The purpose of this paper is to analyze some features of contaminant
flow passing through cracked porous medium, such as the influence of
fracture network on the advection and diffusion of contaminant species,
the impact of adsorption on the overall transport of contaminant wastes.
In order to precisely describe the whole process, we firstly build the
mathematical model to simulate this problem numerically. Taking into
consideration of the characteristics of contaminant flow, we employ
two partial differential equations to formulate the whole problem.
One is flow equation; the other is reactive transport equation.
The first equation is used to describe the total flow of contaminant
wastes, which is based on Darcy law. The second one will characterize
the adsorption, diffusion and convection behavior of contaminant species,
which describes most features of contaminant flow we are interested in.
After the construction of numerical model, we apply locally conservative
and compatible algorithms to solve this mathematical model. Specifically,
we apply Mixed Finite Element (MFE) method to the flow equation and
Discontinuous Galerkin (DG) method for the transport equation. MFE has
a good convergence rate and numerical accuracy for Darcy velocity. DG
is more flexible and can be used to deal with irregular meshes, as well
as little numerical diffusion. With these two numerical means, we
investigate the sensitivity analysis of different features of contaminant
flow in our model, such as diffusion, permeability and fracture density.
In particular, we study $K_d$ values which represent the distribution of
contaminant wastes between the solid and liquid phases. We also make comparisons of two different schemes and discuss the advantages of both methods.
In this paper, a reliable algorithm based on an adaptation of the
standard differential transform method is presented, which is the
multi-step differential transform method (MSDTM). The solutions of
non-linear oscillators were obtained by MSDTM. Figurative comparisons
between the MSDTM and the classical fourth-order Runge-Kutta method (RK4)
reveal that the proposed technique is a promising tool to solve non-linear
A novel, highly efficient and accurate adaptive higher-order finite element
method ($hp$-FEM) is used to simulate a multi-frequency resistivity logging-while-drilling (LWD)
tool response in a borehole environment. Presented in this study are the vector expression
of Maxwell's equations, three kinds of boundary conditions, stability weak formulation of
Maxwell's equations, and automatic $hp$-adaptivity strategy. The new $hp$-FEM can select
optimal refinement and calculation strategies based on the practical formation model and
error estimation. Numerical experiments show that the new $hp$-FEM has an exponential
convergence rate in terms of relative error in a user-prescribed quantity of interest
against the degrees of freedom, which provides more accurate results than those obtained
using the adaptive $h$-FEM. The numerical results illustrate the high efficiency and
accuracy of the method at a given LWD tool structure and parameters in different physical
models, which further confirm the accuracy of the results using the Hermes
library (http://hpfem.org/hermes) with a multi-frequency resistivity LWD tool
response in a borehole environment.
This paper at first shows the details of finite volume-based lattice Boltzmann
method (FV-LBM) for simulation of compressible flows with shock waves. In the
FV-LBM, the normal convective flux at the interface of a cell is evaluated by
using one-dimensional compressible lattice Boltzmann model, while the tangential
flux is calculated using the same way as used in the conventional Euler solvers.
The paper then presents a platform to construct one-dimensional compressible
lattice Boltzmann model for its use in FV-LBM. The platform is formed from the
conservation forms of moments. Under the platform, both the equilibrium
distribution functions and lattice velocities can be determined, and
therefore, non-free parameter model can be developed. The paper particularly
presents three typical non-free parameter models, D1Q3, D1Q4 and D1Q5. The
performances of these three models for simulation of compressible flows are
investigated by a brief analysis and their application to solve some
one-dimensional and two-dimensional test problems. Numerical results
showed that D1Q3 model costs the least computation time and D1Q4 and D1Q5
models have the wider application range of Mach number. From the results,
it seems that D1Q4 model could be the best choice for the FV-LBM simulation
of hypersonic flows.
An acceleration scheme based on stationary
iterative methods is presented for solving linear system of equations. Unlike Chebyshev semi-iterative method
which requires accurate estimation of the bounds for iterative matrix
eigenvalues, we use a wide range of Chebyshev-like polynomials for
the accelerating process without estimating the bounds of the
iterative matrix. A detailed error analysis is presented and convergence rates are obtained.
Numerical experiments are carried out and comparisons with classical Jacobi and Chebyshev semi-iterative methods
The stability and natural vibration of a standing tapered vertical
column under its own weight are studied. Exact stability criteria are
found for the pointy column and numerical stability boundaries are
determined for the blunt tipped column. For vibrations we use an
accurate, efficient initial value numerical method for the first
three frequencies. Four kinds of columns with linear taper are
considered. Both the taper and the cross section shape of the
column have large influences on the vibration frequencies. It is
found that gravity decreases the frequency while the degree of
taper may increase or decrease frequency. Vibrations may occur
in two different planes.
We present the finite difference/element method for a
two-dimensional modified fractional diffusion equation. The analysis
is carried out first for the time semi-discrete scheme, and then for
the full discrete scheme. The time discretization is based on the
$L1$-approximation for the fractional derivative terms and the
second-order backward differentiation formula for the classical
first order derivative term. We use finite element method for the
spatial approximation in full discrete scheme. We show that both the
semi-discrete and full discrete schemes are unconditionally stable
and convergent. Moreover, the optimal convergence rate is obtained.
Finally, some numerical examples are tested in the case of one and
two space dimensions and the numerical results confirm our
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