Abstract: We have developed a new computer program in
Fortran 90, in order to obtain numerical solutions of a system
of Relativistic Magnetohydrodynamics partial differential equations
with predetermined gravitation (GRMHD), capable of simulating
the formation of relativistic jets from the accretion disk of matter
up to his ejection. Initially we carried out a study on numerical
methods of unidimensional Finite Volume, namely Lax-Friedrichs,
Lax-Wendroff, Nessyahu-Tadmor method and Godunov methods
dependent on Riemann problems, applied to equations Euler in
order to verify their main features and make comparisons among
those methods. It was then implemented the method of Finite
Volume Centered of Nessyahu-Tadmor, a numerical schemes that
has a formulation free and without dimensional separation of
Riemann problem solvers, even in two or more spatial dimensions,
at this point, already applied in equations GRMHD. Finally, the
Nessyahu-Tadmor method was possible to obtain stable numerical
solutions - without spurious oscillations or excessive dissipation -
from the magnetized accretion disk process in rotation with respect
to a central black hole (BH) Schwarzschild and immersed in a
magnetosphere, for the ejection of matter in the form of jet over a
distance of fourteen times the radius of the BH, a record in terms
of astrophysical simulation of this kind. Also in our simulations,
we managed to get substructures jets. A great advantage obtained
was that, with the our code, we got simulate GRMHD equations in
a simple personal computer.
Abstract: In this paper, a backward semi-Lagrangian scheme
combined with the second-order backward difference formula
is designed to calculate the numerical solutions of nonlinear
advection-diffusion equations. The primary aims of this paper are
to remove any iteration process and to get an efficient algorithm
with the convergence order of accuracy 2 in time. In order to achieve
these objects, we use the second-order central finite difference and the
B-spline approximations of degree 2 and 3 in order to approximate
the diffusion term and the spatial discretization, respectively. For the
temporal discretization, the second order backward difference formula
is applied. To calculate the numerical solution of the starting point
of the characteristic curves, we use the error correction methodology
developed by the authors recently. The proposed algorithm turns out
to be completely iteration free, which resolves the main weakness
of the conventional backward semi-Lagrangian method. Also, the
adaptability of the proposed method is indicated by numerical
simulations for Burgers’ equations. Throughout these numerical
simulations, it is shown that the numerical results is in good
agreement with the analytic solution and the present scheme offer
better accuracy in comparison with other existing numerical schemes.
Abstract: One of the most important parameters in petroleum reservoirs is the pressure distribution along the reservoir, as the pressure varies with the time and location. A popular method to determine the pressure distribution in a reservoir in the unsteady state regime of flow is applying Darcy’s equation and solving this equation numerically. The numerical simulation of reservoirs is based on these numerical solutions of different partial differential equations (PDEs) representing the multiphase flow of fluids. Pressure profile has obtained in a one dimensional system solving Darcy’s equation explicitly. Changes of pressure profile in three situations are investigated in this work. These situations include section length changes, step time changes and time approach to infinity. The effects of these changes in pressure profile are shown and discussed in the paper.
Abstract: Herpes zoster is a disease that manifests as a dermatological condition. The characteristic of this disease is an irritating skin rash with blisters. This is often limited to one side of body. From the data of Herpes zoster cases in Thailand, we found that age structure effects to the transmission of this disease. In this study, we construct the age structural model of Herpes zoster in Thailand. The local stability analysis of this model is given. The numerical solutions are shown to confirm the analytical results.
Abstract: A novel method based on Genetic Algorithm to solve the boundary value problems (BVPs) of the Falkner–Skan equation over a semi-infinite interval has been presented. In our approach, we use the free boundary formulation to truncate the semi-infinite interval into a finite one. Then we use the shooting method based on Genetic Algorithm to transform the BVP into initial value problems (IVPs). Genetic Algorithm is used to calculate shooting angle. The initial value problems arisen during shooting are computed by Runge-Kutta Fehlberg method. The numerical solutions obtained by the present method are in agreement with those obtained by previous authors.
Abstract: By means of Sidi-Israeli’s quadrature rules, mechanical quadrature methods (MQMs) for solving the first kind boundary integral equations (BIEs) of steady state Stokes problem are presented. The convergence of numerical solutions by MQMs is proved based on Anselone’s collective compact and asymptotical compact theory, and the asymptotic expansions with the odd powers of the errors are provided, which implies that the accuracy of the approximations by MQMs possesses high accuracy order O (h3). Finally, the numerical examples show the efficiency of our methods.
Abstract: In this paper, numerical solutions of the nonlinear Benjamin-Bona-Mahony-Burgers (BBMB) equation are obtained by a method based on collocation of cubic B-splines. Applying the Von-Neumann stability analysis, the proposed method is shown to be unconditionally stable. The method is applied on some test examples, and the numerical results have been compared with the exact solutions. The L∞ and L2 in the solutions show the efficiency of the method computationally.
