Abstract: In this paper, we present a fast and accurate numerical scheme for the solution of a Laplace equation with Dirichlet boundary conditions. The non-standard finite difference scheme (NSFD) is applied to construct the numerical solutions of a Laplace equation with two different Dirichlet boundary conditions. The solutions obtained using NSFD are compared with the solutions obtained using the standard finite difference scheme (SFD). The NSFD scheme is demonstrated to be reliable and efficient.
Abstract: Several numerical schemes utilizing central difference
approximations have been developed to solve the Goursat problem.
However, in a recent years compact discretization methods which
leads to high-order finite difference schemes have been used since it
is capable of achieving better accuracy as well as preserving certain
features of the equation e.g. linearity. The basic idea of the new
scheme is to find the compact approximations to the derivative terms
by differentiating centrally the governing equations. Our primary
interest is to study the performance of the new scheme when applied
to two Goursat partial differential equations against the traditional
finite difference scheme.
Abstract: The paper deals with a mathematical model for fluid dynamic flows on road networks which is based on conservation laws. This nonlinear framework is based on the conservation of cars. We focus on traffic circle, which is a finite number of roads that meet at some junctions. The traffic circle with junctions having either one incoming and two outgoing or two incoming and one outgoing roads. We describe the numerical schemes with the particular boundary conditions used to produce approximated solutions of the problem.
Abstract: An efficient transient flow simulation for gas
pipelines and networks is presented. The proposed transient flow
simulation is based on the transfer function models and MATLABSimulink.
The equivalent transfer functions of the nonlinear
governing equations are derived for different types of the boundary
conditions. Next, a MATLAB-Simulink library is developed and
proposed considering any boundary condition type. To verify the
accuracy and the computational efficiency of the proposed
simulation, the results obtained are compared with those of the
conventional finite difference schemes (such as TVD, method of
lines, and other finite difference implicit and explicit schemes). The
effects of the flow inertia and the pipeline inclination are
incorporated in this simulation. It is shown that the proposed
simulation has a sufficient accuracy and it is computationally more
efficient than the other methods.
Abstract: Flow movement in unsaturated soil can be expressed
by a partial differential equation, named Richards equation. The
objective of this study is the finding of an appropriate implicit
numerical solution for head based Richards equation. Some of the
well known finite difference schemes (fully implicit, Crank Nicolson
and Runge-Kutta) have been utilized in this study. In addition, the
effects of different approximations of moisture capacity function,
convergence criteria and time stepping methods were evaluated. Two
different infiltration problems were solved to investigate the
performance of different schemes. These problems include of vertical
water flow in a wet and very dry soils. The numerical solutions of
two problems were compared using four evaluation criteria and the
results of comparisons showed that fully implicit scheme is better
than the other schemes. In addition, utilizing of standard chord slope
method for approximation of moisture capacity function, automatic
time stepping method and difference between two successive
iterations as convergence criterion in the fully implicit scheme can
lead to better and more reliable results for simulation of fluid
movement in different unsaturated soils.
Abstract: In this article, we aim to discuss the formulation of two explicit group iterative finite difference methods for time-dependent two dimensional Burger-s problem on a variable mesh. For the non-linear problems, the discretization leads to a non-linear system whose Jacobian is a tridiagonal matrix. We discuss the Newton-s explicit group iterative methods for a general Burger-s equation. The proposed explicit group methods are derived from the standard point and rotated point Crank-Nicolson finite difference schemes. Their computational complexity analysis is discussed. Numerical results are given to justify the feasibility of these two proposed iterative methods.
Abstract: In this paper, we have combined some spatial derivatives with the optimised time derivative proposed by Tam and Webb in order to approximate the linear advection equation which is given by = 0. Ôêé Ôêé + Ôêé Ôêé x f t u These spatial derivatives are as follows: a standard 7-point 6 th -order central difference scheme (ST7), a standard 9-point 8 th -order central difference scheme (ST9) and optimised schemes designed by Tam and Webb, Lockard et al., Zingg et al., Zhuang and Chen, Bogey and Bailly. Thus, these seven different spatial derivatives have been coupled with the optimised time derivative to obtain seven different finite-difference schemes to approximate the linear advection equation. We have analysed the variation of the modified wavenumber and group velocity, both with respect to the exact wavenumber for each spatial derivative. The problems considered are the 1-D propagation of a Boxcar function, propagation of an initial disturbance consisting of a sine and Gaussian function and the propagation of a Gaussian profile. It is known that the choice of the cfl number affects the quality of results in terms of dissipation and dispersion characteristics. Based on the numerical experiments solved and numerical methods used to approximate the linear advection equation, it is observed in this work, that the quality of results is dependent on the choice of the cfl number, even for optimised numerical methods. The errors from the numerical results have been quantified into dispersion and dissipation using a technique devised by Takacs. Also, the quantity, Exponential Error for Low Dispersion and Low Dissipation, eeldld has been computed from the numerical results. Moreover, based on this work, it has been found that when the quantity, eeldld can be used as a measure of the total error. In particular, the total error is a minimum when the eeldld is a minimum.