A modified discrete scheme in the Ausas cavitation algorithm

In order to reduce the dependence of accuracy on the number of grids in the Ausas cavitation algorithm, a modified Ausas algorithm was presented. By modifying the mass-conservative Reynolds equation with the concept of linear complementarity problems (LCPs), the coupling of film thickness h and density ratio θ disappeared. The modified equation achieved a new discrete scheme that ensured a complete second-order-accurate central difference scheme for the full film region, avoiding a hybrid-order-accurate discrete scheme. A journal bearing case was studied to show the degree of accuracy improvement and the calculation time compared to a standard LCP solver. The results showed that the modified Ausas algorithm made the asymptotic and convergent behavior with the increase of nodes disappear and allowed for the use of coarse meshes to obtain sufficient accuracy. The calculation time of the modified Ausas algorithm is shorter than the LCP solver (Lemke's pivoting algorithm) for middle and large scale problems.

Numerical Solution of Three-Dimensional Transient Heat Conduction Equation in Cylindrical Coordinates

Heat equation is a partial differential equation used to describe the temperature distribution in a heat-conducting body. The implementation of a numerical solution method for heat equation can vary with the geometry of the body. In this study, a three-dimensional transient heat conduction equation was solved by approximating second-order spatial derivatives by five-point central differences in cylindrical coordinates. The stability condition of the numerical method was discussed. A MATLAB code was developed to implement the numerical method. An example was provided in order to demonstrate the method. The numerical solution by the method was in a good agreement with the exact solution for the example considered. The accuracy of the five-point central difference method was compared with that of the three-point central difference method in solving the heat equation in cylindrical coordinates. The solutions obtained by the numerical method in cylindrical coordinates were displayed in the Cartesian coordinate system graphically. The method requires relatively very small time steps for a given mesh spacing to avoid computational instability. The result of this study can provide insights to use appropriate coordinates and more accurate computational methods in solving physical problems described by partial differential equations.

Pressure-Velocity Coupling Schemes for Bouyancy Driven Flow in a Differentially Heated Cavity Using F.V.M and Matlab

Numerical analysis of fluid flow is anchored on the laws of conservation. A challenge in solving the momentum equation arises due to the unavailability of an explicit pressure equation. To avoid solving the pressure term most researchers have eliminated it by cross differentiating the x and the y two dimensional momentum equations and subtracting them. This method introduces more variables to be solved in comparison to the primitive variables and is restricted to two-dimensional flows as streamlines do not exist in three-dimension. This method thus presents a serious limitation in analysis of fluid flow. In this study an equation for computing pressure has been developed using pressure - velocity coupling and used in solving the governing equations. The performance of three pressure velocity schemes namely; the Semi Implicit Method for Pressure linked Equation (SIMPLE), SIMPLE Revised (SIMPLER) and SIMPLE Consistent (SIMPLEC) for laminar buoyancy driven flow has been tested in order to establish the scheme that gives results consistent with bench mark data. The equations governing the flow are solved iteratively using finite volume method together with the central difference interpolating scheme. The solutions are presented for Rayleigh numbers of 103, 104, and 105. This resulted in the velocity profiles for the SIMPLE, SIMPLER, and SIMPLEC algorithm for a Rayleigh number of 104 and 105 converging to the same path. At a Rayleigh number of 103 however, SIMPLER algorithm undergoes a degradation in convergence with grid refinement at the baffle region. Results predicted by using the SIMPLEC algorithm are thus able to effectively compute the velocity of fluid flow in a differentially heated square enclosure with baffles for both low and higher Rayleigh numbers irrespective of the grid size.

Mathematical Modeling of the Phytoplankton Populations Geographic Dynamics for Possible Scenarios of Changes in the Azov Sea Hydrological Regime

Increased influence of abiotic and anthropogenic factors on the ecological state of coastal systems leads to uncontrollable changes in the overall ecosystem. This paper considers the crucial problem of studying the effect of an increase in the water’s salinity in the Azov Sea and the Taganrog Bay on hydrobiological processes. The main aim of the research is the diagnostic and predictive modeling of the geographic dynamics of the general phytoplankton populations. A mathematical model that describes the dynamics of three types of phytoplankton is proposed, considering the influence of salinity and nutrients on algae development. Discretization is carried out based on a linear combination of Upwind Leapfrog difference schemes and a central difference scheme, which makes it possible to increase the accuracy of solving the biological kinetics problem at large values of the grid Péclet number (Peh > 2). A software package has been developed that implements interrelated models of hydrodynamics and biogeochemical cycles. A modified alternating-triangular method was used to solve large-dimensional systems of linear algebraic equations (SLAE). Based on the scenario approach, several numerical experiments were carried out to simulate the dynamics of the main species of phytoplankton populations at different levels of water salinity in coastal systems. It is shown that with an increase in the salinity of waters, the habitats of phytoplankton populations shift, and marine species invasively replace freshwater species of algae.

Applications of Haar Wavelet-Finite Difference Hybrid Method and Its Convergence for Hyperbolic Nonlinear Schrödinger Equation with Energy and Mass Conversion

This article is concerned with the numerical solution of nonlinear hyperbolic Schro¨dinger equations (NHSEs) via an efficient Haar wavelet collocation method (HWCM). The time derivative is approximated in the governing equations by the central difference scheme, while the space derivatives are replaced by finite Haar series, which transform it to full algebraic form. The experimental rate of convergence follows the theoretical statements of convergence and the conservation laws of energy and mass are also presented, which strengthens the proposed method to be convergent and conservative. The Haar wavelets based on numerical results for solitary wave shape of |φ| are discussed in detail. The proposed approach provides a fast convergent approximation to the NHSEs. The reliability and efficiency of the method are illustrated by computing the maximum error norm and the experimental rate of convergence for different problems. Comparisons are performed with various existing methods in recent literature and better performance of the proposed method is shown in various tables and figures.

A Fast Preconditioned Semi-Implicit Difference Scheme for Strongly Nonlinear Space-Fractional Diffusion Equations

In this paper, we propose a semi-implicit difference scheme for solving one-dimensional nonlinear space-fractional diffusion equations. The method is first-order accurate in time and second-order accurate in space. It uses a fractional central difference formula and the backward Euler method to approximate its space and time derivatives, respectively. Stability and convergence properties of the proposed scheme are proved with the help of a discrete Grönwall inequality. Moreover, we extend the method to the solution of two-dimensional nonlinear models. A fast matrix-free implementation based on preconditioned Krylov subspace methods is presented for solving the discretized linear systems. The resulting fast preconditioned semi-implicit difference scheme reduces the memory requirement of conventional semi-implicit difference schemes from O(Ns2) to O(Ns) and the computational complexity from O(Ns3) to O(NslogNs) in each iterative step, where Ns is the number of space grid points. Experiments with two numerical examples are shown to support the theoretical findings and to illustrate the efficiency of our proposed method.

Numerical simulation of non-Newtonian Carreau fluid in a lid-driven cavity

In this work, the 2D lid-driven cavity flow of non-Newtonian Carreau fluids has been studied by finite difference method on a staggered grid. A finite-difference algorithm on staggered grid based on projection method is adopted to solve the lid-driven cavity flow, which includes a second-order central difference scheme for the non-Newtonian viscous stress term. This study has been conducted for the certain pertinent parameters of Reynolds number (Re=100-1000), power-law index (n=0.6-1.4). The results show that as the Reynolds number increases, the influence of the power-law index on the flow increases. As the power-law index decreases, the flow field becomes more complicated.