Explicit Finite Difference Scheme Research Articles

Convergence rates of finite difference schemes for the linear advection and wave equation with rough coefficient

We prove convergence rates of explicit finite difference schemes for the linear advection and wave equation in one space dimension with H\"older continuous coefficient. The obtained convergence rates explicitly depend on the H\"older regularity of the coefficient and the modulus of continuity of the initial data. We compare the theoretically established rates with the experimental rates of a couple of numerical examples.

Effects of peripheral layer thickness on pulsatile flow of Herschel–Bulkley fluid through a stenotic artery

A study is conducted to analyze two-phase flow model of blood in a flexible stenotic artery. The Newtonian and Herschel–Bulkley constitutive relations are used to model the core and peripheral regions, respectively. The arterial wall is assumed to be elastic. By making appropriate assumptions, the flow problem is transformed into a set of nonlinear partial different equations. These equation together with suitable initial and boundary data are solved using an explicit finite difference scheme. The radial and axial velocities of the streaming blood are computed for several values of the emerging parameters. The axial distribution of flow rate, resistance impedance, and wall shear stress are also shown graphically and discussed in detail. The instantaneous streamlines patterns of the flow are also shown.

Explicit Finite-Difference Scheme for the Numerical Solution of the Model Equation of Nonlinear Hereditary Oscillator with Variable-Order Fractional Derivatives

Abstract The paper deals with the model of variable-order nonlinear hereditary oscillator based on a numerical finite-difference scheme. Numerical experiments have been carried out to evaluate the stability and convergence of the difference scheme. It is argued that the approximation, stability and convergence are of the first order, while the scheme is stable and converges to the exact solution.

Effects of Interstrut Distance on the Transport of Drug and its Retention in the Arterial Tissue

The present investigation is dealt with the effect of interstrut distance on the transport and retention of drug eluted from a well-apposed drug-eluting stent (DES). We consider a two-species model (free and bound), capable of predicting the delivery of drug and its retention in the arterial tissue, in which the specific binding site action is modelled using a reversible chemical reaction. The model also takes into account a time-dependent release kinetics of DES. An axi-symmetric model of drug delivery from a pair of stent struts has been developed in cylindrical polar coordinate system where the transport of free drug is modelled as an unsteady convection–diffusion-reaction process, while the bound drug as an unsteady reaction-diffusion process. An explicit finite-difference scheme has been used to tackle numerically the governing equations of motion together with the physiologically realistic boundary conditions. The effects of quantities of significance such as Peclet number (Pe), equilibrium coefficient (\(K_{eq}\)) and interstrut distance on the transport of free as well as bound drugs are quantitatively investigated graphically. The present results predict a single peak profile of drug concentration when the pair of struts are placed one strut-width apart and also as the interstrut distance increases, discrete peaks form over each strut which is consistent with several existing results in the literature.

A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations

This paper addresses a stable three-level explicit scheme for a class of nonlinear time variable order fractional partial differential equations. The proposed strategy is based on the linear B-spline approximation of the time variable order fractional derivative in the Caputo sense and the Du Fort–Frankel algorithm. The unconditional stability and the convergence of the scheme are established. Several numerical results confirm the accuracy and efficiency of the novel scheme.

A study on the efficiency and stability of high-order numerical methods for Form-II and Form-III of the nonlinear Klein–Gordon equations

In this paper, we discuss the problem of solving nonlinear Klein–Gordon equations (KGEs), which are especially useful to model nonlinear phenomena. In order to obtain more exact solutions, we have derived different fourth- and sixth-order, stable explicit and implicit finite difference schemes for some of the best known nonlinear KGEs. These new higher-order methods allow a reduction in the number of nodes, which is necessary to solve multi-dimensional KGEs. Moreover, we describe how higher-order stable algorithms can be constructed in a similar way following the proposed procedures. For the considered equations, the stability and consistency of the proposed schemes are studied under certain smoothness conditions of the solutions. In addition to that, we present experimental results obtained from numerical methods that illustrate the efficiency of the new algorithms, their stability, and their convergence rate.

Computational Model of Drug-Coated Balloon Delivery in a Patient-Specific Arterial Vessel with Heterogeneous Tissue Composition.

