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1.
This paper presents a new approach to verify the accuracy of computational
simulations. We develop mathematical theorems which can serve as robust a posteriori
error estimation techniques to identify numerical pollution, check the performance of
adaptive meshes, and verify numerical solutions. We demonstrate performance of this
methodology on problems from flow thorough porous media. However, one can extend
it to other models. We construct mathematical properties such that the solutions
to Darcy and Darcy-Brinkman equations satisfy them. The mathematical properties
include the total minimum mechanical power, minimum dissipation theorem, reciprocal
relation, and maximum principle for the vorticity. All the developed theorems
have firm mechanical bases and are independent of numerical methods. So, these can
be utilized for solution verification of finite element, finite volume, finite difference,
lattice Boltzmann methods and so forth. In particular, we show that, for a given set of
boundary conditions, Darcy velocity has the minimum total mechanical power of all
the kinematically admissible vector fields. We also show that a similar result holds for
Darcy-Brinkman velocity. We then show for a conservative body force, the Darcy and
Darcy-Brinkman velocities have the minimum total dissipation among their respective
kinematically admissible vector fields. Using numerical examples, we show that the
minimum dissipation and total mechanical power theorems can be utilized to identify
pollution errors in numerical solutions. The solutions to Darcy and Darcy-Brinkman
equations are shown to satisfy a reciprocal relation, which has the potential to identify
errors in the numerical implementation of boundary conditions. It is also shown
that the vorticity under both steady and transient Darcy-Brinkman equations satisfy
maximum principles if the body force is conservative and the permeability is homogeneous
and isotropic. A discussion on the nature of vorticity under steady and transient
Darcy equations is also presented. Using several numerical examples, we will demonstrate
the predictive capabilities of the proposed a posteriori techniques in assessing the
accuracy of numerical solutions for a general class of problems, which could involve
complex domains and general computational grids. 相似文献
2.
A Strong Stability-Preserving Predictor-Corrector Method for the Simulation of Elastic Wave Propagation in Anisotropic Media
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In this paper, we propose a strong stability-preserving predictor-corrector
(SSPC) method based on an implicit Runge-Kutta method to solve the acoustic- and
elastic-wave equations. We first transform the wave equations into a system of ordinary differential equations (ODEs) and apply the local extrapolation method to discretize the spatial high-order derivatives, resulting in a system of semi-discrete ODEs.
Then we use the SSPC method based on an implicit Runge-Kutta method to solve
the semi-discrete ODEs and introduce a weighting parameter into the SSPC method.
On top of such a structure, we develop a robust numerical algorithm to effectively
suppress the numerical dispersion, which is usually caused by the discretization of
wave equations when coarse grids are used or geological models have large velocity
contrasts between adjacent layers. Meanwhile, we investigate the performance of the
SSPC method including numerical errors and convergence rate, numerical dispersion,
and stability criteria with different choices of the weighting parameter to solve 1-D
and 2-D acoustic- and elastic-wave equations. When the SSPC is applied to seismic
simulations, the computational efficiency is also investigated by comparing the SSPC,
the fourth-order Lax-Wendroff correction (LWC) method, and the staggered-grid (SG)
finite difference method. Comparisons of synthetic waveforms computed by the SSPC
and analytic solutions for acoustic and elastic models are given to illustrate the accuracy and the validity of the SSPC method. Furthermore, several numerical experiments
are conducted for the geological models including a 2-D homogeneous transversely
isotropic (TI) medium, a two-layer elastic model, and the 2-D SEG/EAGE salt model.
The results show that the SSPC can be used as a practical tool for large-scale seismic
simulation because of its effectiveness in suppressing numerical dispersion even in the
situations such as coarse grids, strong interfaces, or high frequencies. 相似文献
3.
Development of an Explicit Symplectic Scheme that Optimizes the Dispersion-Relation Equation of the Maxwell's Equations
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Tony W. H. Sheu L. Y. Liang & J. H. Li 《Communications In Computational Physics》2013,13(4):1107-1133
In this paper an explicit finite-difference time-domain scheme for solving
the Maxwell's equations in non-staggered grids is presented. The proposed scheme
for solving the Faraday's and Ampère's equations in a theoretical manner is aimed to
preserve discrete zero-divergence for the electric and magnetic fields. The inherent local conservation laws in Maxwell's equations are also preserved discretely all the time
using the explicit second-order accurate symplectic partitioned Runge-Kutta scheme.
