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1.
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
2.
A Space-Time Conservative Method for Hyperbolic Systems with Stiff and Non-Stiff Source Terms 下载免费PDF全文
In this article we propose a higher-order space-time conservative method
for hyperbolic systems with stiff and non-stiff source terms as well as relaxation systems. We call the scheme a slope propagation (SP) method. It is an extension of our
scheme derived for homogeneous hyperbolic systems [1]. In the present inhomogeneous
systems the relaxation time may vary from order of one to a very small value. These
small values make the relaxation term stronger and highly stiff. In such situations
underresolved numerical schemes may produce spurious numerical results. However,
our present scheme has the capability to correctly capture the behavior of the physical
phenomena with high order accuracy even if the initial layer and the small relaxation
time are not numerically resolved. The scheme treats the space and time in a unified
manner. The flow variables and their slopes are the basic unknowns in the scheme. The
source term is treated by its volumetric integration over the space-time control volume
and is a direct part of the overall space-time flux balance. We use two approaches
for the slope calculations of the flow variables, the first one results directly from the
flux balance over the control volumes, while in the second one we use a finite difference approach. The main features of the scheme are its simplicity, its Jacobian-free
and Riemann solver-free recipe, as well as its efficiency and high order accuracy. In
particular we show that the scheme has a discrete analog of the continuous asymptotic limit. We have implemented our scheme for various test models available in the
literature such as the Broadwell model, the extended thermodynamics equations, the
shallow water equations, traffic flow and the Euler equations with heat transfer. The
numerical results validate the accuracy, versatility and robustness of the present scheme. 相似文献
3.
Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes 下载免费PDF全文
Walter Boscheri Michael Dumbser & Elena Gaburro 《Communications In Computational Physics》2022,32(1):259-298
We propose a new high order accurate nodal discontinuous Galerkin (DG)
method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical
polynomials of degree $N$ inside each element, in our new approach the discrete solution
is represented by piecewise continuous polynomials of degree $N$ within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon.
We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DG
method on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on each
sub-triangle are defined, as usual, on a universal reference element, hence allowing to
compute universal mass, flux and stiffness matrices for the subgrid triangles once and
for all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructured
nature of the computational grid. High order of accuracy in time is achieved thanks
to the ADER approach, making use of an element-local space-time Galerkin finite element predictor.The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical results
have been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtained
by the corresponding modal DG version of the scheme. 相似文献
4.
Shuangzhang Tu Gordon W. Skelton & Qing Pang 《Communications In Computational Physics》2011,9(2):441-480
This paper presents a novel high-order space-time method for hyperbolic
conservation laws. Two important concepts, the staggered space-time mesh of the
space-time conservation element/solution element (CE/SE) method and the local discontinuous
basis functions of the space-time discontinuous Galerkin (DG) finite element
method, are the two key ingredients of the new scheme. The staggered space-time
mesh is constructed using the cell-vertex structure of the underlying spatial mesh.
The universal definitions of CEs and SEs are independent of the underlying spatial
mesh and thus suitable for arbitrarily unstructured meshes. The solution within each
physical time step is updated alternately at the cell level and the vertex level. For
this solution updating strategy and the DG ingredient, the new scheme here is termed
as the discontinuous Galerkin cell-vertex scheme (DG-CVS). The high order of accuracy
is achieved by employing high-order Taylor polynomials as the basis functions
inside each SE. The present DG-CVS exhibits many advantageous features such as
Riemann-solver-free, high-order accuracy, point-implicitness, compactness, and ease
of handling boundary conditions. Several numerical tests including the scalar advection
equations and compressible Euler equations will demonstrate the performance of
the new method. 相似文献
5.
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 下载免费PDF全文
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
6.
