Explicit Finite Difference Scheme Research Articles

A Reduced-Dimension Weighted Explicit Finite Difference Method Based on the Proper Orthogonal Decomposition Technique for the Space-Fractional Diffusion Equation

A kind of reduced-dimension method based on a weighted explicit finite difference scheme and the proper orthogonal decomposition (POD) technique for diffusion equations with Riemann–Liouville fractional derivatives in space are discussed. The constructed approximation method written in matrix form can not only ensure a sufficient accuracy order but also reduce the degrees of freedom, decrease storage requirements, and accelerate the computation rate. Uniqueness, stabilization, and error estimation are demonstrated by matrix analysis. The procedural steps of the POD algorithm, which reduces dimensionality, are outlined. Numerical simulations to assess the viability and effectiveness of the reduced-dimension weighted explicit finite difference method are given. A comparison between the reduced-dimension method and the classical weighted explicit finite difference scheme is presented, including the error in the L2 norm, the accuracy order, and the CPU time.

Efficient Numerical Solutions for Fuzzy Time Fractional Diffusion Equations Using Two Explicit Compact Finite Difference Methods

This article introduces an extension of classical fuzzy partial differential equations, known as fuzzy fractional partial differential equations. These equations provide a better explanation for certain phenomena. We focus on solving the fuzzy time diffusion equation with a fractional order of 0 < α ≤ 1, using two explicit compact finite difference schemes that are the compact forward time center space (CFTCS) and compact Saulyev’s scheme. The time fractional derivative uses the Caputo definition. The double-parametric form approach is used to transfer the governing equation from an uncertain to a crisp form. To ensure stability, we apply the von Neumann method to show that CFTCS is conditionally stable, while compact Saulyev’s is unconditionally stable. A numerical example is provided to demonstrate the practicality of our proposed schemes.

Efficient Numerical Solutions for Fuzzy Time Fractional Convection Diffusion Equations Using Two Explicit Finite Difference Methods

In this study, we explore fractional partial differential equations as a more generalized version of classical partial differential equations. These fractional equations have shown promise in providing improved descriptions of certain phenomena under specific circumstances. The main focus of this paper comprises the development, analysis, and application of two explicit finite difference schemes to solve an initial boundary value problem involving a fuzzy time fractional convection–diffusion equation with a fractional order in the range of 0≤ ξ ≤ 1. The uniqueness of this problem lies in its consideration of fuzziness within both the initial and boundary conditions. To handle the uncertainty, we propose a computational mechanism based on the double parametric form of fuzzy numbers, effectively converting the problem from an uncertain format to a crisp one. To assess the stability of our proposed schemes, we employ the von Neumann method and find that they demonstrate unconditional stability. To illustrate the feasibility and practicality of our approach, we apply the developed scheme to a specific example.

Explicit P1 Finite Element Solution of the Maxwell-Wave Equation Coupling Problem with Absorbing b. c.

In this paper, we address the approximation of the coupling problem for the wave equation and Maxwell’s equations of electromagnetism in the time domain in terms of electric field by means of a nodal linear finite element discretization in space, combined with a classical explicit finite difference scheme for time discretization. Our study applies to a particular case where the dielectric permittivity has a constant value outside a subdomain, whose closure does not intersect the boundary of the domain where the problem is defined. Inside this subdomain, Maxwell’s equations hold. Outside this subdomain, the wave equation holds, which may correspond to Maxwell’s equations with a constant permittivity under certain conditions. We consider as a model the case of first-order absorbing boundary conditions. First-order error estimates are proven in the sense of two norms involving first-order time and space derivatives under reasonable assumptions, among which lies a CFL condition for hyperbolic equations. The theoretical estimates are validated by numerical computations, which also show that the scheme is globally of the second order in the maximum norm in time and in the least-squares norm in space.