The remaining spatial derivative terms in the semi-discretized Faraday's and Ampère's
equations are then discretized to provide an accurate mathematical dispersion relation
equation that governs the numerical angular frequency and the wavenumbers in two
space dimensions. To achieve the goal of getting the best dispersive characteristics, we
propose a fourth-order accurate space centered scheme which minimizes the difference
between the exact and numerical dispersion relation equations. Through the computational exercises, the proposed dual-preserving solver is computationally demonstrated
to be efficient for use to predict the long-term accurate Maxwell's solutions. 相似文献
4.
This study shows a new way to implement terrain-following σ-coordinate
in a numerical model, which does not lead to the well-known "pressure gradient force
(PGF)" problem. First, the causes of the PGF problem are analyzed with existing methods that are categorized into two different types based on the causes. Then, the new
method that bypasses the PGF problem all together is proposed. By comparing these
three methods and analyzing the expression of the scalar gradient in a curvilinear coordinate system, this study finds out that only when using the covariant scalar equations
of σ-coordinate will the PGF computational form have one term in each momentum
component equation, thereby avoiding the PGF problem completely. A convenient
way of implementing the covariant scalar equations of σ-coordinate in a numerical atmospheric model is illustrated, which is to set corresponding parameters in the scalar
equations of the Cartesian coordinate. Finally, two idealized experiments manifest that
the PGF calculated with the new method is more accurate than using the classic one.
This method can be used for oceanic models as well, and needs to be tested in both the
atmospheric and oceanic models. 相似文献
5.
A Hodge Decomposition Method for Dynamic Ginzburg–Landau Equations in Nonsmooth Domains — A Second Approach
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In a general polygonal domain, possibly nonconvex and multi-connected
(with holes), the time-dependent Ginzburg–Landau equation is reformulated into a
new system of equations. The magnetic field $B$:=∇×A is introduced as an unknown
solution in the new system, while the magnetic potential A is solved implicitly through
its Hodge decomposition into divergence-free part, curl-free and harmonic parts, separately. Global well-posedness of the new system and its equivalence to the original problem are proved. A linearized and decoupled Galerkin finite element method
is proposed for solving the new system. The convergence of numerical solutions is
proved based on a compactness argument by utilizing the maximal $L^p$-regularity of
the discretized equations. Compared with the Hodge decomposition method proposed in [27],the new method has the advantage of approximating the magnetic field
B directly and converging for initial conditions that are incompatible with the external
magnetic field. Several numerical examples are provided to illustrate the efficiency of
the proposed numerical method in both simply connected and multi-connected nonsmooth domains. We observe that even in simply connected domains, the new method
is superior to the method in [27] for approximating the magnetic field. 相似文献
6.
Truncation Errors,Exact and Heuristic Stability Analysis of Two-Relaxation-Times Lattice Boltzmann Schemes for Anisotropic Advection-Diffusion Equation
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Irina Ginzburg 《Communications In Computational Physics》2012,11(5):1439-1502
This paper establishes relations between the stability and the high-order
truncated corrections for modeling of the mass conservation equation with the two-relaxation-times (TRT) collision operator. First we propose a simple method to derive
the truncation errors from the exact, central-difference type, recurrence equations of
the TRT scheme. They also supply its equivalent three-time-level discretization form.
Two different relationships of the two relaxation rates nullify the third (advection) and
fourth (pure diffusion) truncation errors, for any linear equilibrium and any velocity set. However, the two relaxation times alone cannot remove the leading-order
advection-diffusion error, because of the intrinsic fourth-order numerical diffusion.
The truncation analysis is carefully verified for the evolution of concentration waves
with the anisotropic diffusion tensors. The anisotropic equilibrium functions are presented in a simple but general form, suitable for the minimal velocity sets and the
d2Q9, d3Q13, d3Q15 and d3Q19 velocity sets. All anisotropic schemes are complemented by their exact necessary von Neumann stability conditions and equivalent
finite-difference stencils. The sufficient stability conditions are proposed for the most
stable (OTRT) family, which enables modeling at any Peclet numbers with the same
velocity amplitude. The heuristic stability analysis of the fourth-order truncated corrections extends the optimal stability to larger relationships of the two relaxation rates,
in agreement with the exact (one-dimensional) and numerical (multi-dimensional) stability analysis. A special attention is put on the choice of the equilibrium weights. By
combining accuracy and stability predictions, several strategies for selecting the relaxation and free-tunable equilibrium parameters are suggested and applied to the
evolution of the Gaussian hill. 相似文献
7.
