共查询到20条相似文献,搜索用时 31 毫秒
1.
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
2.
Runge-Kutta Discontinuous Galerkin Method Using WENO-Type Limiters: Three-Dimensional Unstructured Meshes 下载免费PDF全文
This paper further considers weighted essentially non-oscillatory (WENO) and Hermite weighted essentially non-oscillatory (HWENO) finite volume methods as limiters for Runge-Kutta discontinuous Galerkin (RKDG) methods to solve problems involving nonlinear hyperbolic conservation laws. The application discussed here is the solution of 3-D problems on unstructured meshes. Our numerical tests again demonstrate this is a robust and high order limiting procedure, which simultaneously achieves high order accuracy and sharp non-oscillatory shock transitions. 相似文献
3.
On the Order of Accuracy and Numerical Performance of Two Classes of Finite Volume WENO Schemes 下载免费PDF全文
Rui Zhang Mengping Zhang & Chi-Wang Shu 《Communications In Computational Physics》2011,9(3):807-827
In this paper we consider two commonly used classes of finite volume
weighted essentially non-oscillatory (WENO) schemes in two dimensional Cartesian
meshes. We compare them in terms of accuracy, performance for smooth and shocked
solutions, and efficiency in CPU timing. For linear systems both schemes are high
order accurate, however for nonlinear systems, analysis and numerical simulation results
verify that one of them (Class A) is only second order accurate, while the other
(Class B) is high order accurate. The WENO scheme in Class A is easier to implement
and costs less than that in Class B. Numerical experiments indicate that the resolution
for shocked problems is often comparable for schemes in both classes for the same
building blocks and meshes, despite of the difference in their formal order of accuracy.
The results in this paper may give some guidance in the application of high order finite
volume schemes for simulating shocked flows. 相似文献
4.
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
5.
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
6.
Jonas P. Berberich Roger Kä ppeli Praveen Chandrashekar & Christian Klingenberg 《Communications In Computational Physics》2021,30(3):666-708
We introduce novel high order well-balanced finite volume methods for the
full compressible Euler system with gravity source term. They require no à priori
knowledge of the hydrostatic solution which is to be well-balanced and are not restricted to certain classes of hydrostatic solutions. In one spatial dimension we construct a method that exactly balances a high order discretization of any hydrostatic
state. The method is extended to two spatial dimensions using a local high order approximation of a hydrostatic state in each cell. The proposed simple, flexible, and
robust methods are not restricted to a specific equation of state. Numerical tests verify
that the proposed method improves the capability to accurately resolve small perturbations on hydrostatic states. 相似文献
7.
A Sufficient and Necessary Condition of the Existence of WENO-Like Linear Combination for Finite Difference Schemes 下载免费PDF全文
Jian Kang & Xinliang Li 《Communications In Computational Physics》2021,29(2):534-570
In the finite difference WENO (weighted essentially non-oscillatory) method, the final scheme on the whole stencil was constructed by linear combinations of
highest order accurate schemes on sub-stencils, all of which share the same total count
of grid points. The linear combination method which the original WENO applied was
generalized to arbitrary positive-integer-order derivative on an arbitrary (uniform or
non-uniform) mesh, still applying finite difference method. The possibility of expressing the final scheme on the whole stencil as a linear combination of highest order accurate schemes on WENO-like sub-stencils was investigated. The main results include:
(a) the highest order of accuracy a finite difference scheme can achieve and (b) a sufficient and necessary condition that the linear combination exists. This is a sufficient
and necessary condition for all finite difference schemes in a set (rather than a specific
finite difference scheme) to have WENO-like linear combinations. After the proofs
of the results, some remarks on the WENO schemes and TENO (targeted essentially
non-oscillatory) schemes were given. 相似文献
8.
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
9.
High Order Fixed-Point Sweeping WENO Methods for Steady State of Hyperbolic Conservation Laws and Its Convergence Study 下载免费PDF全文
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
10.
Michael Dumbser Ariunaa Uuriintsetseg & Olindo Zanotti 《Communications In Computational Physics》2013,14(2):301-327
In this article we present a new family of high order accurate Arbitrary
Lagrangian-Eulerian one-step WENO finite volume schemes for the solution of stiff
hyperbolic balance laws. High order accuracy in space is obtained with a standard
WENO reconstruction algorithm and high order in time is obtained using the local
space-time discontinuous Galerkin method recently proposed in [20]. In the Lagrangian
framework considered here, the local space-time DG predictor is based on a weak
formulation of the governing PDE on a moving space-time element. For the space-time basis and test functions we use Lagrange interpolation polynomials defined by
tensor-product Gauss-Legendre quadrature points. The moving space-time elements
are mapped to a reference element using an isoparametric approach, i.e. the space-time mapping is defined by the same basis functions as the weak solution of the PDE.
