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1.
In this paper, we will develop a fast iterative solver for the system of linear
equations arising from the local discontinuous Galerkin (LDG) spatial discretization
and additive Runge-Kutta (ARK) time marching method for the KdV type equations.
Being implicit in time, the severe time step ($∆t$=$\mathcal{O}(∆x^k)$, with the $k$-th order of the
partial differential equations (PDEs)) restriction for explicit methods will be removed.
The equations at the implicit time level are linear and we demonstrate an efficient,
practical multigrid (MG) method for solving the equations. In particular, we numerically
show the optimal or sub-optimal complexity of the MG solver and a two-level
local mode analysis is used to analyze the convergence behavior of the MG method.
Numerical results for one-dimensional, two-dimensional and three-dimensional cases
are given to illustrate the efficiency and capability of the LDG method coupled with
the multigrid method for solving the KdV type equations. 相似文献
2.
An Efficient Parallel/Unstructured-Multigrid Implicit Method for Simulating 3D Fluid-Structure Interaction 下载免费PDF全文
X. Lv Y. Zhao X. Y. Huang G. H. Xia & X. H. Su 《Communications In Computational Physics》2008,4(2):350-377
A finite volume (FV) method for simulating 3D Fluid-Structure Interaction (FSI) is presented in this paper. The fluid flow is simulated using a parallel unstructured multigrid preconditioned implicit compressible solver, whist a 3D matrix-free implicit unstructured multigrid finite volume solver is employed for the structural dynamics. The two modules are then coupled using a so-called immersed membrane method (IMM). Large-Eddy Simulation (LES) is employed to predict turbulence. Results from several moving boundary and FSI problems are presented to validate proposed methods and demonstrate their efficiency. 相似文献
3.
Adaptive Fully Implicit Simulator with Multilevel Schwarz Methods for Gas Reservoir Flows in Fractured Porous Media 下载免费PDF全文
Large-scale reservoir modeling and simulation of gas reservoir flows in fractured porous media is currently an important topic of interest in petroleum engineering. In this paper, the dual-porosity dual-permeability (DPDP) model coupled with
the Peng-Robinson equation of state (PR-EoS) is used for the mathematical model of
the gas reservoir flow in fractured porous media. We develop and study a parallel and
highly scalable reservoir simulator based on an adaptive fully implicit scheme and
an inexact Newton type method to solve this dual-continuum mathematical model.
In the approach, an explicit-first-step, single-diagonal-coefficient, diagonally implicit
Runge–Kutta (ESDIRK) method with adaptive time stepping is proposed for the fully
implicit discretization, which is second-order and L-stable. And then we focus on the
family of Newton–Krylov methods for the solution of a large sparse nonlinear system
of equations arising at each time step. To accelerate the convergence and improve the
scalability of the solver, a class of multilevel monolithic additive Schwarz methods is
employed for preconditioning. Numerical results on a set of ideal as well as realistic
flow problems are used to demonstrate the efficiency and the robustness of the proposed methods. Experiments on a supercomputer with several thousand processors
are also carried out to show that the proposed reservoir simulator is highly scalable. 相似文献
4.
A Jacobian-Free Newton Krylov Implicit-Explicit Time Integration Method for Incompressible Flow Problems 下载免费PDF全文
We have introduced a fully second order IMplicit/EXplicit (IMEX) time integration technique for solving the compressible Euler equations plus nonlinear heat conduction problems (also known as the radiation hydrodynamics problems) in Kadioglu et al., J. Comp. Physics [22,24]. In this paper, we study the implications when this method is applied to the incompressible Navier-Stokes (N-S) equations. The IMEX method is applied to the incompressible flow equations in the following manner. The hyperbolic terms of the flow equations are solved explicitly exploiting the well understood explicit schemes. On the other hand, an implicit strategy is employed for the non-hyperbolic terms. The explicit part is embedded in the implicit step in such a way that it is solved as part of the non-linear function evaluation within the framework of the Jacobian-Free Newton Krylov (JFNK) method [8,29,31]. This is done to obtain a self-consistent implementation of the IMEX method that eliminates the potential order reduction in time accuracy due to the specific operator separation. We employ a simple yet quite effective fractional step projection methodology (similar to those in [11,19,21,30]) as our preconditioner inside the JFNK solver. We present results from several test calculations. For each test, we show second order time convergence. Finally, we present a study for the algorithm performance of the JFNK solver with the new projection method based preconditioner. 相似文献
5.