Laplace-Transform Finite Element Solution of Nonlocal and Localized Stochastic Moment Equations of Transport
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Eric Morales-Casique & Shlomo P. Neuman 《Communications In Computational Physics》2009,6(1):131-161
Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1238-1255) developed
exact first and second nonlocal moment equations for advective-dispersive transport
in finite, randomly heterogeneous geologic media. The velocity and concentration
in these equations are generally nonstationary due to trends in heterogeneity, conditioning
on site data and the influence of forcing terms. Morales-Casique et al. (Adv.
Water Res., 29 (2006), pp. 1399-1418) solved the Laplace transformed versions of these
equations recursively to second order in the standard deviation σY of (natural) log hydraulic
conductivity, and iteratively to higher-order, by finite elements followed by
numerical inversion of the Laplace transform. They did the same for a space-localized
version of the mean transport equation. Here we recount briefly their theory and algorithms;
compare the numerical performance of the Laplace-transform finite element
scheme with that of a high-accuracy ULTIMATE-QUICKEST algorithm coupled with
an alternating split operator approach; and review some computational results due to
Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1399-1418) to shed light on
the accuracy and computational efficiency of their recursive and iterative solutions in
comparison to conditional Monte Carlo simulations in two spatial dimensions. 相似文献
8.
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
9.
A Numerical Methodology for Enforcing Maximum Principles and the Non-Negative Constraint for Transient Diffusion Equations
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K. B. Nakshatrala H. Nagarajan & M. Shabouei 《Communications In Computational Physics》2016,19(1):53-93
Transient diffusion equations arise in many branches of engineering and
applied sciences (e.g., heat transfer and mass transfer), and are parabolic partial differential
equations. It is well-known that these equations satisfy important mathematical
properties like maximum principles and the non-negative constraint, which have implications
in mathematical modeling. However, existing numerical formulations for
these types of equations do not, in general, satisfy maximum principles and the non-negative
constraint. In this paper, we present a methodology for enforcing maximum
principles and the non-negative constraint for transient anisotropic diffusion equation.
The proposed methodology is based on the method of horizontal lines in which
the time is discretized first. This results in solving steady anisotropic diffusion equation
with decay equation at every discrete time-level. We also present other plausible
temporal discretizations, and illustrate their shortcomings in meeting maximum principles
and the non-negative constraint. The proposed methodology can handle general
computational grids with no additional restrictions on the time-step. We illustrate the
performance and accuracy of the proposed methodology using representative numerical
examples. We also perform a numerical convergence analysis of the proposed
methodology. For comparison, we also present the results from the standard single-field
semi-discrete formulation and the results from a popular software package, which
all will violate maximum principles and the non-negative constraint. 相似文献
10.
A. Shamaly G. S. Christensen M. E. El-Hawary 《Optimal control applications & methods.》1981,2(1):81-87
The computational solution of the optimality equations for torque and voltage control of a large-scale turbo-alternator is considered. The equations are extremely ill-conditioned as a consequence of the ill-conditioning of the linearized system model. Our main result is to point out that using the Ricatti equation to estimate the initial values of the Lagrange multipliers renders the equations in a form that can easily be solved as an initial value problem for the non-linear system. 相似文献
11.
A Gas-Kinetic Unified Algorithm for Non-Equilibrium Polyatomic Gas Flows Covering Various Flow Regimes
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Wen-Qiang Hu Zhi-Hui Li Ao-Ping Peng & Xin-Yu Jiang 《Communications In Computational Physics》2021,30(1):144-189
In this paper, a gas-kinetic unified algorithm (GKUA) is developed to investigate the non-equilibrium polyatomic gas flows covering various regimes. Based
on the ellipsoidal statistical model with rotational energy excitation, the computable
modelling equation is presented by unifying expressions on the molecular collision relaxing parameter and the local equilibrium distribution function. By constructing the
corresponding conservative discrete velocity ordinate method for this model, the conservative properties during the collision procedure are preserved at the discrete level
by the numerical method, decreasing the computational storage and time. Explicit
and implicit lower-upper symmetric Gauss-Seidel schemes are constructed to solve
the discrete hyperbolic conservation equations directly. Applying the new GKUA,
some numerical examples are simulated, including the Sod Riemann problem, homogeneous flow rotational relaxation, normal shock structure, Fourier and Couette flows,
supersonic flows past a circular cylinder, and hypersonic flow around a plate placed
normally. The results obtained by the analytic, experimental, direct simulation Monte
Carlo method, and other measurements in references are compared with the GKUA
results, which are in good agreement, demonstrating the high accuracy of the present
algorithm. Especially, some polyatomic gas non-equilibrium phenomena are observed
and analysed by solving the Boltzmann-type velocity distribution function equation
covering various flow regimes. 相似文献
12.
