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1.
High Order Fixed-Point Sweeping WENO Methods for Steady State of Hyperbolic Conservation Laws and Its Convergence Study
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Liang Wu Yong-Tao Zhang Shuhai Zhang & Chi-Wang Shu 《Communications In Computational Physics》2016,20(4):835-869
Fixed-point iterative sweeping methods were developed in the literature to
efficiently solve static Hamilton-Jacobi equations. This class of methods utilizes the
Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence
rate. They take advantage of the properties of hyperbolic partial differential equations
(PDEs) and try to cover a family of characteristics of the corresponding Hamilton-Jacobi
equation in a certain direction simultaneously in each sweeping order. Different
from other fast sweeping methods, fixed-point iterative sweeping methods have the
advantages such as that they have explicit forms and do not involve inverse operation
of nonlinear local systems. In principle, it can be applied to solving very general
equations using any monotone numerical fluxes and high order approximations easily.
In this paper, based on the recently developed fifth order WENO schemes which improve
the convergence of the classical WENO schemes by removing slight post-shock
oscillations, we design fifth order fixed-point sweeping WENO methods for efficient
computation of steady state solution of hyperbolic conservation laws. Especially, we
show that although the methods do not have linear computational complexity, they
converge to steady state solutions much faster than regular time-marching approach
by stability improvement for high order schemes with a forward Euler time-marching. 相似文献
2.
In this paper, we present a new type of Hermite weighted essentially non-oscillatory
(HWENO) schemes for solving the Hamilton-Jacobi equations on the finite
volume framework. The cell averages of the function and its first one (in one dimension)
or two (in two dimensions) derivative values are together evolved via time
approaching and used in the reconstructions. And the major advantages of the new
HWENO schemes are their compactness in the spacial field, purely on the finite volume
framework and only one set of small stencils is used for different type of the
polynomial reconstructions. Extensive numerical tests are performed to illustrate the
capability of the methodologies. 相似文献
3.
Numerical Solution of 3D Poisson-Nernst-Planck Equations Coupled with Classical Density Functional Theory for Modeling Ion and Electron Transport in a Confined Environment
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Da Meng Bin Zheng Guang Lin & Maria L. Sushko 《Communications In Computational Physics》2014,16(5):1298-1322
We have developed efficient numerical algorithms for solving 3D steady-state
Poisson-Nernst-Planck (PNP) equations with excess chemical potentials described
by the classical density functional theory (cDFT). The coupled PNP equations are discretized
by a finite difference scheme and solved iteratively using the Gummel method
with relaxation. The Nernst-Planck equations are transformed into Laplace equations
through the Slotboom transformation. Then, the algebraic multigrid method is
applied to efficiently solve the Poisson equation and the transformed Nernst-Planck
equations. A novel strategy for calculating excess chemical potentials through fast
Fourier transforms is proposed, which reduces computational complexity from $\mathcal{O}$($N^2$) to $\mathcal{O}$($NlogN$), where $N$ is the number of grid points. Integrals involving the Dirac
delta function are evaluated directly by coordinate transformation, which yields more
accurate results compared to applying numerical quadrature to an approximated delta
function. Numerical results for ion and electron transport in solid electrolyte for lithium-ion
(Li-ion) batteries are shown to be in good agreement with the experimental data
and the results from previous studies. 相似文献
4.