Balloon angioplasty followed by local delivery of antiproliferative drugs to target tissue is increasingly being considered for the treatment of obstructive arterial disease, and yet there is much to appreciate regarding pharmacokinetics in arteries of non-uniform disease. We developed a computational model capable of simulating drug-coated balloon delivery to arteries of heterogeneous tissue composition comprising healthy tissue, as well as regions of fibrous, fibro-fatty, calcified and necrotic core lesions. Image processing using an unsupervised clustering technique was used to reconstruct an arterial geometry from a single, patient-specific color image obtained from intravascular ultrasound-derived virtual histology. Transport of free drug was modeled using a time-dependent reaction-diffusion model and the bound, immobilized drug using the time-dependent reaction equation. The governing equations representing the transport of free as well as bound drug along with a set of initial settings and boundary conditions were solved numerically using an explicit finite difference scheme that satisfied the Courant-Friedrichs-Lewy stability criterion. Our results support previous findings related to the transport and binding of drug in arteries where tissue retention is strongly dependent on local pharmacologic properties. Additionally, modeling results indicate that non-uniform disease composition leads to heterogeneous arterial drug distribution patterns, although further validation using animal studies is required to fully appreciate pharmacokinetics in disease-laden arteries.

Development of an explicit non-staggered scheme for solving three-dimensional Maxwell’s equations

An explicit finite-difference scheme for solving the three-dimensional Maxwell’s equations in non-staggered grids is presented. We aspire to obtain time-dependent solutions of the Faraday’s and Ampère’s equations and predict the electric and magnetic fields within the discrete zero-divergence context (or Gauss’s law). The local conservation laws in Maxwell’s equations are numerically preserved using the explicit second-order accurate symplectic partitioned Runge–Kutta temporal scheme. Following the method of lines, the spatial derivative terms in the semi-discretized Faraday’s and Ampère’s equations are approximated theoretically to obtain a highly accurate numerical phase velocity. The proposed fourth-order accurate space-centered finite difference scheme minimizes the discrepancy between the exact and numerical phase velocities. This minimization process considerably reduces the dispersion and anisotropy errors normally associated with finite difference time-domain methods. The computational efficiency of getting the same level of accuracy at less computing time and the ability of preserving the symplectic property have been numerically demonstrated through several test problems.

Modeling Heat Transfer During Blanching of Cubic Particles of Loche (Cucurbita moschata Duch.) and Potato (Solanum tuberosum L.) Using Finite Difference Method

AbstractThe aim of this study was to model heat transfer during blanching of cubic particles of loche (Cucurbita moschata Duch.) and potato (Solanum tuberosum L.). The model included the variation of thermal properties based on temperature and the solution of the model was implemented by explicit finite difference method. Cube particles of 1 × 1 × 1 cm3, 2 × 2 × 2 cm3, and 3 × 3 × 3 cm3 were subjected to blanching at temperatures of 70, 80, and 90°C, each one, for 5 min. The heat transfer coefficient (h) was determined experimentally during heating of different sizes of aluminum cubes. The variation of the thermal diffusivity (α) was also determined according to the temperature increase, finding the minimum and maximum value of (α) for loche were: 1.55–1.61 × 10−7 m2 s−1 and for potato were: 1.40–1.46 × 10−7 m2 s−1. A computer program in Visual Basic language was developed. The program includes the variation of the thermal diffusivity with respect to the increase of temperature with a second degree polynomial function. Experimental temperature profiles were compared with simulated ones, showing that an efficient convergence was achieved (RMSE: 0.329–5.119°C) for cubic particle of both vegetables.Practical ApplicationsThe explicit finite difference scheme in three‐dimensions (3D) developed in this research can be used to simulate heat transfer during heating of cubic particles of vegetables with variable thermal properties. Thermal properties of Loche (peruvian pumpkin) was reported for the first time and can be used for design and optimization of processes involving heating.

An explicit finite-difference scheme for one-dimensional Generalized Porous Medium Equations: Interface tracking and the hole filling problem

Based on results of E. DiBenedetto and D. Hoff we propose an explicit finite-difference scheme for one dimensional Generalized Porous Medium Equations ∂ t u = ∂ xx 2 Φ(u) . The scheme allows to track the moving free boundaries, and captures the so-called hole filling phenomenon when free boundaries collide. We prove the convergence of the discrete solution when the mesh parameter Δx → 0. Finally, we provide numerical evidence of the convergence of the scheme.

