Semi-implicit Finite Difference Method Research Articles

Thermal Stability and Reactant Consumption Analysis in a Reactive Cylinder of Variable Thermal Conductivity

In this article, we consider thermal stability and oxygen consumption rate in a reactive stockpile material and at the same time spontaneously combusts. Spontaneous combustion usually takes place when a material containing carbon or hydrocarbon reacts with oxygen that is trapped within the stockpile due to low-temperature oxidation. Heat is one of the products of combustion in any exothermic chemical reaction, and in this case, complete combustion is assumed with the reactant oxygen consumption. The study is done theoretically with the application of energy and mass transfer equations, to give an understanding of the complicated process of combustion. The nonlinear differential equations governing the problem are solved numerically using semi-implicit finite difference method (FDM). The reactant and the temperature behaviors are depicted graphically, and results are discussed accordingly.

On Heat Transfer Analysis for a Sphere of Combustible Material of Variable Thermal Conductivity

This article considers an exothermic chemical reaction taking place in a stockpile of reactive material with a thermal conductivity that is temperature dependent. The study is modeled in a spherical domain whose carbon-containing material reacts spontaneously with the oxygen trapped within the system. The combustion process results with a complicated process that is nonlinear in nature, and the energy equation used to govern the problem is tackled numerically with the semi-implicit finite difference method (FDM). The results are depicted graphically and discussed to give a theoretical understanding of the heat transfer analysis during combustion.

A Numerical Design of Marine Current for Predicting Velocity and Kinetic Energy

<span id="m_3697106264798858821m_4578206257817085836gmail-result_box" class="m_3697106264798858821m_4578206257817085836gmail-" lang="en"><span class="m_3697106264798858821m_4578206257817085836gmail-"><br /></span></span><em><span lang="EN-US">One of marine current power plant equipment is turbine. In the design of marine current turbines required variables such as velocity and kinetic energy. This paper presents numerical study on a numerical design of marine current for predicting velocity and kinetic energy in the Bangka strait, North Sulawesi, Indonesia that developed from Yaxum/3D model. This study includes the simulation of velocity and kinetic energy distributions at low and high tide current conditions by two flow rates of <span>0.1 and 0.3 Sv</span> respectively. The numerical method used to design computational program. Semi-implicit finite difference method used to 3D shallow water flow. It found distributions of velocity and kinetic energy at high tide current condition greater than low tide current condition. The future, these results will be developed in marine current power plant project in the Bangka strait, North Sulawesi, Indonesia.</span></em>

Thermal decomposition analysis in a sphere of combustible materials

In this article, we look at spontaneous combustion due to exothermic chemical reaction taking place within a stockpile of combustible material. The model includes mass and energy balance equations in a spherical domain. The complicated chemical reaction is simplified by considering a one-dimensional process. The differential equations governing the problem are solved using semi-implicit finite difference method. The effects of kinetic parameters embedded within the system are analyzed and the results are expressed graphically and discussed accordingly.

Comparison between two meshless methods based on collocation technique for the numerical solution of four-species tumor growth model

As is said in [27], the tumor-growth model is the incorporation of nutrient within the mixture as opposed to being modeled with an auxiliary reaction-diffusion equation. The formulation involves systems of highly nonlinear partial differential equations of surface effects through diffuse-interface models [27]. Simulations of this practical model using numerical methods can be applied for evaluating it. The present paper investigates the solution of the tumor growth model with meshless techniques. Meshless methods are applied based on the collocation technique which employ multiquadrics (MQ) radial basis function (RBFs) and generalized moving least squares (GMLS) procedures. The main advantages of these choices come back to the natural behavior of meshless approaches. As well as, a method based on meshless approach can be applied easily for finding the solution of partial differential equations in high-dimension using any distributions of points on regular and irregular domains. The present paper involves a time-dependent system of partial differential equations that describes four-species tumor growth model. To overcome the time variable, two procedures will be used. One of them is a semi-implicit finite difference method based on Crank–Nicolson scheme and another one is based on explicit Runge–Kutta time integration. The first case gives a linear system of algebraic equations which will be solved at each time-step. The second case will be efficient but conditionally stable. The obtained numerical results are reported to confirm the ability of these techniques for solving the two and three-dimensional tumor-growth equations.