Mathematical formulation and computation of the dynamics of blood flow, heat and mass transfer during MRI scanning

Computational modeling of arterial blood flow, heat and mass transfer during MRI scanning is studied. The flow is assumed to be unsteady, in-compressible, and asymmetric. Mathematical formulation considers the presence of stenosis, joule heating viscous dissipation and chemical reaction. The explicit finite difference scheme is used to numerically solve the model equations. The MATLAB software was used to plot the graphical results. The study reveals that, during MRI scanning, both radial and axial velocities diminish with increase in the strength of magnetic fields. Besides, the study found that, Eckert number and Hartman number enhance the blood’s temperature and the same, diminishes with increase in Prandtl and Reynolds numbers. Concentration profile is observed to decline with increase in chemical reaction parameter, Schmidt number and Reynolds number. Soret number on the other hand, is observed to positively influence the concentration.

Explicit scheme for solving variable-order time-fractional initial boundary value problems

The creation of an explicit finite difference scheme with the express purpose of resolving initial boundary value issues with linear and semi-linear variable-order temporal fractional properties is presented in this study. The rationale behind the utilization of the Caputo derivative in this scheme stems from its known importance in fractional calculus, an area of study that has attracted significant interest in the mathematical sciences and physics. Because of its special capacity to accurately represent physical memory and inheritance, the Caputo derivative is a relevant and appropriate option for representing the fractional features present in the issues this study attempts to address. Moreover, a detailed Fourier analysis of the explicit finite difference scheme’s stability is shown, demonstrating its conditional stability. Finally, certain numerical example solutions are reviewed and MATLAB-based graphic presentations are made.

Pulsatile flow of EMHD non-Newtonian nano-blood through an inclined tapered porous artery with combination of stenosis and aneurysm under body acceleration and slip effects

Research on blood flow in diseased arteries has become a priority in recent years. Stenosis/aneurysm is the most common arterial diseases. This work investigates the mixed effect of electric and magnetic fields on the unsteady, two-dimensional, laminar, non-Newtonian pulsatile flow of blood in an axisymmetrically inclined tapered porous artery with stenosis and aneurysm containing gold nanoparticles, subjected to body acceleration and slip effect. Thermal radiation and heat sources are also included. To immobilize the effect of the vessel wall, we use a radial coordinate transformation. The nonlinear partial differential equations governing the present problem, together with the prescribed boundary conditions, are solved using an explicit finite-difference scheme. The current analysis could help to understand the improvement in mass distribution and heat transfer in diseased blood vessels, which might be useful for drug delivery in the therapy of stenosis and aneurysm.

Numerical Investigation of Three-Dimensional Magnetohydrodynamic Flow of Ag 􀀀 H2O Nanofluid Over an Oscillating Surface in a Rotating Porous Medium

Objective: To investigate the three-dimensional flow of a nanofluid (Ag-water) over a stretchable vertical oscillatory sheet. This study involves considering fluctuating temperatures on the sheet and comparing them to the free stream temperature. The formulation of the unsteady boundary layer equations leading to the flow of nanofluid also takes into consideration the occurrence of the heterogeneous-homogeneous chemical reaction and thermal radiation. Method: The governing equations and the boundary conditions have been derived in a dimensionless form by using the appropriate transformations, and they are then solved using an EFDS (Explicit Finite Difference Scheme) in Matlab software. The Von-Neumann stability analysis is used to determine the method’s stability requirements for constant sizes of the grid. Findings: The physical factors impact on the concentration fields, temperature distribution, and velocity distribution were obtained and are studied by graphs and described in extensive detail. Convergence and stability requirements are attained in order to achieve accurate solutions. Novelty: In this study fluctuations in the temperature and stretching velocity of sheet on three-dimensional magnetohydrodynamic flow of Ag − H2O nanofluid over an oscillating surface through rotating porous are taken into account. Impacts of porous media permeability, velocity slip, magnetic fields, nanoparticle volume fraction, heat radiation, rotation, and homogeneous and heterogeneous chemical reaction parameters had all been attempted to be determined. Keywords: Oscillatory Surface, Heat transmission, Nonlinear PDE, Explicit Finite Difference Scheme, Nanoparticle

Ion-concentration gradients induced by synaptic input increase the voltage depolarization in dendritic spines