Numerical modeling of 1D heterogeneous combustion in porous media under free convection taking into account dependence of permeability on porosity

Using numerical experiment the one-dimensional unsteady process of heterogeneous combustion in porous object under free convection is considered when the dependence of permeability on porosity is taken into account. The combustion is due to exothermic reaction between the fuel in the solid porous medium and oxidizer contained in the gas flowing through the porous object. In the present work the process is considered under natural convection, i.e. when the flow rate and velocity of the gas at the inlet to the porous objects are unknown, but the gas pressure at object boundaries is known. The influence of changing of permeability due to the changing of porosity on the solution is investigated using original numerical method, which is based on a combination of explicit and implicit finite-difference schemes. It was shown that taking into account the dependence of permeability on porosity, which is described by some known equations, can significantly change the solution in one-dimensional case. The changing of permeability due to the changing of porosity leads to the speed increasing of both cocurrent and the countercurrent combustion waves, and to the temperature increasing in the combustion zone of countercurrent combustion wave.

Interaction of geometry and mechanical property of trapezoidal sedimentary basins with incident SH waves

This paper investigated the effects of basin geometry and material property on the response of 2D trapezoidal sediment-filled basin to incident plane SH waves. Ten basin configurations with different geometries were developed, and then their seismic responses to both Ricker wavelets and seismic records were simulated by using an explicit finite difference scheme. The definition of deep/shallow basin, the precondition for the observation of prominent surface waves and the influential area of edge effects of the shallow basin were discussed quantitatively in this study. The followings were concluded: in the common velocity contrast range (v s1/v s2 3.0 can be estimated approximately by 1D theory. The complexity of peak ground acceleration distribution pattern, the width of the most affected section as well as the amplitude of ground motion in the Edge Region increase with incident frequency. Prominent surface waves can only be observed when the incident wavelength is shorter than the critical wavelength λ c . The interaction between incident wave and basin dynamic property plays a dominant role on the peak ground acceleration amplitude while the interaction between incident wave and geometry plays a more significant role on the peak ground acceleration distribution. For very shallow basin, different areas along the basin width are affected to different extents. Only a limited area close to the basin edge is influenced significantly. It is more feasible to propose spectral aggravation factor for different surface zones respectively than a uniform constant as a tool to calibrate the 1D-based design spectrum so as to take the basin effects into account.

Finite difference schemes for linear stochastic integro-differential equations

We study the rate of convergence of an explicit and an implicit–explicit finite difference scheme for linear stochastic integro-differential equations of parabolic type arising in non-linear filtering of jump–diffusion processes. We show that the rate is of order one in space and order one-half in time.

Application of a three-dimensional aeroelastic model to study the wind-induced response of bridge stay cables in unsteady wind conditions

The possibility of bridge stay cables experiencing violent dry inclined cable galloping raises great concern in the engineering community. Numerous experimental and analytical studies have been conducted to investigate this phenomenon, most of which were in the context of steady wind past a rigid cylindrical body. Real stay cables however, are generally long and flexible. They are exposed to more “broad” range of atmospheric boundary layer type of wind velocity profile which is also unsteady and turbulent by nature. To better understand the physics underlying this type of wind-induced cable vibration and to elucidate various contributing factors, a more realistic analytical model which is capable of addressing the above elements is imperative. In the current paper, a three-dimensional aeroelastic model is proposed to study the aerodynamic response of an inclined and/or yawed slender flexible cylindrical body subjected to unsteady mean wind, with practical application to wind-induced vibrations of bridge stay cables under no precipitation condition. The non-linear aerodynamic forces derived in the present study are combined with the cable free vibration equations available in literature to obtain the equations of motion for the wind-induced vibration of stay cables, which are solved numerically by an explicit finite difference scheme. The proposed three-dimensional aeroelastic model and numerical solution technique are validated by comparing the predicted cable free vibration responses with existing data in the literature. The mechanism which triggers dry inclined cable galloping and the required conditions for its growth are explored. In addition, the impact of different initial conditions and various unsteady mean wind scenarios on this violent cable motion are investigated. Results show that the occurrence of dry inclined cable galloping is associated with an opposite-phase relation between the relative wind speed and the aerodynamic force along the direction of cable motion in the critical flow regime. The cable would respond most violently should it be exposed to uniformly distributed steady wind in the critical Reynolds number range for a sufficiently long duration. The presence of multi-mode oscillation in a cable prior to its exposure to the critical flow condition would lead to a considerable reduction in cable response. Furthermore, the boundary layer type velocity profile would have a sizable impact on the cable aerodynamic response provided only a portion of the cable is subjected to the critical flow condition, and the unsteadiness in wind is found to have a stabilizing effect on cable aerodynamic response.