NUMERICAL SIMULATION STUDY OF HYDRODYNAMIC IMPACT OF SEA-CROSSING BRIDGE

This paper used the Sea-Crossing Bridge in Quanzhou Bay as an example to simulate its tidal flow field after construction by a semi-implicit Lagrangian-Eulerian finite difference method. Triangle or quadrilateral meshes were used to refine a mesh for the sea area close to the bridge in the model. Each bridge pier was treated as land. The minimum side length of the mesh was approximately 5 meters. The simulation results showed that slow flow areas were formed in front of bridge piers due to the influence of rising and falling tides; at the back of bridge piers, slow flow areas were also formed after rising and falling tides passed the piers. The tide race direction of rise and fall in the main bridge area was basically perpendicular to the bridge site line. The influence domain of upstream and downstream flow along the bridge site line could extend approximately 1 km away. The variation of annual siltation intensity was mainly located in the water area around the bridge piers. The annual siltation intensity of suspended sediment around the main bridge pier increased about 2-20 cm/year. The annual siltation intensities between piers decreased about 1-2 cm/year. The movement of tidal flow and the variation of sediment back-siltation rule would have certain impacts on the marine ecosystem, regional flood control, and navigation in Quanzhou Bay.

An energy preserving finite difference scheme for the Poisson–Nernst–Planck system

In this paper, we construct a semi-implicit finite difference method for the time dependent Poisson–Nernst–Planck system. Although the Poisson–Nernst–Planck system is a nonlinear system, the numerical method presented in this paper only needs to solve a linear system at each time step, which can be done very efficiently. The rigorous proof for the mass conservation and electric potential energy decay are shown. Moreover, mesh refinement analysis shows that the method is second order convergent in space and first order convergent in time. Finally we point out that our method can be easily extended to the case of multi-ions.

Thermal Stability Investigation in a Reactive Sphere of Combustible Material

An investigation of thermal stability in a stockpile of combustible material is considered. The combustible material is any carbon containing material that can react with oxygen trapped in a stockpile due to exothermic chemical reaction. The complicated process is modelled in a sphere and one-dimensional energy equation is used to solve the problem. The semi-implicit finite difference method (FDM) is applied to tackle the nonlinear differential equation governing the problem. Graphical solutions are displayed to describe effects of embedded kinetic parameters on the temperature of the system.

Analysis of shear banding phenomena in non-isothermal flow of fluids governed by the diffusive Johnson–Segalman model

This article investigates phenomena associated with shear banded flows of viscoelastic fluids in both pipe flow and coaxial annular flow. We in particular computationally analyze the effects of stress diffusion, non-isothermal flow conditions and annular gap size on the shear stress path selection (and the selected shear stress value) in shear banded flow of viscoelastic fluids described by the non-local Diffusive Johnson–Segalman (DJS) constitutive model. The DJS model is one of such constitutive models which allows us for a non-zero inter-facial layer between regions of high and low shear rates within the flow field. Temperature effects due to polymer orientation changes, entropic effects (resulting from stress work), conduction heat transfer, Arrhenius chemical kinetics as well as the influence of slip and related frictional heating on the wall are all accounted for. The time dependent model formulated to capture gap effect in the axial annular flow problem (under certain values of the relevant parameters), reduces to the system of equations governing a pipe flow setup. Semi-implicit finite difference methods are employed for the solution process of the coupled nonlinear time dependent partial differential equations governing the flow problem. We discuss with graphical representations the effect of temperature, stress diffusion, wall slip, suction/injection and annular gap size on the related shear rate path selection phenomena for certain material parameter values.

Radiation effects on an unsteady MHD natural convective flow of a nanofluid past a vertical plate

Numerical analysis is carried out on an unsteady MHD natural convective boundary layer flow of a nanofluid past an isothermal vertical plate in the presence of thermal radiation. The governing partial differential equations are solved numerically by an efficient, iterative, tri-diagonal, semi-implicit finite-difference method. In particular, we investigate the effects of radiation, magnetic field and nanoparticle volume fraction on the flow and heat transfer characteristics. The nanofluids containing nanoparticles of aluminium oxide, copper, titanium oxide and silver with nanoparticle volume fraction range less than or equal to 0.04 are considered. The numerical results indicate that in the presence of radiation and magnetic field, an increase in the nanoparticle volume fraction will decrease the velocity boundary layer thickness while increasing the thickness of the thermal boundary layer. Meanwhile, an increase in the magnetic field or nanoparticle volume fraction decreases the average skin-friction at the plate. Excellent validation of the present results has been achieved with the published results in the literature in the absence of the nanoparticle volume fraction.

Uniform point-wise error estimates of semi-implicit compact finite difference methods for the nonlinear Schrödinger equation perturbed by wave operator

This study is devoted to analysis of semi-implicit compact finite difference (SICFD) methods for the nonlinear Schrödinger equation (NLS) perturbed by the wave operator (NLSW) with a perturbation strength described by a dimensionless parameter ε∈(0,1]. Uniform l∞-norm error bounds of the proposed SICFD schemes are built to give immediate insight on point-wise error occurring as time increases, and the explicit dependence of the mesh size and time step on the parameter ε is also figured out. In the small ε regime, highly oscillations arise in time with O(ε2)-wavelength. This highly oscillatory nature in time as well as the difficulty raised by the compact FD discretization make establishing the l∞-norm error bounds uniformly in ε of the SICFD methods for NLSW to be a very interesting and challenging issue. The uniform l∞-norm error bounds in ε are proved to be of O(h4+τ) and O(h4+τ2/3) with time step τ and mesh size h for well-prepared and ill-prepared initial data. Finally, numerical results are reported to verify the error estimates and show the sharpness of the convergence rates in the respectively parameter regimes.