High Order Finite Difference WENO Methods with Unequal-Sized Sub-Stencils for the Degasperis-Procesi Type Equations
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Jianfang Lin Yan Xu Huiwen Xue & Xinghui Zhong 《Communications In Computational Physics》2022,31(3):913-946
In this paper, we develop two finite difference weighted essentially
non-oscillatory (WENO) schemes with unequal-sized sub-stencils for solving the
Degasperis-Procesi (DP) and $\mu$-Degasperis-Procesi ($\mu$DP) equations, which contain
nonlinear high order derivatives, and possibly peakon solutions or shock waves. By
introducing auxiliary variable(s), we rewrite the DP equation as a hyperbolic-elliptic
system, and the $\mu$DP equation as a first order system. Then we choose a linear finite
difference scheme with suitable order of accuracy for the auxiliary variable(s), and
two finite difference WENO schemes with unequal-sized sub-stencils for the primal
variable. One WENO scheme uses one large stencil and several smaller stencils, and
the other WENO scheme is based on the multi-resolution framework which uses a series of unequal-sized hierarchical central stencils. Comparing with the classical WENO
scheme which uses several small stencils of the same size to make up a big stencil, both
WENO schemes with unequal-sized sub-stencils are simple in the choice of the stencil
and enjoy the freedom of arbitrary positive linear weights. Another advantage is that
the final reconstructed polynomial on the target cell is a polynomial of the same degree as the polynomial over the big stencil, while the classical finite difference WENO
reconstruction can only be obtained for specific points inside the target interval. Numerical tests are provided to demonstrate the high order accuracy and non-oscillatory
properties of the proposed schemes. 相似文献
5.
José Antonio Carrillo Shi Jin & Yijia Tang 《Communications In Computational Physics》2022,31(4):997-1019
We consider in this paper random batch particle methods for efficiently solving the homogeneous Landau equation in plasma physics. The methods are stochastic
variations of the particle methods proposed by Carrillo et al. [J. Comput. Phys.: X 7:
100066, 2020] using the random batch strategy. The collisions only take place inside
the small but randomly selected batches so that the computational cost is reduced to $\mathcal{O}(N)$ per time step. Meanwhile, our methods can preserve the conservation of mass,
momentum, energy and the decay of entropy. Several numerical examples are performed to validate our methods. 相似文献
6.
Three Discontinuous Galerkin Methods for One- and Two-Dimensional Nonlinear Dirac Equations with a Scalar Self-Interaction
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Shu-Cun Li & Huazhong Tang 《Communications In Computational Physics》2021,30(4):1150-1184
This paper develops three high-order accurate discontinuous Galerkin (DG)
methods for the one-dimensional (1D) and two-dimensional (2D) nonlinear Dirac
(NLD) equations with a general scalar self-interaction. They are the Runge-Kutta DG
(RKDG) method and the DG methods with the one-stage fourth-order Lax-Wendroff
type time discretization (LWDG) and the two-stage fourth-order accurate time discretization (TSDG). The RKDG method uses the spatial DG approximation to discretize
the NLD equations and then utilize the explicit multistage high-order Runge-Kutta
time discretization for the first-order time derivatives, while the LWDG and TSDG
methods, on the contrary, first give the one-stage fourth-order Lax-Wendroff type and
the two-stage fourth-order time discretizations of the NLD equations, respectively, and
then discretize the first- and higher-order spatial derivatives by using the spatial DG
approximation. The $L^2$ stability of the 2D semi-discrete DG approximation is proved
in the RKDG methods for a general triangulation, and the computational complexities of three 1D DG methods are estimated. Numerical experiments are conducted to
validate the accuracy and the conservation properties of the proposed methods. The
interactions of the solitary waves, the standing and travelling waves are investigated
numerically and the 2D breathing pattern is observed. 相似文献
7.
A Diagonal Sweeping Domain Decomposition Method with Source Transfer for the Helmholtz Equation
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Wei Leng & Lili Ju 《Communications In Computational Physics》2021,29(2):357-395
In this paper, we propose and test a novel diagonal sweeping domain
decomposition method (DDM) with source transfer for solving the high-frequency
Helmholtz equation in$\mathbb{R}^n$. In the method the computational domain is partitioned into
overlapping checkerboard subdomains for source transfer with the perfectly matched
layer (PML) technique, then a set of diagonal sweeps over the subdomains are specially
designed to solve the system efficiently. The method improves the additive overlapping DDM [43] and the L-sweeps method [50] by employing a more efficient subdomain solving order. We show that the method achieves the exact solution of the global
PML problem with $2^n$ sweeps in the constant medium case. Although the sweeping
usually implies sequential subdomain solves, the number of sequential steps required
for each sweep in the method is only proportional to the $n$-th root of the number of
subdomains when the domain decomposition is quasi-uniform with respect to all directions, thus it is very suitable for parallel computing of the Helmholtz problem with
multiple right-hand sides through the pipeline processing. Extensive numerical experiments in two and three dimensions are presented to demonstrate the effectiveness
and efficiency of the proposed method. 相似文献
8.