Unsteady Free Convection Fluid Flow over an Inclined Plate in the Presence of a Magnetic Field with Thermally Stratified High Porosity Medium

MHD free convection over an inclined plate in a thermally stratified high porous medium in the presence of a magnetic field has been studied. The dimensionless momentum and temperature equations have been solved numerically by explicit finite difference technique with the help of a computer programming language Compaq Visual Fortran 6.6a. The obtained results of these studies have been discussed for the different values of well known parameters with different time steps. Also, the stability conditions and convergence criteria of the explicit finite difference scheme has been analyzed for finding the restriction of the values of various parameters to get more accuracy. The effects of various governing parameters on the fluid velocity, temperature, local and average shear stress and Nusselt number has been investigated and presented graphically.

Convergence rate of free boundary of numerical scheme for American option

Based on the optimal estimate of convergence rate $O(\Delta x)$ of the value function ofan explicit finite difference scheme for the American put option problem in [6], an $O(\sqrt{\Delta x})$ rate of convergence of the free boundary resulting from a general compatible numerical scheme to the true free boundary is proven. A new criterion for the compatibility of a generic numerical scheme to the PDE problem is presented. A numerical example is also included.

Development of a 3D staggered FDTD scheme for solving Maxwell’s equations in Drude medium

An explicit finite-difference scheme is developed to solve the three-dimensional Maxwell’s equations in Drude medium. Our aim of developing this scheme in time domain is to compute solutions in staggered grids from the Faraday’s and Ampère’s equations. The electric and magnetic field solutions are sought subject to the discrete zero-divergence condition (Gauss’s law). The local conservation laws in ideal Maxwell’s equations are also numerically preserved all the time using the explicit second-order accurate symplectic partitioned Runge–Kutta temporal scheme. The spatial derivative terms in the Faraday’s and Ampère’s equations are discretized to obtain fourth-order accuracy using the developed scheme underlying the concept of minimizing the discrepancy between the exact and the derived numerical phase velocities. Dispersion and anisotropy errors have been much reduced through the procedure of minimizing phase velocity error. In addition to performing the fundamental analysis on the proposed scheme, the computational efficiency and long-term accurate properties embedded in the proposed symplectic dispersion-error reduction centered scheme are numerically demonstrated through several test problems investigated in ideal and Drude media.

Prediction of the intramembranous tissue formation during perisprosthetic healing with uncertainties. Part 1. Effect of the variability of each biochemical factor

A stochastic model is proposed to predict the intramembranous process in periprosthetic healing in the early post-operative period. The methodology was validated by a canine experimental model. In this first part, the effects of each individual uncertain biochemical factor on the bone-implant healing are examined, including the coefficient of osteoid synthesis, the coefficients of haptotactic and chemotactic migration of osteoblastic population and the radius of the drill hole. A multi-phase reactive model solved by an explicit finite difference scheme is combined with the polynomial chaos expansion to solve the stochastic system. In the second part, combined biochemical factors are considered to study a real configuration of clinical acts.

PENENTUAN HARGA OPSI DAN NILAI HEDGE MENGGUNAKAN PERSAMAAN NON-LINEAR BLACK-SCHOLES

Option are contracts that give the right to sell and buy the asset at a price and a certain period of time. In addition investors use option as a means of hedge against asset owned. Many methods are used to determine the price of option, one of them by using the Black-Scholes equation. But its use these in the assumption that the value for the constant volatility. On market assumption are not appropriates, so many researchers proposed using a volatility calculation option that is non-constant Black-Scholes equation modelled using the volatility is not constant in the range so as to produce a non-linear equation of Black-Scholes. In addition to determine the value of hedge ratio. On completions of this study, for the numerical solution of non-linear Black-Scholes equation using method of explicit finite difference scheme. Option use in research us a stock YAHOO!inc. as the underlying asset. The result showed that the price of the option is calculated using non-linear Black-Scholes equation price close on the market. Therefore, it can produce hedge ration for a risk-free portfolio containing of the option and stock.

Heat and mass transfer to blood flowing through a tapered overlapping stenosed artery

The effects of heat and mass transfer on blood flow through a tapered artery with overlapping stenosis are analyzed in this article. An appropriate mathematical relation is used for the geometry of the stenosed artery. The rheological behavior of streaming blood in the artery is characterized by the Cross non-Newtonian model. The constitutive equation in conjunction with equation of motion is used to analyze momentum transfer while the heat and mass transfer effects are modeled through heat conduction and diffusion equations, respectively. The simplified form of momentum, heat and mass transfer equations are obtained by employing the mild stenosis condition. The emerging unsteady non-linear coupled partial differential equations are solved numerically using a well-tested explicit finite difference scheme. The influence of various emerging parameters on blood flow and on heat and mass transfer is evaluated via various plots. The instantaneous global behavior of temperature and concentration is also illustrated through contour plots for pertinent parameters.