An Implicit LU-SGS Scheme for the Spectral Volume Method on Unstructured Tetrahedral Grids 下载免费PDF全文
Takanori Haga Keisuke Sawada & Z. J. Wang 《Communications In Computational Physics》2009,6(5):978-996
An efficient implicit lower-upper symmetric Gauss-Seidel (LU-SGS) solution
approach has been applied to a high order spectral volume (SV) method for unstructured
tetrahedral grids. The LU-SGS solver is preconditioned by the block element
matrix, and the system of equations is then solved with a LU decomposition.
The compact feature of SV reconstruction facilitates the efficient solution algorithm
even for high order discretizations. The developed implicit solver has shown more
than an order of magnitude of speed-up relative to the Runge-Kutta explicit scheme
for typical inviscid and viscous problems. A convergence to a high order solution for
high Reynolds number transonic flow over a 3D wing with a one equation turbulence
model is also indicated. 相似文献
6.
Extrapolation Cascadic Multigrid Method for Cell-Centered FV Discretization of Diffusion Equations with Strongly Discontinuous and Anisotropic Coefficients 下载免费PDF全文
Kejia Pan Xiaoxin Wu Yunlong Yu Zhiqiang Sheng & Guangwei Yuan 《Communications In Computational Physics》2022,31(5):1561-1584
Extrapolation cascadic multigrid (EXCMG) method with conjugate gradient
smoother is very efficient for solving the elliptic boundary value problems with linear
finite element discretization. However, it is not trivial to generalize the vertex-centred
EXCMG method to cell-centered finite volume (FV) methods for diffusion equations
with strongly discontinuous and anisotropic coefficients, since a non-nested hierarchy
of grid nodes are used in the cell-centered discretization. For cell-centered FV schemes,
the vertex values (auxiliary unknowns) need to be approximated by cell-centered ones
(primary unknowns). One of the novelties is to propose a new gradient transfer (GT)
method of interpolating vertex unknowns with cell-centered ones, which is easy to implement and applicable to general diffusion tensors. The main novelty of this paper is
to design a multigrid prolongation operator based on the GT method and splitting extrapolation method, and then propose a cell-centered EXCMG method with BiCGStab
smoother for solving the large linear system resulting from linear FV discretization
of diffusion equations with strongly discontinuous and anisotropic coefficients. Numerical experiments are presented to demonstrate the high efficiency of the proposed
method. 相似文献
7.
Mathematical and Numerical Aspects of the Adaptive Fast Multipole Poisson-Boltzmann Solver 下载免费PDF全文
Bo Zhang Benzhuo Lu Xiaolin Cheng Jingfang Huang Nikos P. Pitsianis Xiaobai Sun & J. Andrew McCammon 《Communications In Computational Physics》2013,13(1):107-128
This paper summarizes the mathematical and numerical theories and computational elements of the adaptive fast multipole Poisson-Boltzmann (AFMPB) solver.
We introduce and discuss the following components in order: the Poisson-Boltzmann
model, boundary integral equation reformulation, surface mesh generation, the node-patch discretization approach, Krylov iterative methods, the new version of fast multipole methods (FMMs), and a dynamic prioritization technique for scheduling parallel
operations. For each component, we also remark on feasible approaches for further
improvements in efficiency, accuracy and applicability of the AFMPB solver to large-scale long-time molecular dynamics simulations. The potential of the solver is demonstrated with preliminary numerical results. 相似文献
8.
Craig Collins Jie Shen & Steven M. Wise 《Communications In Computational Physics》2013,13(4):929-957
We present an unconditionally energy stable and uniquely solvable finite
difference scheme for the Cahn-Hilliard-Brinkman (CHB) system, which is comprised
of a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation modeling fluid flow. The CHB system is a generalization of the Cahn-Hilliard-Stokes model
and describes two phase very viscous flows in porous media. The scheme is based on
a convex splitting of the discrete CH energy and is semi-implicit. The equations at the
implicit time level are nonlinear, but we prove that they represent the gradient of a
strictly convex functional and are therefore uniquely solvable, regardless of time step
size. Owing to energy stability, we show that the scheme is stable in the time and space
discrete$ℓ^∞$(0,$T$;$H^1_h$) and $ℓ^2$(0,$T$;$H^2_h$) norms. We also present an efficient, practical nonlinear multigrid method – comprised of a standard FAS method for the Cahn-Hilliard
part, and a method based on the Vanka smoothing strategy for the Brinkman part – for
solving these equations. In particular, we provide evidence that the solver has nearly
optimal complexity in typical situations. The solver is applied to simulate spinodal
decomposition of a viscous fluid in a porous medium, as well as to the more general
problems of buoyancy- and boundary-driven flows. 相似文献
9.