Abstract: In this chapter, we have studied Variation of velocity in incompressible fluid over a moving surface. The boundary layer equations are on a fixed or continuously moving flat plate in the same or opposite direction to the free stream with suction and injection. The boundary layer equations are transferred from partial differential equations to ordinary differential equations. Numerical solutions are obtained by using Runge-Kutta and Shooting methods. We have found numerical solution to velocity and skin friction coefficient.
Abstract: A numerical study of flow in a horizontally channel
partially filled with a porous screen with non-uniform inlet has been
performed by lattice Boltzmann method (LBM). The flow in porous
layer has been simulated by the Brinkman-Forchheimer model.
Numerical solutions have been obtained for variable porosity models
and the effects of Darcy number and porosity have been studied in
detail. It is found that the flow stabilization is reliant on the Darcy
number. Also the results show that the stabilization of flow field and
heat transfer is depended to Darcy number. Distribution of stream
field becomes more stable by decreasing Darcy number. Results
illustrate that the effect of variable porosity is significant just in the
region of the solid boundary. In addition, difference between constant
and variable porosity models is decreased by decreasing the Darcy
number.
Abstract: A fully implicit finite-difference method has been proposed for the numerical solutions of one dimensional coupled nonlinear Burgers’ equations on the uniform mesh points. The method forms a system of nonlinear difference equations which is to be solved at each iteration. Newton’s iterative method has been implemented to solve this nonlinear assembled system of equations. The linear system has been solved by Gauss elimination method with partial pivoting algorithm at each iteration of Newton’s method. Three test examples have been carried out to illustrate the accuracy of the method. Computed solutions obtained by proposed scheme have been compared with analytical solutions and those already available in the literature by finding L2 and L∞ errors.
Abstract: The simulation of extrusion process is studied widely
in order to both increase products and improve quality, with broad
application in wire coating. The annular tube-tooling extrusion was
set up by a model that is termed as Navier-Stokes equation in
addition to a rheological model of differential form based on singlemode
exponential Phan-Thien/Tanner constitutive equation in a twodimensional
cylindrical coordinate system for predicting the
contraction point of the polymer melt beyond the die. Numerical
solutions are sought through semi-implicit Taylor-Galerkin pressurecorrection
finite element scheme. The investigation was focused on
incompressible creeping flow with long relaxation time in terms of
Weissenberg numbers up to 200. The isothermal case was considered
with surface tension effect on free surface in extrudate flow and no
slip at die wall. The Stream Line Upwind Petrov-Galerkin has been
proposed to stabilize solution. The structure of mesh after die exit
was adjusted following prediction of both top and bottom free
surfaces so as to keep the location of contraction point around one
unit length which is close to experimental results. The simulation of
extrusion process is studied widely in order to both increase products
and improve quality, with broad application in wire coating. The
annular tube-tooling extrusion was set up by a model that is termed
as Navier-Stokes equation in addition to a rheological model of
differential form based on single-mode exponential Phan-
Thien/Tanner constitutive equation in a two-dimensional cylindrical
coordinate system for predicting the contraction point of the polymer
melt beyond the die. Numerical solutions are sought through semiimplicit
Taylor-Galerkin pressure-correction finite element scheme.
The investigation was focused on incompressible creeping flow with
long relaxation time in terms of Weissenberg numbers up to 200. The
isothermal case was considered with surface tension effect on free
surface in extrudate flow and no slip at die wall. The Stream Line
Upwind Petrov-Galerkin has been proposed to stabilize solution. The
structure of mesh after die exit was adjusted following prediction of
both top and bottom free surfaces so as to keep the location of
contraction point around one unit length which is close to
experimental results.
Abstract: The equations governing the flow of an electrically conducting, incompressible viscous fluid over an infinite flat plate in the presence of a magnetic field are investigated using the homotopy perturbation method (HPM) with Padé approximants (PA) and 4th order Runge–Kutta method (4RKM). Approximate analytical and numerical solutions for the velocity field and heat transfer are obtained and compared with each other, showing excellent agreement. The effects of the magnetic parameter and Prandtl number on velocity field, shear stress, temperature and heat transfer are discussed as well.
Abstract: The aim of this paper is to investigate twodimensional unsteady flow of a viscous incompressible fluid about stagnation point on permeable stretching sheet in presence of time dependent free stream velocity. Fluid is considered in the influence of transverse magnetic field in the presence of radiation effect. Rosseland approximation is use to model the radiative heat transfer. Using time-dependent stream function, partial differential equations corresponding to the momentum and energy equations are converted into non-linear ordinary differential equations. Numerical solutions of these equations are obtained by using Runge-Kutta Fehlberg method with the help of Newton-Raphson shooting technique. In the present work the effect of unsteadiness parameter, magnetic field parameter, radiation parameter, stretching parameter and the Prandtl number on flow and heat transfer characteristics have been discussed. Skin-friction coefficient and Nusselt number at the sheet are computed and discussed. The results reported in the paper are in good agreement with published work in literature by other researchers.