The Hamiltonian Field Theory of the Von Mises Wave Equation: Analytical and Computational Issues
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Christian Cherubini & Simonetta Filippi 《Communications In Computational Physics》2016,19(3):758-769
The Von Mises quasi-linear second order wave equation, which completely
describes an irrotational, compressible and barotropic classical perfect fluid, can be derived
from a nontrivial least action principle for the velocity scalar potential only, in
contrast to existing analog formulations which are expressed in terms of coupled density
and velocity fields. In this article, the classical Hamiltonian field theory specifically
associated to such an equation is developed in the polytropic case and numerically
verified in a simplified situation. The existence of such a mathematical structure suggests
new theoretical schemes possibly useful for performing numerical integrations of
fluid dynamical equations. Moreover, it justifies possible new functional forms for Lagrangian
densities and associated Hamiltonian functions in other theoretical classical
physics contexts. 相似文献
13.
High-Order Gas-Kinetic Scheme in Curvilinear Coordinates for the Euler and Navier-Stokes Solutions
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Liang Pan & Kun Xu 《Communications In Computational Physics》2020,28(4):1321-1351
The high-order gas-kinetic scheme (HGKS) has achieved success in simulating compressible flows with Cartesian meshes. To study the flow problems in general
geometries, such as the flow over a wing-body, the development of HGKS in general
curvilinear coordinates becomes necessary. In this paper, a two-stage fourth-order gas-kinetic scheme is developed for the Euler and Navier-Stokes solutions in the curvilinear coordinates from one-dimensional to three-dimensional computations. Based on
the coordinate transformation, the kinetic equation is transformed first to the computational space, and the flux function in the gas-kinetic scheme is obtained there and
is transformed back to the physical domain for the update of flow variables inside
each control volume. To achieve the expected order of accuracy, the dimension-by-dimension reconstruction based on the WENO scheme is adopted in the computational domain, where the reconstructed variables are the cell averaged Jacobian and the
Jacobian-weighted conservative variables. In the two-stage fourth-order gas-kinetic
scheme, the point values as well as the spatial derivatives of conservative variables at
Gaussian quadrature points have to be used in the evaluation of the time dependent
flux function. The point-wise conservative variables are obtained by ratio of the above
reconstructed data, and the spatial derivatives are reconstructed through orthogonalization in physical space and chain rule. A variety of numerical examples from the
accuracy tests to the solutions with strong discontinuities are presented to validate the
accuracy and robustness of the current scheme for both inviscid and viscous flows.
The precise satisfaction of the geometrical conservation law in non-orthogonal mesh is
also demonstrated through the numerical example. 相似文献
14.
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
15.
This paper develops a high-order accurate gas-kinetic scheme in the framework
of the finite volume method for the one- and two-dimensional flow simulations,
which is an extension of the third-order accurate gas-kinetic scheme [Q.B. Li, K. Xu, and
S. Fu, J. Comput. Phys., 229(2010), 6715-6731] and the second-order accurate gas-kinetic
scheme [K. Xu, J. Comput. Phys., 171(2001), 289-335]. It is formed by two parts: quartic
polynomial reconstruction of the macroscopic variables and fourth-order accurate flux
evolution. The first part reconstructs a piecewise cell-center based quartic polynomial
and a cell-vertex based quartic polynomial according to the "initial" cell average approximation
of macroscopic variables to recover locally the non-equilibrium and equilibrium
single particle velocity distribution functions around the cell interface. It is in
view of the fact that all macroscopic variables become moments of a single particle velocity
distribution function in the gas-kinetic theory. The generalized moment limiter
is employed there to suppress the possible numerical oscillation. In the second part,
the macroscopic flux at the cell interface is evolved in fourth-order accuracy by means
of the simple particle transport mechanism in the microscopic level, i.e. free transport
and the Bhatnagar-Gross-Krook (BGK) collisions. In other words, the fourth-order
flux evolution is based on the solution (i.e. the particle velocity distribution function)
of the BGK model for the Boltzmann equation. Several 1D and 2D test problems are
numerically solved by using the proposed high-order accurate gas-kinetic scheme. By
comparing with the exact solutions or the numerical solutions obtained the second-order
or third-order accurate gas-kinetic scheme, the computations demonstrate that
our scheme is effective and accurate for simulating invisid and viscous fluid flows,
and the accuracy of the high-order GKS depends on the choice of the (numerical) collision
time. 相似文献
16.