The vast majority of excitatory synaptic connections occur on dendritic spines. Due to their extremely small volume and spatial segregation from the dendrite, even moderate synaptic currents can significantly alter ionic concentrations. This results in chemical potential gradients between the dendrite and the spine head, leading to measurable electrical currents. In modeling electric signals in spines, different formalisms were previously used. While the cable equation is fundamental for understanding the electrical potential along dendrites, it only considers electrical currents as a result of gradients in electrical potential. The Poisson-Nernst-Planck (PNP) equations offer a more accurate description for spines by incorporating both electrical and chemical potential. However, solving PNP equations is computationally complex. In this work, diffusion currents are incorporated into the cable equation, leveraging an analogy between chemical and electrical potential. For simulating electric signals based on this extension of the cable equation, a straightforward numerical solver is introduced. The study demonstrates that this set of equations can be accurately solved using an explicit finite difference scheme. Through numerical simulations, this study unveils a previously unrecognized mechanism involving diffusion currents that amplify electric signals in spines. This discovery holds crucial implications for both numerical simulations and experimental studies focused on spine neck resistance and calcium signaling in dendritic spines.

Stability analysis of a finite element approximation for the Navier-Stokes equation with free surface

In this work we study the numerical approximation of incompressible Navier-Stokes equations with free surface. The evolution of the free surface is driven by the kinematic boundary condition, and an Arbitrary Lagrangian Eulerian (ALE) approach is used to derive a (formal) weak formulation which involves three fields, namely, velocity, pressure, and the function describing the free surface. This formulation is discretised using finite elements in space and a time-advancing explicit finite difference scheme in time. In fact, the domain tracking algorithm is explicit: first, we solve the equation for the free surface, then move the mesh according to the sigma transform, and finally we compute the velocity and pressure in the updated domain. This explicit strategy is built in such a way that global conservation can be proven, which plays a pivotal role in the proof of stability of the discrete problem. The well-posedness and stability results are independent of the viscosity of the fluid, but while the proof of stability for the velocity is valid for all time steps, and all geometries, the stability for the free surface requires a CFL condition. The performance of the current approach is presented via numerical results and comparisons with the characteristics finite element method.

Discretization of Laplacian Operator on 19 Points Stencil Using Cylindrical Mesh System with the Help of Explicit Finite Difference Scheme

Discretization of Laplacian Operator on 19 Points Stencil Using Cylindrical Mesh System with the Help of Explicit Finite Difference Scheme

Data assimilation in 2D hyperbolic/parabolic systems using a stabilized explicit finite difference scheme run backward in time

ABSTRACTAn artificial example of a coupled system of three nonlinear partial differential equations generalizing 2D thermoelastic vibrations, is used to demonstrate the effectiveness, as well as the limitations, of a non iterative direct procedure in data assimilation. A stabilized explicit finite difference scheme, run backward in time, is used to find initial values, [u(.,0),v(.,0),w(.,0)], that can evolve into a useful approximation to a hypothetical target result [u∗(.,Tmax),v∗(.,Tmax),w∗(Tmax)], at some realistic 0 $ ]]>Tmax>0. Highly non smooth target data are considered, that may not correspond to actual solutions at time Tmax. Stabilization is achieved by applying a compensating smoothing operator at each time step. Such smoothing leads to a distortion away from the true solution, but that distortion is small enough to allow for useful results. Data assimilation is illustrated using 512×512 pixel images. Such images are associated with highly irregular non smooth intensity data that severely challenge ill-posed reconstruction procedures. Computational experiments show that efficient FFT-synthesized smoothing operators, based on (−Δ)q with real q>3, can be successfully applied, even in nonlinear problems in non-rectangular domains. However, an example of failure illustrates the limitations of the method in problems where Tmax, and/or the nonlinearity, are not sufficiently small.