Linghua Kong Jialin Hong & Jingjing Zhang 《Communications In Computational Physics》2013,14(1):219-241
The local one-dimensional multisymplectic scheme (LOD-MS) is developed for the three-dimensional (3D) Gross-Pitaevskii (GP) equation in Bose-Einstein condensates. The idea is originated from the advantages of multisymplectic integrators and from the cheap computational cost of the local one-dimensional (LOD) method. The 3D GP equation is split into three linear LOD Schrödinger equations and an exactly solvable nonlinear Hamiltonian ODE. The three linear LOD Schrödinger equations are multisymplectic which can be approximated by multisymplectic integrator (MI). The conservative properties of the proposed scheme are investigated. It is mass-preserving. Surprisingly, the scheme preserves the discrete local energy conservation laws and global energy conservation law if the wave function is variable separable. This is impossible for conventional MIs in nonlinear Hamiltonian context. The numerical results show that the LOD-MS can simulate the original problems very well. They are consistent with the numerical analysis. 相似文献
9.
A Hermite WENO Method with Modified Ghost Fluid Method for Compressible Two-Medium Flow Problems
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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
10.
Shi Jin & Xiantao Li 《Communications In Computational Physics》2020,28(5):1907-1936
Random batch algorithms are constructed for quantum Monte Carlo simulations. The main objective is to alleviate the computational cost associated with the
calculations of two-body interactions, including the pairwise interactions in the potential energy, and the two-body terms in the Jastrow factor. In the framework of variational Monte Carlo methods, the random batch algorithm is constructed based on the
over-damped Langevin dynamics, so that updating the position of each particle in an $N$-particle system only requires$\mathcal{O}(1)$ operations, thus for each time step the computational cost for $N$ particles is reduced from$\mathcal{O}(N^2)$ to$\mathcal{O}(N)$. For diffusion Monte Carlo
methods, the random batch algorithm uses an energy decomposition to avoid the computation of the total energy in the branching step. The effectiveness of the random
batch method is demonstrated using a system of liquid $^4$He atoms interacting with a
graphite surface. 相似文献
11.
A Novel Efficient Numerical Solution of Poisson's Equation for Arbitrary Shapes in Two Dimensions
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Zu-Hui Ma Weng Cho Chew & Li Jun Jiang 《Communications In Computational Physics》2016,20(5):1381-1404
Even though there are various fast methods and preconditioning techniques
available for the simulation of Poisson problems, little work has been done for solving
Poisson's equation by using the Helmholtz decomposition scheme. To bridge this
issue, we propose a novel efficient algorithm to solve Poisson's equation in irregular
two dimensional domains for electrostatics through a quasi-Helmholtz decomposition
technique–the loop-tree basis decomposition. It can handle Dirichlet, Neumann or
mixed boundary problems in which the filling media can be homogeneous or inhomogeneous.
A novel point of this method is to first find the electric flux efficiently by
applying the loop-tree basis functions. Subsequently, the potential is obtained by finding
the inverse of the gradient operator. Furthermore, treatments for both Dirichlet
and Neumann boundary conditions are addressed. Finally, the validation and efficiency are illustrated by several numerical examples. Through these simulations, it
is observed that the computational complexity of our proposed method almost scales
as $\mathcal{O}$$(N)$, where $N$ is the triangle patch number of meshes. Consequently, this new
algorithm is a feasible fast Poisson solver. 相似文献
12.