A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws 下载免费PDF全文
Rapha& euml l Loub& egrave re Michael Dumbser & Steven Diot 《Communications In Computational Physics》2014,16(3):718-763
In this paper, we investigate the coupling of the Multi-dimensional Optimal
Order Detection (MOOD) method and the Arbitrary high order DERivatives (ADER)
approach in order to design a new high order accurate, robust and computationally
efficient Finite Volume (FV) scheme dedicated to solving nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and
three space dimensions, respectively. The Multi-dimensional Optimal Order Detection
(MOOD) method for 2D and 3D geometries has been introduced in a recent series of
papers for mixed unstructured meshes. It is an arbitrary high-order accurate Finite
Volume scheme in space, using polynomial reconstructions with a posteriori detection
and polynomial degree decrementing processes to deal with shock waves and other
discontinuities. In the following work, the time discretization is performed with an
elegant and efficient one-step ADER procedure. Doing so, we retain the good properties of the MOOD scheme, that is to say, the optimal high-order of accuracy is reached
on smooth solutions, while spurious oscillations near singularities are prevented. The
ADER technique not only reduces the cost of the overall scheme as shown
on a set of numerical tests in 2D and 3D, but also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO
schemes and the new ADER-MOOD approach has been carried out for high-order
schemes in space and time in terms of cost, robustness, accuracy and efficiency. The
main finding of this paper is that the combination of ADER with MOOD generally
outperforms the one of ADER and WENO either because at given accuracy MOOD isless expensive (memory and/or CPU time), or because it is more accurate for a given
grid resolution. A large suite of classical numerical test problems has been solved
on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations
of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations
(RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic
partial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements. 相似文献
7.
This paper is concerned with a new version of the Osher-Solomon Riemann
solver and is based on a numerical integration of the path-dependent dissipation matrix.
The resulting scheme is much simpler than the original one and is applicable to
general hyperbolic conservation laws, while retaining the attractive features of the original
solver: the method is entropy-satisfying, differentiable and complete in the sense
that it attributes a different numerical viscosity to each characteristic field, in particular
to the intermediate ones, since the full eigenstructure of the underlying hyperbolic system
is used. To illustrate the potential of the proposed scheme we show applications
to the following hyperbolic conservation laws: Euler equations of compressible gasdynamics
with ideal gas and real gas equation of state, classical and relativistic MHD
equations as well as the equations of nonlinear elasticity. To the knowledge of the authors,
apart from the Euler equations with ideal gas, an Osher-type scheme has never
been devised before for any of these complicated PDE systems. Since our new general
Riemann solver can be directly used as a building block of high order finite volume
and discontinuous Galerkin schemes we also show the extension to higher order of
accuracy and multiple space dimensions in the new framework of PNPM schemes on
unstructured meshes recently proposed in [9]. 相似文献
8.
Comparison of Fifth-Order WENO Scheme and Finite Volume WENO-Gas-Kinetic Scheme for Inviscid and Viscous Flow Simulation 下载免费PDF全文
The development of high-order schemes has been mostly concentrated on
the limiters and high-order reconstruction techniques. In this paper, the effect of the
flux functions on the performance of high-order schemes will be studied. Based on the
same WENO reconstruction, two schemes with different flux functions, i.e., the fifth-order WENO method and the WENO-Gas-Kinetic scheme (WENO-GKS), will be compared. The fifth-order finite difference WENO-SW scheme is a characteristic variable
reconstruction based method which uses the Steger-Warming flux splitting for inviscid terms, the sixth-order central difference for viscous terms, and three stages Runge-Kutta time stepping for the time integration. On the other hand, the finite volume
WENO-GKS is a conservative variable reconstruction based method with the same
WENO reconstruction. But it evaluates a time dependent gas distribution function
along a cell interface, and updates the flow variables inside each control volume by
integrating the flux function along the boundary of the control volume in both space
and time. In order to validate the robustness and accuracy of the schemes, both methods are tested under a wide range of flow conditions: vortex propagation, Mach 3
step problem, and the cavity flow at Reynolds number 3200. Our study shows that
both WENO-SW and WENO-GKS yield quantitatively similar results and agree with
each other very well provided a sufficient grid resolution is used. With the reduction of mesh points, the WENO-GKS behaves to have less numerical dissipation and
present more accurate solutions than those from the WENO-SW in all test cases. For
the Navier-Stokes equations, since the WENO-GKS couples inviscid and viscous terms
in a single flux evaluation, and the WENO-SW uses an operator splitting technique, it
appears that the WENO-SW is more sensitive to the WENO reconstruction and boundary treatment. In terms of efficiency, the finite volume WENO-GKS is about 4 times
slower than the finite difference WENO-SW in two dimensional simulations. The current study clearly shows that besides high-order reconstruction, an accurate gas evolution model or flux function in a high-order scheme is also important in the capturing of physical solutions. In a physical flow, the transport, stress deformation, heat conduction, and viscous heating are all coupled in a single gas evolution process. Therefore,
it is preferred to develop such a scheme with multi-dimensionality, and unified treatment of inviscid and dissipative terms. A high-order scheme does prefer a high-order
gas evolution model. Even with the rapid advances of high-order reconstruction techniques, the first-order dynamics of the Riemann solution becomes the bottleneck for
the further development of high-order schemes. In order to avoid the weakness of the
low order flux function, the development of high-order schemes relies heavily on the
weak solution of the original governing equations for the update of additional degree
of freedom, such as the non-conservative gradients of flow variables, which cannot be
physically valid in discontinuous regions. 相似文献
9.