Computational Dynamics of Arterial Blood Flow in the Presence of Magnetic Field and Thermal Radiation Therapy

We conduct a numerical study to determine the influence of magnetic field and thermal radiation on both velocity and temperature distributions in a single blood vessel. The model here assumes that blood is a Newtonian incompressible conducting fluid with radially varying viscosity due to hematocrit variation. The transient equations of momentum and energy transport governing the flow in an axisymmetric configuration are solved numerically using a semi-implicit finite difference method. Results are presented graphically and discussed both qualitatively and quantitatively from the physiological point of view. The results of this work may enhance current understanding of the factors that determine the effects of hyperthermia treatment on tumor tissues.

Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrödinger-Poisson System

AbstractWe study compact finite difference methods for the Schrodinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave function and external potential V(x). The Crank-Nicolson compact finite difference method and the semi-implicit compact finite difference method are both of order Ҩ(h4 + τ2) in discrete l2,H1 and l∞ norms with mesh size h and time step t. For the errors ofcompact finite difference approximation to the second derivative andPoisson potential are nonlocal, thus besides the standard energy method and mathematical induction method, the key technique in analysisis to estimate the nonlocal approximation errors in discrete l∞ and H1 norm by discrete maximum principle of elliptic equation and properties of some related matrix. Also some useful inequalities are established in this paper. Finally, extensive numerical re-sults are reported to support our error estimates of the numerical methods.

Viscoelastic modeling of the diffusion of polymeric pollutants injected into a pipe flow

This study focuses on the transient analysis of nonlinear dispersion of a polymeric pollutant ejected by an external source into a laminar pipe flow of a Newtonian liquid under axi-symmetric conditions. The influence of density variation with pollutant concentration is approximated according to the Boussinesq approximation and the nonlinear governing equations of momentum, pollutant concentration are obtained together with and Oldroyd-B constitutive model for the polymer stress. The problem is solved numerically using a semi-implicit finite difference method. Solutions are presented in graphical form for various parameter values and given in terms of fluid velocity, pollutant concentration, polymer stress components, skin friction and wall mass transfer rate. The model can be a useful tool in understanding the dynamics of industrial pollution situations arising from improper discharge of hydrocarbon pollutants into, say, water bodies. The model can also be quite useful for available necessary early warning methods for detecting or predicting the scale of pollution and hence help mitigate related damage downstream by earlier instituting relevant decontamination measures.

Uniform Error Estimates of Finite Difference Methods for the Nonlinear Schrödinger Equation with Wave Operator

We establish uniform error estimates of finite difference methods for the nonlinear Schrodinger equation (NLS) perturbed by the wave operator (NLSW) with a perturbation strength described by a dimensionless parameter $\varepsilon$ ($\varepsilon\in(0,1]$). When $\varepsilon\to0^+$, NLSW collapses to the standard NLS. In the small perturbation parameter regime, i.e., $0<\varepsilon\ll1$, the solution of NLSW is perturbed from that of NLS with a function oscillating in time with $O(\varepsilon^2)$-wavelength at $O(\varepsilon^4)$ and $O(\varepsilon^2)$ amplitudes for well-prepared and ill-prepared initial data, respectively. This high oscillation of the solution in time brings significant difficulties in establishing error estimates uniformly in $\varepsilon$ of the standard finite difference methods for NLSW, such as the conservative Crank-Nicolson finite difference (CNFD) method, and the semi-implicit finite difference (SIFD) method. We obtain error bounds uniformly in $\varepsilon$, at the order of $O(h^2+\tau)$ and $O(h^2+\tau^{2/3})$ with time step $\tau$ and mesh size $h$ for well-prepared and ill-prepared initial data, respectively, for both CNFD and SIFD in the $l^2$-norm and discrete semi-$H^1$ norm. Our error bounds are valid for general nonlinearity in NLSW and for one, two, and three dimensions. To derive these uniform error bounds, we combine $\varepsilon$-dependent error estimates of NLSW, $\varepsilon$-dependent error bounds between the numerical approximate solutions of NLSW and the solution of NLS, together with error bounds between the solutions of NLSW and NLS. Other key techniques in the analysis include the energy method, cut-off of the nonlinearity, and a posterior bound of the numerical solutions by using the inverse inequality and discrete semi-$H^1$ norm estimate. Finally, numerical results are reported to confirm our error estimates of the numerical methods and show that the convergence rates are sharp in the respective parameter regimes.