Jian Deng Cristina Anton & Yau Shu Wong 《Communications In Computational Physics》2014,16(1):169-200
The construction of symplectic numerical schemes for stochastic Hamiltonian systems is studied. An approach based on generating functions method is proposed to generate the stochastic symplectic integration of any desired order. In general, the proposed symplectic schemes are fully implicit, and they become computationally
expensive for mean square orders greater than two. However, for stochastic Hamiltonian systems preserving Hamiltonian functions, the high-order symplectic methods
have simpler forms than the explicit Taylor expansion schemes. A theoretical analysis
of the convergence and numerical simulations are reported for several symplectic integrators. The numerical case studies confirm that the symplectic methods are efficient
computational tools for long-term simulations. 相似文献
13.
Multi-Valued Solution and Level Set Methods in Computational High Frequency Wave Propagation
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We review the level set methods for computing multi-valued solutions to a class of nonlinear first order partial differential equations, including Hamilton-Jacobi equations, quasi-linear hyperbolic equations, and conservative transport equations with multi-valued transport speeds. The multivalued solutions are embedded as the zeros of a set of scalar functions that solve the initial value problems of a time dependent partial differential equation in an augmented space. We discuss the essential ideas behind the techniques, the coupling of these techniques to the projection of the interaction of zero level sets and a collection of applications including the computation of the semiclassical limit for Schrödinger equations and the high frequency geometrical optics limits of linear wave equations. 相似文献
14.
Construction of the Local Structure-Preserving Algorithms for the General Multi-Symplectic Hamiltonian System
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Many partial differential equations can be written as a multi-symplectic Hamiltonian system, which has three local conservation laws, namely multi-symplectic conservation law, local energy conservation law and local momentum conservation law. In this paper, we systematically give a unified framework to construct the local structure-preserving algorithms for general conservative partial differential equations starting from the multi-symplectic formulation and using the concatenating method. We construct four multi-symplectic algorithms, two local energy-preserving algorithms and two local momentum-preserving algorithms, which are independent of the boundary conditions and can be used to integrate any partial differential equations written in multi-symplectic Hamiltonian form. Among these algorithms, some have been discussed and widely used before while most are novel schemes. These algorithms are illustrated by the nonlinear Schrödinger equation and the Klein-Gordon-Schrödinger equation. Numerical experiments are conducted to show the good performance of the proposed methods. 相似文献
15.
Runge-Kutta Discontinuous Galerkin Method Using WENO-Type Limiters: Three-Dimensional Unstructured Meshes
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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
16.
A Class of Hybrid DG/FV Methods for Conservation Laws III: Two-Dimensional Euler Equations
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Laiping Zhang Wei Liu Lixin He & Xiaogang Deng 《Communications In Computational Physics》2012,12(1):284-314
A concept of "static reconstruction" and "dynamic reconstruction" was introduced for higher-order (third-order or more) numerical methods in our previous
work. Based on this concept, a class of hybrid DG/FV methods had been developed
for one-dimensional conservation law using a "hybrid reconstruction" approach, and
extended to two-dimensional scalar equations on triangular and Cartesian/triangular
hybrid grids. In the hybrid DG/FV schemes, the lower-order derivatives of the piecewise polynomial are computed locally in a cell by the traditional DG method (called
as "dynamic reconstruction"), while the higher-order derivatives are reconstructed by
the "static reconstruction" of the FV method, using the known lower-order derivatives
in the cell itself and in its adjacent neighboring cells. In this paper, the hybrid DG/FV
schemes are extended to two-dimensional Euler equations on triangular and Cartesian/triangular hybrid grids. Some typical test cases are presented to demonstrate
the performance of the hybrid DG/FV methods, including the standard vortex evolution problem with exact solution, isentropic vortex/weak shock wave interaction,
subsonic flows past a circular cylinder and a three-element airfoil (30P30N), transonic
flow past a NACA0012 airfoil. The accuracy study shows that the hybrid DG/FV
method achieves the desired third-order accuracy, and the applications demonstrate
that they can capture the flow structure accurately, and can reduce the CPU time and
memory requirement greatly than the traditional DG method with the same order of
accuracy. 相似文献
17.