Modelling Infectious Disease Dynamics: A Robust Computational Approach for Stochastic SIRS with Partial Immunity and an Incidence Rate

For decades, understanding the dynamics of infectious diseases and halting their spread has been a major focus of mathematical modelling and epidemiology. The stochastic SIRS (susceptible–infectious–recovered–susceptible) reaction–diffusion model is a complicated but crucial computational scheme due to the combination of partial immunity and an incidence rate. Considering the randomness of individual interactions and the spread of illnesses via space, this model is a powerful instrument for studying the spread and evolution of infectious diseases in populations with different immunity levels. A stochastic explicit finite difference scheme is proposed for solving stochastic partial differential equations. The scheme is comprised of predictor–corrector stages. The stability and consistency in the mean square sense are also provided. The scheme is applied to diffusive epidemic models with incidence rates and partial immunity. The proposed scheme with space’s second-order central difference formula solves deterministic and stochastic models. The effect of transmission rate and coefficient of partial immunity on susceptible, infected, and recovered people are also deliberated. The deterministic model is also solved by the existing Euler and non-standard finite difference methods, and it is found that the proposed scheme forms better than the existing non-standard finite difference method. Providing insights into disease dynamics, control tactics, and the influence of immunity, the computational framework for the stochastic SIRS reaction–diffusion model with partial immunity and an incidence rate has broad applications in epidemiology. Public health and disease control ultimately benefit from its application to the study and management of infectious illnesses in various settings.

Entropy generation in the ternary nanofluid flow through an atherosclerotic vessel under periodic body acceleration

This article delves into the examination of the entropy generation within the slip flow of a ternary nanofluid flowing through a stenotic blood vessel. To capture the non-Newtonian rheology of blood, the constitutive equation of the Sutterby fluid model is employed. The governing equations are modeled in cylindrical polar coordinates. These equations are made dimensionless by a suitable choice of dimensionless parameters. Additionally, the dimensionless governing equations are simplified by applying the mild stenosis assumption. With the help of prescribed initial and boundary conditions, the numerical solution of the formulated problem is acquired by applying the explicit finite difference scheme. The outcomes of numerous governing parameters on the velocity profile, temperature profile, and entropy generation are explored through graphical forms. The velocity profile rises as the slip parameter and the inclination parameter are increased. The temperature profile rises with the growing values of the source parameter and decreases with enhancing values of the shape size parameter. The increasing strength of the applied magnetic field causes a reduction in the entropy generation. A comparison of the current study with the existing results is also presented. The plot of the Nusselt number shows that the height of stenosis plays an important role in monitoring its temporal variation. The current findings have been refined to provide elasticity-worthy material for researchers in the field of biomedical science who are interested in assessing blood flow under stenosis conditions and developing effective therapies for a wide range of diseases.

Numerical study of irreversibility analysis on pulsatile flow of blood through a ω-shaped stenotic artery

The mathematical modeling of blood flow in a stenotic blood vessel is very important for understanding the effects of various geometric and rheological parameters on the normal flow of blood. The present study is concerned with the heat and mass transfer in the blood flow of a non-Newtonian fluid through a ω-shaped stenotic arterial segment. The non-Newtonian behavior of blood is described mathematically by employing the constitutive equation of the Herschel-Bulkley fluid. The wall of the artery is considered to be rigid having ω-shaped constriction in its lumen. The mild stenosis assumption is used to simplify the fully coupled momentum, energy, and concentration equations as well as the constitutive equation of the Herschel-Bulkley fluid. Numerical solutions for fluid velocity, temperature, mass concentration, and entropy generation are evaluated by using the explicit finite difference scheme. The effects of various emerging parameters on the velocity, wall shear stress, flow rate, temperature, concentration, and entropy distributions have been analyzed in detail. A comparison between different fluid models has also been presented. It is found that the temperature in the Herschel-Bulkley fluid is marginally lower than that of the Newtonian, Power law, and Bingham fluids whereas an opposite trend is noticed in the case of mass concentration.

A Numerical Approximation of the Stochastic Ito-Volterra Integral Equation

A stochastic differential equation (SDE) is a differential equation in which one or more of the terms and the solution are stochastic processes. Numerous studies have employed orthogonal polynomials, however most of them focus on deterministic rather than stochastic systems. This is the reason why in this study, we looked into a numerical solution for the stochastic Ito-Volterra integral equation using the explicit finite difference scheme and Bernstein polynomials as trial functions. The equidistant collocation procedure was used to calculate the unknown constant parameters in between and reach the desired approximation. The method was evaluated and contrasted with the Block Pulse method for approximate answers based on the aforementioned method, which were obtained and compared with others in the literature.