A Finite Element Solution of the Mathematical Model for Smoke Dispersion from Two Sources

Smoke discharging is a main reason of air pollution problem from industrial plants. The obstacle of a building has an affect with the air pollutant discharge. In this research, a mathematical model of the smoke dispersion from two sources and one source with a structural obstacle is considered. The governing equation of the model is an isothermal mass transfer model in a viscous fluid. The finite element method is used to approximate the solutions of the model. The triangular linear elements have been used for discretising the domain, and time integration has been carried out by semi-implicit finite difference method. The simulations of smoke dispersion in cases of one chimney and two chimneys are presented. The maximum calculated smoke concentration of both cases are compared. It is then used to make the decision for smoke discharging and air pollutant control problems on industrial area. Keywords—Air pollution, Smoke dispersion, Finite element method, Stream function, Vorticity equation, Convection-diffusion equation, Semi-implicit method

Transient analysis of pollutant dispersion in a cylindrical pipe with a nonlinear waste discharge concentration

In this study we investigate the transient dispersion of a pollutant ejected by an external source into a laminar flow of an incompressible fluid in a cylindrical pipe. The Boussinesq assumption is applied and the nonlinear governing equations of momentum and pollutant concentration are obtained. The problem is solved numerically using a semi-implicit finite difference method. Solutions are presented in graphical form and given in terms of fluid velocity, pollutant concentration, skin friction and wall mass transfer rate for various parametric values. The model can be a useful tool for understanding the polluting situations of an improper discharge incident and evaluating the effects of decontaminating measures for the water body.

Numerical investigation of transient heat transfer to hydromagnetic channel flow with radiative heat and convective cooling

This present study consists of a numerical investigation of transient heat transfer in channel flow of an electrically conducting variable viscosity Boussinesq fluid in the presence of a magnetic field and thermal radiation. The temperature dependent nature of viscosity is assumed to follow an exponentially model and the system exchanges heat with the ambient following Newton’s law of cooling. The governing nonlinear equations of momentum and energy transport are solved numerically using a semi-implicit finite difference method. Solutions are presented in graphical form and given in terms of fluid velocity, fluid temperature, skin friction and heat transfer rate for various parametric values. Our results reveal that combined effect of thermal radiation, magnetic field, viscosity variation and convective cooling have significant impact in controlling the rate of heat transfer in the boundary layer region.

Analysis of Nonlinear Dispersion of a Pollutant Ejected by an External Source into a Channel Flow

This paper focuses on the transient analysis of nonlinear dispersion of a pollutant ejected by an external source into a laminar flow of an incompressible fluid in a channel. The influence of density variation with pollutant concentration is approximated according to the Boussinesq approximation, and the nonlinear governing equations of momentum and pollutant concentration are obtained. The problem is solved numerically using a semi‐implicit finite difference method. Solutions are presented in graphical form and given in terms of fluid velocity, pollutant concentration, skin friction, and wall mass transfer rate for various parametric values. The model can be a useful tool for understanding the polluting situations of an improper discharge incident and evaluating the effects of decontaminating measures for the water body.

Rotating Rayleigh–Bénard convection under the influence of transverse magnetic field

In the present numerical study the effect of constant transverse magnetic field on convection of low Prandtl number liquid metal rotating in a cubical cavity with an aspect-ratio of 8:8:1 has been investigated. The bottom wall is heated while the top-wall is cooled and all the other walls are kept thermally insulated. The governing equations of mass, momentum, energy and magneto-hydrodynamic for a frame rotating with the enclosure, subject to Boussinesq approximation applied to gravity and centrifugal force terms, have been solved on a collocated grid using a semi-implicit finite difference method. The simulations have been carried out for liquid metal flows having a fixed Prandtl number Pr=0.01, Rayleigh number Ra=107, and magnetic Prandtl number Pm=4.0×10-4 while Chandrasekhar number Q varies from 5.0625×104 to 1.21×106 and non-dimensional rotation rate Ω is varied from zero to 105.The increase in strength of transverse magnetic field (from Ql=5.0625×104 to Qh=1.21×106) till Q≃Ta leads to slight increase in convective heat transfer as well as formation of two-dimensional coherent structures aligned along the direction of magnetic field. For cases pertaining to Q<Ta the two-dimensionality of the flow breaks down and the rolls distort in their alignment which leads to decrease in magnitude of vertical heat transfer. For cases where Q≪Ta, the increased Coriolis forces lead to generation of large-scale circulation which forms a large cylindrical rotating column of fluid in consonance with Taylor–Proudman theorem. On increasing the strength of magnetic field the component of rms velocity in the direction of magnetic field gets suppressed while there is increase in other two components.