Langhua Hu Siyang Yang & Guo-Wei Wei 《Communications In Computational Physics》2014,16(5):1201-1238
The solution of systems of hyperbolic conservation laws remains an interesting and challenging task due to the diversity of physical origins and complexity of
the physical situations. The present work introduces the use of the partial differential
equation (PDE) transform, paired with the Fourier pseudospectral method (FPM), as
a new approach for hyperbolic conservation law problems. The PDE transform, based
on the scheme of adaptive high order evolution PDEs, has recently been applied to decompose signals, images, surfaces and data to various target functional mode functions
such as trend, edge, texture, feature, trait, noise, etc. Like wavelet transform, the PDE
transform has controllable time-frequency localization and perfect reconstruction. A
fast PDE transform implemented by the fast Fourier Transform (FFT) is introduced to
avoid stability constraint of integrating high order PDEs. The parameters of the PDE
transform are adaptively computed to optimize the weighted total variation during
the time integration of conservation law equations. A variety of standard benchmark
problems of hyperbolic conservation laws is employed to systematically validate the
performance of the present PDE transform based FPM. The impact of two PDE transform parameters, i.e., the highest order and the propagation time, is carefully studied
to deliver the best effect of suppressing Gibbs' oscillations. The PDE orders of 2-6
are used for hyperbolic conservation laws of low oscillatory solutions, while the PDE
orders of 8-12 are often required for problems involving highly oscillatory solutions,
such as shock-entropy wave interactions. The present results are compared with those
in the literature. It is found that the present approach not only works well for problems that favor low order shock capturing schemes, but also exhibits superb behavior
for problems that require the use of high order shock capturing methods. 相似文献
10.
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
11.
In this paper, we present an adaptive moving mesh technique for solving
the incompressible viscous flows using the vorticity stream-function formulation. The
moving mesh strategy is based on the approach proposed by Li et al. [J. Comput. Phys.,
170 (2001), pp. 562–588] to separate the mesh-moving and evolving PDE at each time
step. The Navier-Stokes equations are solved in the vorticity stream-function form by
a finite-volume method in space, and the mesh-moving part is realized by solving the
Euler-Lagrange equations to minimize a certain variation in conjunction with a more
sophisticated monitor function. A conservative interpolation is used to redistribute
the numerical solutions on the new meshes. This paper discusses the implementation
of the periodic boundary conditions, where the physical domain is allowed to deform
with time while the computational domain remains fixed and regular throughout. Numerical results demonstrate the accuracy and effectiveness of the proposed algorithm. 相似文献
12.
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
13.
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
14.
Performance Enhancement for High-Order Gas-Kinetic Scheme Based on WENO-Adaptive-Order Reconstruction 下载免费PDF全文
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
15.
The Brinkman model describes flow of fluid in complex porous media with
a high-contrast permeability coefficient such that the flow is dominated by Darcy in
some regions and by Stokes in others. A weak Galerkin (WG) finite element method
for solving the Brinkman equations in two or three dimensional spaces by using polynomials
is developed and analyzed. The WG method is designed by using the generalized
functions and their weak derivatives which are defined as generalized distributions.