The immersed boundary method has been extensively used to simulate the
motion of elastic structures immersed in a viscous fluid. For some applications, such as
modeling biological materials, capturing internal boundary viscosity is important. We
present numerical methods for simulating Kelvin-Voigt and standard linear viscoelastic structures immersed in zero Reynolds number flow. We find that the explicit time
immersed boundary update is unconditionally unstable above a critical boundary to
fluid viscosity ratio for a Kelvin-Voigt material. We also show there is a severe time
step restriction when simulating a standard linear boundary with a small relaxation
time scale using the same explicit update. A stable implicit method is presented to
overcome these computation challenges. 相似文献
10.
An Efficient Neural-Network and Finite-Difference Hybrid Method for Elliptic Interface Problems with Applications 下载免费PDF全文
Wei-Fan Hu Te-Sheng Lin Yu-Hau Tseng & Ming-Chih Lai 《Communications In Computational Physics》2023,33(4):1090-1105
A new and efficient neural-network and finite-difference hybrid method is
developed for solving Poisson equation in a regular domain with jump discontinuities
on embedded irregular interfaces. Since the solution has low regularity across the interface, when applying finite difference discretization to this problem, an additional
treatment accounting for the jump discontinuities must be employed. Here, we aim to
elevate such an extra effort to ease our implementation by machine learning methodology. The key idea is to decompose the solution into singular and regular parts. The
neural network learning machinery incorporating the given jump conditions finds the
singular solution, while the standard five-point Laplacian discretization is used to obtain the regular solution with associated boundary conditions. Regardless of the interface geometry, these two tasks only require supervised learning for function approximation and a fast direct solver for Poisson equation, making the hybrid method easy
to implement and efficient. The two- and three-dimensional numerical results show
that the present hybrid method preserves second-order accuracy for the solution and
its derivatives, and it is comparable with the traditional immersed interface method in
the literature. As an application, we solve the Stokes equations with singular forces to
demonstrate the robustness of the present method. 相似文献
11.
A Weighted Runge-Kutta Discontinuous Galerkin Method for 3D Acoustic and Elastic Wave-Field Modeling 下载免费PDF全文
Numerically solving 3D seismic wave equations is a key requirement for
forward modeling and inversion. Here, we propose a weighted Runge-Kutta discontinuous Galerkin (WRKDG) method for 3D acoustic and elastic wave-field modeling. For this method, the second-order seismic wave equations in 3D heterogeneous anisotropic media are transformed into a first-order hyperbolic system, and
then we use a discontinuous Galerkin (DG) solver based on numerical-flux formulations for spatial discretization. The time discretization is based on an implicit diagonal Runge-Kutta (RK) method and an explicit iterative technique, which avoids
solving a large-scale system of linear equations. In the iterative process, we introduce
a weighting factor. We investigate the numerical stability criteria of the 3D method in
detail for linear and quadratic spatial basis functions. We also present a 3D analysis of
numerical dispersion for the full discrete approximation of acoustic equation, which
demonstrates that the WRKDG method can efficiently suppress numerical dispersion
on coarse grids. Numerical results for several different 3D models including homogeneous and heterogeneous media with isotropic and anisotropic cases show that the 3D
WRKDG method can effectively suppress numerical dispersion and provide accurate
wave-field information on coarse mesh. 相似文献
12.
A Numerical Investigation for a Model of the Solid-Gas Phase in a Crystal Growth Apparatus 下载免费PDF全文
We present discretization and solver methods for a model of the solid-gas
phase in a crystal growth apparatus. The model equations are coupled Eulerian and
heat-transfer equations with flux boundary conditions. For a more detailed discussion
we consider simpler equations and present time- and space-decomposition methods as
solver methods to decouple the multi-physics processes. We present the error analysis
for the discretization and solver methods. Numerical experiments are performed for
the Eulerian and heat-transfer equation using decomposition methods. We present
a real-life application of a crystal growth apparatus, based on underlying stationary
heat conduction. Finally we discuss further error analysis and application to a more
complex model of crystal growth. 相似文献
13.
