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1.
High-Order Gas-Kinetic Scheme in Curvilinear Coordinates for the Euler and Navier-Stokes Solutions 下载免费PDF全文
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
2.
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
3.
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
4.
In this paper, a high-order moment-based multi-resolution Hermite
weighted essentially non-oscillatory (HWENO) scheme is designed for hyperbolic conservation laws. The main idea of this scheme is derived from our previous work [J.
Comput. Phys., 446 (2021) 110653], in which the integral averages of the function and
its first order derivative are used to reconstruct both the function and its first order
derivative values at the boundaries. However, in this paper, only the function values at
the Gauss-Lobatto points in the one or two dimensional case need to be reconstructed
by using the information of the zeroth and first order moments. In addition, an extra
modification procedure is used to modify those first order moments in the troubled-cells, which leads to an improvement of stability and an enhancement of resolution
near discontinuities. To obtain the same order of accuracy, the size of the stencil required by this moment-based multi-resolution HWENO scheme is still the same as the
general HWENO scheme and is more compact than the general WENO scheme. Moreover, the linear weights are not unique and are independent of the node position, and
the CFL number can still be 0.6 whether for the one or two dimensional case, which has
to be 0.2 in the two dimensional case for other HWENO schemes. Extensive numerical
examples are given to demonstrate the stability and resolution of such moment-based
multi-resolution HWENO scheme. 相似文献
5.
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
6.
A Compact Third-Order Gas-Kinetic Scheme for Compressible Euler and Navier-Stokes Equations 下载免费PDF全文
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
7.
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
8.
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
9.
Jun Zhu Xinghui Zhong Chi-Wang Shu & Jianxian Qiu 《Communications In Computational Physics》2016,19(4):944-969
In this paper, we propose a new type of weighted essentially non-oscillatory
(WENO) limiter, which belongs to the class of Hermite WENO (HWENO) limiters, for
the Runge-Kutta discontinuous Galerkin (RKDG) methods solving hyperbolic conservation
laws. This new HWENO limiter is a modification of the simple WENO limiter
proposed recently by Zhong and Shu [29]. Both limiters use information of the DG
solutions only from the target cell and its immediate neighboring cells, thus maintaining
the original compactness of the DG scheme. The goal of both limiters is to obtain
high order accuracy and non-oscillatory properties simultaneously. The main novelty
of the new HWENO limiter in this paper is to reconstruct the polynomial on the target
cell in a least square fashion [8] while the simple WENO limiter [29] is to use the entire
polynomial of the original DG solutions in the neighboring cells with an addition of
a constant for conservation. The modification in this paper improves the robustness
in the computation of problems with strong shocks or contact discontinuities, without
changing the compact stencil of the DG scheme. Numerical results for both one and
two dimensional equations including Euler equations of compressible gas dynamics
are provided to illustrate the viability of this modified limiter. 相似文献
10.
High-Order Runge-Kutta Discontinuous Galerkin Methods with a New Type of Multi-Resolution WENO Limiters on Tetrahedral Meshes 下载免费PDF全文
In this paper, the second-order and third-order Runge-Kutta discontinuous
Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory
(WENO) limiters are proposed on tetrahedral meshes. The multi-resolution WENO
limiter is an extension of a finite volume multi-resolution WENO scheme developed
in [81], which serves as a limiter for RKDG methods on tetrahedral meshes. This new
WENO limiter uses information of the DG solution essentially only within the troubled cell itself which is identified by a new modified version of the original KXRCF
indicator [42], to build a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to the second or third degree of the DG solution. The second-order and
third-order RKDG methods with the associated multi-resolution WENO limiters are
developed as examples for general high-order RKDG methods, which could maintain
the original order of accuracy in smooth regions and keep essentially non-oscillatory
property near strong discontinuities by gradually degrading from the optimal order
to the first order. The linear weights inside the procedure of the new multi-resolution
WENO limiters can be set as any positive numbers on the condition that they sum
to one. A series of polynomials of different degrees within the troubled cell itself
are applied in a WENO fashion to modify the DG solutions in the troubled cell on
tetrahedral meshes. These new WENO limiters are very simple to construct, and can
be easily implemented to arbitrary high-order accuracy on tetrahedral meshes. Such
spatial reconstruction methodology improves the robustness in the simulation on the
same compact spatial stencil of the original DG methods on tetrahedral meshes. Extensive one-dimensional (run as three-dimensional problems on tetrahedral meshes)
and three-dimensional tests are performed to demonstrate the good performance of
the RKDG methods with new multi-resolution WENO limiters. 相似文献
11.