The variational form we considered in this paper is based on two gradient operators
which is different from the usual gradient-divergence operators for Brinkman
equations. The WG method is highly flexible by allowing the use of discontinuous
functions on arbitrary polygons or polyhedra with certain shape regularity. Optimal-order
error estimates are established for the corresponding WG finite element solutions
in various norms. Some computational results are presented to demonstrate the
robustness, reliability, accuracy, and flexibility of the WG method for the Brinkman
equations. 相似文献
16.
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
17.
张舒 《中华关节外科杂志(电子版)》2019,13(3):365-367
目的了解膝关节置换术(TKA)后关节僵硬与对侧膝关节置换之间的关系。 方法回顾性分析2016年06月至2017年06月在北京市东城区第一人民医院骨科行TKA的病例及随访资料,根据TKA术后12个月西安大略和曼彻斯特大学骨关节炎指数(WOMAC)僵硬评分,分为非僵硬组、僵硬组。比较2组患者对侧膝关节行TKA的时机,进行生存分析,行log-rank、Wilcoxon检验,以P<0.05为差异具有统计学意义。 结果非僵硬组、僵硬组分别在初次TKA术后(31.1±1.4)个月、(23.9±1.5)个月行对侧膝关节TKA,两组患者行对侧TKA时间间隔的差异具有统计学意义(log-rank检验χ2=26.130,P<0.05;Wilcoxon检验χ2=18.286,P<0.05)。 结论初次TKA膝关节后关节僵硬的患者,行对侧TKA的时机会提前。 相似文献
18.
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
19.
Learning to Discretize: Solving 1D Scalar Conservation Laws via Deep Reinforcement Learning 下载免费PDF全文
Yufei Wang Ziju Shen Zichao Long & Bin Dong 《Communications In Computational Physics》2020,28(5):2158-2179
Conservation laws are considered to be fundamental laws of nature. It has
broad applications in many fields, including physics, chemistry, biology, geology, and
engineering. Solving the differential equations associated with conservation laws is a
major branch in computational mathematics. The recent success of machine learning,
especially deep learning in areas such as computer vision and natural language processing, has attracted a lot of attention from the community of computational mathematics and inspired many intriguing works in combining machine learning with traditional methods. In this paper, we are the first to view numerical PDE solvers as an
MDP and to use (deep) RL to learn new solvers. As proof of concept, we focus on
1-dimensional scalar conservation laws. We deploy the machinery of deep reinforcement learning to train a policy network that can decide on how the numerical solutions should be approximated in a sequential and spatial-temporal adaptive manner.
We will show that the problem of solving conservation laws can be naturally viewed
as a sequential decision-making process, and the numerical schemes learned in such a
way can easily enforce long-term accuracy. Furthermore, the learned policy network
is carefully designed to determine a good local discrete approximation based on the
current state of the solution, which essentially makes the proposed method a meta-learning approach. In other words, the proposed method is capable of learning how to
discretize for a given situation mimicking human experts. Finally, we will provide details on how the policy network is trained, how well it performs compared with some
state-of-the-art numerical solvers such as WENO schemes, and supervised learning
based approach L3D and PINN, and how well it generalizes. 相似文献
20.
Yongyue Jiang Ping Lin Zhenlin Guo & Shuangling Dong 《Communications In Computational Physics》2015,18(1):180-202
In this paper, we compute a phase field (diffuse interface) model of Cahn-Hilliard
type for moving contact line problems governing the motion of isothermal
multiphase incompressible fluids. The generalized Navier boundary condition proposed
by Qian et al. [1] is adopted here. We discretize model equations using a continuous
finite element method in space and a modified midpoint scheme in time. We
apply a penalty formulation to the continuity equation which may increase the stability
in the pressure variable. Two kinds of immiscible fluids in a pipe and droplet displacement
with a moving contact line under the effect of pressure driven shear flow
are studied using a relatively coarse grid. We also derive the discrete energy law for
the droplet displacement case, which is slightly different due to the boundary conditions.
The accuracy and stability of the scheme are validated by examples, results and
estimate order. 相似文献