Meng Zhao Pedro Anjos John Lowengrub Wenjun Ying & Shuwang Li 《Communications In Computational Physics》2023,33(2):399-424
We investigate the nonlinear dynamics of a moving interface in a Hele-Shaw
cell subject to an in-plane applied electric field. We develop a spectrally accurate numerical method for solving a coupled integral equation system. Although the stiffness
due to the high order spatial derivatives can be removed using a small scale decomposition technique, the long-time simulation is still expensive since the evolving velocity of the interface drops dramatically as the interface expands. We remove this
physically imposed stiffness by employing a rescaling scheme, which accelerates the
slow dynamics and reduces the computational cost. Our nonlinear results reveal that
positive currents restrain finger ramification and promote the overall stabilization of
patterns. On the other hand, negative currents make the interface more unstable and
lead to the formation of thin tail structures connecting the fingers and a small inner
region. When no fluid is injected, and a negative current is utilized, the interface tends
to approach the origin and break up into several drops. We investigate the temporal
evolution of the smallest distance between the interface and the origin and find that it
obeys an algebraic law $(t_∗−t)^b,$ where $t_∗$ is the estimated pinch-off time. 相似文献
18.
A fully discrete discontinuous Galerkin method is introduced for solving
time-dependent Maxwell's equations. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), in
our scheme, discontinuous Galerkin methods are used to discretize not only the spatial
domain but also the temporal domain. The proposed numerical scheme is proved to be
unconditionally stable, and a convergent rate $\mathcal{O}((∆t)^{r+1}+h^{k+1/2})$ is established under the $L^2$ -norm when polynomials of degree at most $r$ and $k$ are used for temporal and
spatial approximation, respectively. Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction. An ultra-convergence of order $(∆t)^{2r+1}$ in
time step is observed numerically for the numerical fluxes w.r.t. temporal variable at
the grid points. 相似文献
19.
A novel unified approach to two‐degrees‐of‐freedom control is devised and applied to a classical chemical reactor model. The scheme is constructed from the optimal control point of view and along the lines of the Hamiltonian formalism for nonlinear processes. The proposed scheme optimizes both the feedforward and the feedback components of the control variable with respect to the same cost objective. The original Hamiltonian function governs the feedforward dynamics, and its derivatives are part of the gain for the feedback component. The optimal state trajectory is generated online, and is tracked by a combination of deterministic and stochastic optimal tools. The relevant numerical data to manipulate all stages come from a unique off‐line calculation, which provides design information for a whole family of related control problems. This is possible because a new set of PDEs (the variational equations) allow to recover the initial value of the costate variable, and the Hamilton equations can then be solved as an initial‐value problem. Perturbations from the optimal trajectory are abated through an optimal state estimator and a deterministic regulator with a generalized Riccati gain. Both gains are updated online, starting with initial values extracted from the solution to the variational equations. The control strategy is particularly useful in driving nonlinear processes from an equilibrium point to an arbitrary target in a finite‐horizon optimization context. Copyright © 2010 John Wiley & Sons, Ltd. 相似文献
20.
Directional $\mathcal{H}^2$ Compression Algorithm: Optimisations and Application to a Discontinuous Galerkin BEM for the Helmholtz Equation
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Nadir-Alexandre Messaï Sebastien Pernet & Abdesselam Bouguerra 《Communications In Computational Physics》2022,31(5):1585-1635
This study aimed to specialise a directional $\mathcal{H}^2
(\mathcal{D}\mathcal{H}^2)$ compression to matrices arising from the discontinuous Galerkin (DG) discretisation of the hypersingular
equation in acoustics. The significant finding is an algorithm that takes a DG stiffness matrix and finds a near-optimal $\mathcal{D}\mathcal{H}^2$ approximation for low and high-frequency
problems. We introduced the necessary special optimisations to make this algorithm
more efficient in the case of a DG stiffness matrix. Moreover, an automatic parameter
tuning strategy makes it easy to use and versatile. Numerical comparisons with a classical Boundary Element Method (BEM) show that a DG scheme combined with a $\mathcal{D}\mathcal{H}^2$ gives better computational efficiency than a classical BEM in the case of high-order finite elements and $hp$ heterogeneous meshes. The results indicate that DG is suitable
for an auto-adaptive context in integral equations. 相似文献