A Higher Order Interpolation Scheme of Finite Volume Method for Compressible Flow on Curvilinear Grids 下载免费PDF全文
Zhen-Hua Jiang Xi Deng Feng Xiao Chao Yan & Jian Yu 《Communications In Computational Physics》2020,28(4):1609-1638
A higher order interpolation scheme based on a multi-stage BVD (Boundary Variation Diminishing) algorithm is developed for the FV (Finite Volume) method
on non-uniform, curvilinear structured grids to simulate the compressible turbulent
flows. The designed scheme utilizes two types of candidate interpolants including
a higher order linear-weight polynomial as high as eleven and a THINC (Tangent of
Hyperbola for INterface Capturing) function with the adaptive steepness. We investigate not only the accuracy but also the efficiency of the methodology through the cost
efficiency analysis in comparison with well-designed mapped WENO (Weighted Essentially Non-Oscillatory) scheme. Numerical experimentation including benchmark
broadband turbulence problem as well as real-life wall-bounded turbulent flows has
been carried out to demonstrate the potential implementation of the present higher
order interpolation scheme especially in the ILES (Implicit Large Eddy Simulation) of
compressible turbulence. 相似文献
12.
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
13.
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
14.
Arbitrarily High-Order (Weighted) Essentially Non-Oscillatory Finite Difference Schemes for Anelastic Flows on Staggered Meshes 下载免费PDF全文
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
15.
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
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.
A Multi-Domain Hybrid DG and WENO Method for Hyperbolic Conservation Laws on Hybrid Meshes 下载免费PDF全文
In [SIAM J. Sci. Comput., 35(2)(2013), A1049–A1072], a class of multi-domain
hybrid DG and WENO methods for conservation laws was introduced. Recent applications of this method showed that numerical instability may encounter if the DG flux
with Lagrangian interpolation is applied as the interface flux during the moment of
conservative coupling. In this continuation paper, we present a more robust approach
in the construction of DG flux at the coupling interface by using WENO procedures of
reconstruction. Based on this approach, such numerical instability is overcome very
well. In addition, the procedure of coupling a DG method with a WENO-FD scheme
on hybrid meshes is disclosed in detail. Typical testing cases are employed to demonstrate the accuracy of this approach and the stability under the flexibility of using either
WENO-FD flux or DG flux at the moment of requiring conservative coupling. 相似文献
18.
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
19.
Divergence-Free WENO Reconstruction-Based Finite Volume Scheme for Solving Ideal MHD Equations on Triangular Meshes 下载免费PDF全文
Zhiliang Xu Dinshaw S. Balsara & Huijing Du 《Communications In Computational Physics》2016,19(4):841-880
In this paper, we introduce a high-order accurate constrained transport
type finite volume method to solve ideal magnetohydrodynamic equations on two-dimensional
triangular meshes. A new divergence-free WENO-based reconstruction
method is developed to maintain exactly divergence-free evolution of the numerical
magnetic field. In this formulation, the normal component of the magnetic field at each
face of a triangle is reconstructed uniquely and with the desired order of accuracy. Additionally,
a new weighted flux interpolation approach is also developed to compute
the z-component of the electric field at vertices of grid cells. We also present numerical
examples to demonstrate the accuracy and robustness of the proposed scheme. 相似文献
20.
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