We show some computational examples in one space-dimension for non-stiff and for
stiff balance laws, in particular for the Euler equations of compressible gas dynamics,
for the resistive relativistic MHD equations, and for the relativistic radiation hydrodynamics equations. Numerical convergence results are presented for the stiff case up to
sixth order of accuracy in space and time and for the non-stiff case up to eighth order
of accuracy in space and time. 相似文献
11.
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
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.
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
14.
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
15.
Simulations of Compressible Two-Medium Flow by Runge-Kutta Discontinuous Galerkin Methods with the Ghost Fluid Method 下载免费PDF全文
Jianxian Qiu Tiegang Liu & Boo Cheong Khoo 《Communications In Computational Physics》2008,3(2):479-504
The original ghost fluid method (GFM) developed in [13] and the modified GFM (MGFM) in [26] have provided a simple and yet flexible way to treat two-medium flow problems. The original GFM and MGFM make the material interface
”invisible” during computations and the calculations are carried out as for a single
medium such that its extension to multi-dimensions becomes fairly straightforward.
The Runge-Kutta discontinuous Galerkin (RKDG) method for solving hyperbolic conservation laws is a high order accurate finite element method employing the useful
features from high resolution finite volume schemes, such as the exact or approximate
Riemann solvers, TVD Runge-Kutta time discretizations, and limiters. In this paper,
we investigate using RKDG finite element methods for two-medium flow simulations
in one and two dimensions in which the moving material interfaces are treated via nonconservative methods based on the original GFM and MGFM. Numerical results for
both gas-gas and gas-water flows are provided to show the characteristic behaviors of
these combinations. 相似文献
16.
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
17.
Jun Zhu & Jianxian Qiu 《Communications In Computational Physics》2020,27(3):897-920
In this paper, a new type of third-order and fourth-order weighted essentially non-oscillatory (WENO) schemes is designed for simulating the Hamilton-Jacobi equations on triangular meshes. We design such schemes with the use of the nodal information defined on five unequal-sized spatial stencils, the application of monotone Hamiltonians as a building block, the artificial set of positive linear weights to make up high-order approximations in smooth regions simultaneously avoiding spurious oscillations nearby discontinuities of the derivatives of the solutions. The spatial reconstructions are convex combinations of the derivatives of a modified cubic/quartic polynomial defined on a big spatial stencil and four quadratic polynomials defined on small spatial stencils, and a third-order TVD Runge-Kutta method is used for the time discretization. The main advantages of these WENO schemes are their efficiency, simplicity, and can be easily implemented to higher dimensional unstructured meshes. Extensive numerical tests are performed to illustrate the good performance of such new WENO schemes. 相似文献
18.
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
19.
A Geometry-Preserving Finite Volume Method for Compressible Fluids on Schwarzschild Spacetime 下载免费PDF全文
We consider the relativistic Euler equations governing spherically symmetric, perfect fluid flows on the outer domain of communication of Schwarzschild spacetime, and we introduce a version of the finite volume method which is formulated
from the geometric formulation (and thus takes the geometry into account at the discretization level) and is well-balanced, in the sense that it preserves steady solutions to
the Euler equations on the curved geometry under consideration. In order to formulate our method, we first derive a closed formula describing all steady and spherically
symmetric solutions to the Euler equations posed on Schwarzschild spacetime. Second, we describe a geometry-preserving, finite volume method which is based on the family of steady solutions to the Euler system. Our scheme is second-order accurate and, as required, preserves the family of steady solutions at the discrete level.
Numerical experiments are presented which demonstrate the efficiency and robustness of the proposed method even for solutions containing shock waves and nonlinear
interacting wave patterns. As an application, we investigate the late-time asymptotics
of perturbed steady solutions and demonstrate its convergence for late time toward
another steady solution, taking the overall effect of the perturbation into account. 相似文献
20.
Bhuvana Srinivasan Ammar Hakim & Uri Shumlak 《Communications In Computational Physics》2011,10(1):183-215
The finite volume wave propagation method and the finite element RungeKutta
discontinuous Galerkin (RKDG) method are studied for applications to balance
laws describing plasma fluids. The plasma fluid equations explored are dispersive and
not dissipative. The physical dispersion introduced through the source terms leads to
the wide variety of plasma waves. The dispersive nature of the plasma fluid equations
explored separates the work in this paper from previous publications. The linearized
Euler equations with dispersive source terms are used as a model equation system to
compare the wave propagation and RKDG methods. The numerical methods are then
studied for applications of the full two-fluid plasma equations. The two-fluid equations
describe the self-consistent evolution of electron and ion fluids in the presence
of electromagnetic fields. It is found that the wave propagation method, when run
at a CFL number of 1, is more accurate for equation systems that do not have disparate
characteristic speeds. However, if the oscillation frequency is large compared
to the frequency of information propagation, source splitting in the wave propagation
method may cause phase errors. The Runge-Kutta discontinuous Galerkin method
provides more accurate results for problems near steady-state as well as problems with
disparate characteristic speeds when using higher spatial orders. 相似文献