Benchmark Computations of the Phase Field Crystal and Functionalized Cahn-Hilliard Equations via Fully Implicit,Nesterov Accelerated Schemes 下载免费PDF全文
Jea-Hyun Park Abner J. Salgado & Steven M. Wise 《Communications In Computational Physics》2023,33(2):367-398
We introduce a fast solver for the phase field crystal (PFC) and functionalized Cahn-Hilliard (FCH) equations with periodic boundary conditions on a rectangular domain that features the preconditioned Nesterov’s accelerated gradient descent
(PAGD) method. We discretize these problems with a Fourier collocation method
in space, and employ various second-order schemes in time. We observe a significant speedup with this solver when compared to the preconditioned gradient descent
(PGD) method. With the PAGD solver, fully implicit, second-order-in-time schemes
are not only feasible to solve the PFC and FCH equations, but also do so more efficiently than some semi-implicit schemes in some cases where accuracy issues are
taken into account. Benchmark computations of four different schemes for the PFC
and FCH equations are conducted and the results indicate that, for the FCH experiments, the fully implicit schemes (midpoint rule and BDF2 equipped with the PAGD
as a nonlinear time marching solver) perform better than their IMEX versions in terms
of computational cost needed to achieve a certain precision. For the PFC, the results
are not as conclusive as in the FCH experiments, which, we believe, is due to the fact
that the nonlinearity in the PFC is milder nature compared to the FCH equation. We
also discuss some practical matters in applying the PAGD. We introduce an averaged
Newton preconditioner and a sweeping-friction strategy as heuristic ways to choose good
preconditioner parameters. The sweeping-friction strategy exhibits almost as good
a performance as the case of the best manually tuned parameters. 相似文献
14.
In this paper, a novel implementation of immersed interface method combined
with Stokes solver on a MAC staggered grid for solving the steady two-fluid
Stokes equations with interfaces. The velocity components along the interface are introduced
as two augmented variables and the resulting augmented equation is then
solved by the GMRES method. The augmented variables and/or the forces are related
to the jumps in pressure and the jumps in the derivatives of both pressure and velocity,
and are interpolated using cubic splines and are then applied to the fluid through
the jump conditions. The Stokes equations are discretized on a staggered Cartesian
grid via a second order finite difference method and solved by the conjugate gradient
Uzawa-type method. The numerical results show that the overall scheme is second order
accuracy. The major advantages of the present IIM-Stokes solver are the efficiency
and flexibility in terms of types of fluid flow and different boundary conditions. The
proposed method avoids solution of the pressure Poisson equation, and comparisons
are made to show the advantages of time savings by the present method. The generalized
two-phase Stokes solver with correction terms has also been applied to incompressible
two-phase Navier-Stokes flow. 相似文献
15.
A Comparative Study of Rosenbrock-Type and Implicit Runge-Kutta Time Integration for Discontinuous Galerkin Method for Unsteady 3D Compressible Navier-Stokes equations 下载免费PDF全文
Xiaodong Liu Yidong Xia Hong Luo & Lijun Xuan 《Communications In Computational Physics》2016,20(4):1016-1044
A comparative study of two classes of third-order implicit time integration
schemes is presented for a third-order hierarchical WENO reconstructed discontinuous
Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes
equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3)
scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential
algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme,
a remarkable feature of the ROW schemes is that, they only require one approximate
Jacobian matrix calculation every time step, thus considerably reducing the overall
computational cost. A variety of test cases, ranging from inviscid flows to DNS of
turbulent flows, are presented to assess the performance of these schemes. Numerical
experiments demonstrate that the third-order ROW scheme for the DAEs of index-2
can not only achieve the designed formal order of temporal convergence accuracy in
a benchmark test, but also require significantly less computing time than its ESDIRK3
counterpart to converge to the same level of discretization errors in all of the flow
simulations in this study, indicating that the ROW methods provide an attractive alternative
for the higher-order time-accurate integration of the unsteady compressible
Navier-Stokes equations. 相似文献
16.