Julian Koellermeier & Marvin Rominger 《Communications In Computational Physics》2020,28(3):1038-1084
Shallow Water Moment Equations allow for vertical changes in the horizontal velocity, so that complex shallow flows can be described accurately. However, we
show that these models lack global hyperbolicity and the loss of hyperbolicity already
occurs for small deviations from equilibrium. This leads to instabilities in a numerical
test case. We then derive new Hyperbolic Shallow Water Moment Equations based on
a modification of the system matrix. The model can be written in analytical form and
hyperbolicity can be proven for a large number of equations. A second variant of this
model is obtained by generalizing the modification with the help of additional parameters. Numerical tests of a smooth periodic problem and a dam break problem using
the new models yield accurate and fast solutions while guaranteeing hyperbolicity. 相似文献
17.
In this paper, we study the Camassa-Holm equation and the Degasperis-Procesi
equation. The two equations are in the family of integrable peakon equations,
and both have very rich geometric properties. Based on these geometric structures, we
construct the geometric numerical integrators for simulating their soliton solutions.
The Camassa-Holm equation and the Degasperis-Procesi equation have many common
properties, however, they also have the significant differences, for example, there
exist the shock wave solutions for the Degasperis-Procesi equation. By using the symplectic
Fourier pseudo-spectral integrator, we simulate the peakon solutions of the two
equations. To illustrate the smooth solitons and shock wave solutions of the DP equation,
we use the splitting technique and combine the composition methods. In the
numerical experiments, comparisons of these two kinds of methods are presented in
terms of accuracy, computational cost and invariants preservation. 相似文献
18.
Yongping Cheng Haiyun Dong Maojun Li & Weizhi Xian 《Communications In Computational Physics》2020,28(4):1437-1463
In this paper, we focus on the numerical simulation of the two-layer shallow water equations over variable bottom topography. Although the existing numerical schemes for the single-layer shallow water equations can be extended to two-layer
shallow water equations, it is not a trivial work due to the complexity of the equations.
To achieve the well-balanced property of the numerical scheme easily, the two-layer
shallow water equations are reformulated into a new form by introducing two auxiliary variables. Since the new equations are only conditionally hyperbolic and their
eigenstructure cannot be easily obtained, we consider the utilization of the central discontinuous Galerkin method which is free of Riemann solvers. By choosing the values
of the auxiliary variables suitably, we can prove that the scheme can exactly preserve
the still-water solution, and thus it is a truly well-balanced scheme. To ensure the
non-negativity of the water depth, a positivity-preserving limiter and a special approximation to the bottom topography are employed. The accuracy and validity of the
numerical method will be illustrated through some numerical tests. 相似文献
19.
S. C. Fu R. M. C. So & W. W. F. Leung 《Communications In Computational Physics》2011,9(5):1257-1283
The objective of this paper is to seek an alternative to the numerical simulation
of the Navier-Stokes equations by a method similar to solving the BGK-type
modeled lattice Boltzmann equation. The proposed method is valid for both gas and
liquid flows. A discrete flux scheme (DFS) is used to derive the governing equations
for two distribution functions; one for mass and another for thermal energy. These
equations are derived by considering an infinitesimally small control volume with a
velocity lattice representation for the distribution functions. The zero-order moment
equation of the mass distribution function is used to recover the continuity equation,
while the first-order moment equation recovers the linear momentum equation. The
recovered equations are correct to the first order of the Knudsen number (Kn); thus,
satisfying the continuum assumption. Similarly, the zero-order moment equation of
the thermal energy distribution function is used to recover the thermal energy equation.
For aerodynamic flows, it is shown that the finite difference solution of the DFS
is equivalent to solving the lattice Boltzmann equation (LBE) with a BGK-type model
and a specified equation of state. Thus formulated, the DFS can be used to simulate a
variety of aerodynamic and hydrodynamic flows. Examples of classical aeroacoustics,
compressible flow with shocks, incompressible isothermal and non-isothermal Couette
flows, stratified flow in a cavity, and double diffusive flow inside a rectangle are used
to demonstrate the validity and extent of the DFS. Very good to excellent agreement
with known analytical and/or numerical solutions is obtained; thus lending evidence
to the DFS approach as an alternative to solving the Navier-Stokes equations for fluid
flow simulations. 相似文献
20.
In this article, we address both recent advances and open questions in some mathematical and computational issues in geophysical fluid dynamics (GFD) and climate dynamics. The main focus is on 1) the primitive equations (PEs) models and their related mathematical and computational issues, 2) climate variability, predictability and successive bifurcation, and 3) a new dynamical systems theory and its applications to GFD and climate dynamics. 相似文献