》2023,33(3):692-732
Non-equilibrium hyperbolic traffic models can be derived as continuum
approximations of car-following models and in many cases the resulting continuum
models are non-conservative. This leads to numerical difficulties, which seem to have
discouraged further development of complex behavioral continuum models, which is
a significant research need.In this paper, we develop a robust numerical scheme that solves hyperbolic traffic
flow models based on their non-conservative form. We develop a fifth-order alternative weighted essentially non-oscillatory (A-WENO) finite-difference scheme based
on the path-conservative central-upwind (PCCU) method for several non-equilibrium
traffic flow models. In order to treat the non-conservative product terms, we use a
path-conservative technique. To this end, we first apply the recently proposed second-order finite-volume PCCU scheme to the traffic flow models, and then extend this
scheme to the fifth-order of accuracy via the finite-difference A-WENO framework.
The designed schemes are applied to three different traffic flow models and tested on
a number of challenging numerical examples. Both schemes produce quite accurate results though the resolution achieved by the fifth-order A-WENO scheme is higher. The
proposed scheme in this paper sets the stage for developing more robust and complex
continuum traffic flow models with respect to human psychological factors. 相似文献
15.
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
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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
16.
High Order Finite Difference WENO Methods with Unequal-Sized Sub-Stencils for the Degasperis-Procesi Type Equations
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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
17.
The concept of diffusion regulation (DR) was originally proposed by
Jaisankar for traditional second order finite volume Euler solvers. This was used to
decrease the inherent dissipation associated with using approximate Riemann solvers.
In this paper, the above concept is extended to the high order spectral volume (SV)
method. The DR formulation was used in conjunction with the Rusanov flux to handle the inviscid flux terms. Numerical experiments were conducted to compare and
contrast the original and the DR formulations. These experiments demonstrated (i)
retention of high order accuracy for the new formulation, (ii) higher fidelity of the DR
formulation, when compared to the original scheme for all orders and (iii) straightforward extension to Navier Stokes equations, since the DR does not interfere with
the discretization of the viscous fluxes. In general, the 2D numerical results are very
promising and indicate that the approach has a great potential for 3D flow problems. 相似文献
18.
Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes
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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
19.
On the Order of Accuracy and Numerical Performance of Two Classes of Finite Volume WENO Schemes
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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
20.
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