Abstract: In this work, we consider a deterministic model for
the transmission of leptospirosis which is currently spreading in the
Thai population. The SIR model which incorporates the features of
this disease is applied to the epidemiological data in Thailand. It is
seen that the numerical solutions of the SIR equations are in good
agreement with real empirical data. Further improvements are
discussed.
Abstract: In this article the homotopy continuation method (HCM) to solve the forward kinematic problem of the 3-PRS parallel manipulator is used. Since there are many difficulties in solving the system of nonlinear equations in kinematics of manipulators, the numerical solutions like Newton-Raphson are inevitably used. When dealing with any numerical solution, there are two troublesome problems. One is that good initial guesses are not easy to detect and another is related to whether the used method will converge to useful solutions. Results of this paper reveal that the homotopy continuation method can alleviate the drawbacks of traditional numerical techniques.
Abstract: Bubble columns have a variety of applications in
absorption, bio-reactions, catalytic slurry reactions, and coal
liquefaction; because they are simple to operate, provide good heat
and mass transfer, having less operational cost. The use of
Computational Fluid Dynamics (CFD) for bubble column becomes
important, since it can describe the fluid hydrodynamics on both local
and global scale. Euler- Euler two-phase fluid model has been used to
simulate two-phase (air and water) transient up-flow in bubble
column (15cm diameter) using FLUENT6.3. These simulations and
experiments were operated over a range of superficial gas velocities
in the bubbly flow and churn turbulent regime (1 to16 cm/s) at
ambient conditions. Liquid velocity was varied from 0 to 16cm/s. The
turbulence in the liquid phase is described using the standard k-ε
model. The interactions between the two phases are described
through drag coefficient formulations (Schiller Neumann). The
objectives are to validate CFD simulations with experimental data,
and to obtain grid-independent numerical solutions. Quantitatively
good agreements are obtained between experimental data for hold-up
and simulation values. Axial liquid velocity profiles and gas holdup
profiles were also obtained for the simulation.
Abstract: The effect of a time delay on the transmission on
dengue fever is studied. The time delay is due to the presence of an
incubation period for the dengue virus to develop in the mosquito
before the mosquito becomes infectious. The conditions for the
existence of a Hopf bifurcation to limit cycle behavior are
established. The conditions are different from the usual one and they
are based on whether a particular third degree polynomial has
positive real roots. A theorem for determining whether for a given
set of parameter values, a critical delay time exist is given. It is
found that for a set of realistic values of the parameters in the model,
a Hopf bifurcation can not occur. For a set of unrealistic values of
some of the parameters, it is shown that a Hopf bifurcation can occur.
Numerical solutions using this last set show the trajectory of two of
the variables making a transition from a spiraling orbit to a limit
cycle orbit.
Abstract: In the present analysis an unsteady laminar
forced convection water boundary layer flow is considered.
The fluid properties such as viscosity and Prandtl number are
taken as variables such that those are inversely proportional to
temperature. By using quasi-linearization technique the nonlinear
coupled partial differential equations are linearized and
the numerical solutions are obtained by using implicit finite
difference scheme with the appropriate selection of step sizes.
Non-similar solutions have been obtained from the starting
point of the stream-wise coordinate to the point where skin
friction value vanishes. The effect non-uniform mass transfer
along the surface of the cylinder through slot is studied on the
skin friction and heat transfer coefficients.
Abstract: Semilinear elliptic equations are ubiquitous in natural sciences. They give rise to a variety of important phenomena in quantum mechanics, nonlinear optics, astrophysics, etc because they have rich multiple solutions. But the nontrivial solutions of semilinear equations are hard to be solved for the lack of stabilities, such as Lane-Emden equation, Henon equation and Chandrasekhar equation. In this paper, bifurcation method is applied to solving semilinear elliptic equations which are with homogeneous Dirichlet boundary conditions in 2D. Using this method, nontrivial numerical solutions will be computed and visualized in many different domains (such as square, disk, annulus, dumbbell, etc).
Abstract: Leptospirosis occurs worldwide (except the
poles of the earth), urban and rural areas, developed and
developing countries, especially in Thailand. It can be
transmitted to the human by rats through direct and indirect
ways. Human can be infected by either touching the infected rats
or contacting with water, soil containing urine from the infected
rats through skin, eyes and nose. The data of the people who
are infected with this disease indicates that most of the
patients are adults. The transmission of this disease is studied
through mathematical model. The population is separated into human
and rat. The human is divided into two classes, namely juvenile
and adult. The model equation is constructed for each class. The
standard dynamical modeling method is then used for
analyzing the behaviours of solutions. In addition, the
conditions of the parameters for the disease free and endemic
states are obtained. Numerical solutions are shown to support the
theoretical predictions. The results of this study guide the way to
decrease the disease outbreak.