A Conservative and Monotone Characteristic Finite Element Solver for Three-Dimensional Transport and Incompressible Navier-Stokes Equations on Unstructured Grids 下载免费PDF全文
Bassou Khouya Mofdi El-Amrani & Mohammed Seaid 《Communications In Computational Physics》2022,31(1):224-256
We propose a mass-conservative and monotonicity-preserving characteristic finite element method for solving three-dimensional transport and incompressible
Navier-Stokes equations on unstructured grids. The main idea in the proposed algorithm consists of combining a mass-conservative and monotonicity-preserving modified method of characteristics for the time integration with a mixed finite element
method for the space discretization. This class of computational solvers benefits from
the geometrical flexibility of the finite elements and the strong stability of the modified method of characteristics to accurately solve convection-dominated flows using
time steps larger than its Eulerian counterparts. In the current study, we implement
three-dimensional limiters to convert the proposed solver to a fully mass-conservative
and essentially monotonicity-preserving method in addition of a low computational
cost. The key idea lies on using quadratic and linear basis functions of the mesh element where the departure point is localized in the interpolation procedures. The
proposed method is applied to well-established problems for transport and incompressible Navier-Stokes equations in three space dimensions. The numerical results
illustrate the performance of the proposed solver and support its ability to yield accurate and efficient numerical solutions for three-dimensional convection-dominated
flow problems on unstructured tetrahedral meshes. 相似文献
17.
J. Vides B. Braconnier E. Audit C. Berthon & B. Nkonga 《Communications In Computational Physics》2014,15(1):46-75
We present a new numerical method to approximate the solutions of an
Euler-Poisson model, which is inherent to astrophysical flows where gravity plays an
important role. We propose a discretization of gravity which ensures adequate coupling of the Poisson and Euler equations, paying particular attention to the gravity
source term involved in the latter equations. In order to approximate this source term,
its discretization is introduced into the approximate Riemann solver used for the Euler equations. A relaxation scheme is involved and its robustness is established. The
method has been implemented in the software HERACLES [29] and several numerical
experiments involving gravitational flows for astrophysics highlight the scheme. 相似文献
18.
Wenqiang Feng Xiaoming He Yanping Lin & Xu Zhang 《Communications In Computational Physics》2014,15(4):1045-1067
This article is to discuss the bilinear and linear immersed finite element (IFE)
solutions generated from the algebraic multigrid solver for both stationary and moving
interface problems. For the numerical methods based on finite difference formulation
and a structured mesh independent of the interface, the stiffness matrix of the linear
system is usually not symmetric positive-definite, which demands extra efforts to design
efficient multigrid methods. On the other hand, the stiffness matrix arising from
the IFE methods is naturally symmetric positive-definite. Hence the IFE-AMG algorithm
is proposed to solve the linear systems of the bilinear and linear IFE methods
for both stationary and moving interface problems. The numerical examples demonstrate
the features of the proposed algorithms, including the optimal convergence in
both $L^2$ and semi-$H^1$ norms of the IFE-AMG solutions, the high efficiency with proper
choice of the components and parameters of AMG, the influence of the tolerance and
the smoother type of AMG on the convergence of the IFE solutions for the interface
problems, and the relationship between the cost and the moving interface location. 相似文献
19.
Bo Gong Jiayu Han Jiguang Sun & Zhimin Zhang 《Communications In Computational Physics》2020,27(1):251-273
A shifted-inverse iteration is proposed for the finite element discretization
of the elastic eigenvalue problem. The method integrates the multigrid scheme and
adaptive algorithm to achieve high efficiency and accuracy. Error estimates and optimal convergence for the proposed method are proved. Numerical examples show that
the proposed method inherits the advantages of both ingredients and can compute low
regularity eigenfunctions effectively. 相似文献
20.
Extended hydrodynamic models for carrier transport are derived from the
semiconductor Boltzmann equation with relaxation time approximation of the scattering
term, by using the globally hyperbolic moment method and the moment-dependent
relaxation time. Incorporating the microscopic relaxation time and the applied voltage
bias, a formula is proposed to determine the relaxation time for each moment equation,
which sets different relaxation rates for different moments such that higher moments
damp faster. The resulting models would give more satisfactory results of macroscopic
quantities of interest with a high-order convergence to those of the underlying Boltzmann
equation as the involved moments increase, in comparison to the corresponding
moment models using a single relaxation time. In order to simulate the steady states
efficiently, a multigrid solver is developed for the derived moment models. Numerical
simulations of an $n^+-n-n^+$ silicon diode are carried out to demonstrate the validation
of the presented moment models, and the robustness and efficiency of the designed
multigrid solver. 相似文献