The splitting-based semi-implicit finite-difference schemes for simulation of blood flow in arteries

This paper is devoted to construction and analysis of splitting-based finite-difference schemes for simulation of blood flow in applications of one-dimensional models. Proposed schemes are constructed for inviscid and viscid models of blood. The proposed approach is based on the modification of averaged incompressibility condition, rewritten for the square root of a cross-sectional area. It is approximated by an absolutely stable scheme with a skew-symmetric operator. The constructed schemes can be classified as semi-implicit. The stability of proposed schemes is analyzed by the method of energy inequalities. Sufficient stability conditions are obtained. For the problems with analytical solutions, it is demonstrated that a second-order convergence rate is achieved in practice.The proposed schemes are compared with the widely used explicit finite-difference schemes, such as Lax–Wendroff, Lax–Friedrichs and McCormack schemes. The benchmark problems for simulation of flows in the following vascular systems are considered: human carotid artery, human aorta with bifurcation, vessel with bifurcation and absorbing outflow conditions, and networks with 31, 63, and 127 vessels. It is demonstrated that the computations based on the splitting-based semi-implicit schemes can be performed much faster than for the explicit schemes with close values of relative errors.

RBF based some implicit–explicit finite difference schemes for pricing option under extended jump-diffusion model

In this manuscript, we presented some efficient and accurate radial basis function-based finite difference (RBF-FD) implicit–explicit (IMEX) numerical techniques for pricing the option when the underlying asset follows the jump-diffusion process with local volatility. For the time semi-discretization, we present three numerical techniques Crank–Nikolson Leap-Frog (CNLF), Crank–Nikolson AdamBashforth (CNAB), and Backward difference formula of order two (BDF2), incorporated with the radial basis function based finite difference (RBF-FD) method. The stabilities of time semi-discretized schemes are also proved. The computational methods developed for the European option are extended for the American option. We amalgamate the RBF-FD implicit–explicit methods with an operator splitting (OS) method for solving the linear complementarity problem (LCP) with variable parameters that determines the price of an American option. In order to demonstrate the effectiveness and precision of the current techniques, numerical data for European and American put options under the Merton and Kou models are presented.

EFFICIENT EXPLICIT NUMERICAL TECHNIQUE FOR MODELING ADVECTION DIFFUSION REACTION FOR A WATER QUALITY MODEL IN AN OPENED UNIFORM FLOW – A 1D PERSPECTIVE

This research paper presents a novel and efficient explicit numerical technique for modeling advection diffusion reactions in an opened uniform flow from a one-dimensional perspective. The proposed hybrid scheme combines the benefits of explicit finite difference schemes, resulting in an accurate and fast solution for the advection-diffusion equation in water stream problems. The effectiveness of the scheme is demonstrated through its successful implementation in the solution of the water quality problems, where the advection-diffusion equation plays a crucial role. The results obtained using this technique show improved accuracy and computational efficiency. Overall, this research offers a valuable contribution to the field of numerical modeling in water quality and provides a useful tool for researchers and practitioners working in the area of approximating the one-dimensional diffusion equation for the measurement of pollutant concentration.

Numerical Simulation of One-Dimensional Advection Diffusion Equation by New Hybrid Explicit Finite Difference Schemes

In this paper, the aim is to develop two Hybrid Explicit Schemes based on the Finite Difference method for a one- dimensional Advection Diffusion Equation. Moreover, the study considered the advection-diffusion equation as an initial boundary value problem (IBVP) for numerical solutions obtained from various second-order explicit methods along with the solution by proposed methods. Von-Neumann stability analysis is used to analyze the stability of the developed schemes graphically. In the numerical analysis of errors, the L 2 has been computed to compare proposed methods with existing methods in literature and has been carried out in different conditions and step sizes. Proposed methods are robust explicit methods for the purpose of solution of the 1-D advection-diffusion equation.