Forward Finite Difference Method Research Articles

Performance analysis of linearization schemes for modelling multi-phase flow in porous media

Reservoir simulation is crucial for understanding the flow response in underground reservoirs, and it significantly helps reduce uncertainties in geological characterization and optimize methodologies for field development strategies. However, providing efficient and accurate solutions for the strong heterogeneity remains challenging, as most of the discretization methods cannot handle this complexity. In this work, we perform a comprehensive assessment of various numerical linearization techniques employed in reservoir simulation, particularly focusing on the performance of the nonlinear solver for problem dealing with fluid flow in porous media. The primary linearization methods examined are finite difference central (FDC), finite forward difference (FDF), and operator-based linearization (OBL). These methods are rigorously analyzed and compared in terms of their accuracy, computational efficiency, and adaptability to changing reservoir conditions. The results demonstrate that each method has distinct strengths and limitations. The FDC method is more accurate particularly in complex simulations where strong heterogeneity are introduced but is generally slower in convergence. The OBL on the other hand, is more efficient and converges quickly, which makes it suitable for scenarios with limited computational resources and simple physics, while the FDF method provides a balanced combination of precision and computational speed, contingent upon careful step size management of the derivative estimations. This paper aims to guide the selection of appropriate linearization techniques for enhancing nonlinear solvers’ accuracy and efficiency in reservoir simulation .

The nonlocal coupled system of Caputo–Fabrizio fractional q‐integro differential equation

This scheme's main goal is to examine the existence, uniqueness, and continuous dependence of solutions for a nonlinear coupled system of fractional q‐integro‐differential equations involving the derivation and integration of fractional Caputo–Fabrizio. The numerical technique methodology of the proposed problem will be introduced. Proving the existence theorem depends on Schauder's fixed‐point theorem. To drive the numerical method, we use the definitions of the fractional derivative and integral of Caputo–Fabrizio and the q‐integral of the Riemann–Liouville type. Then, the integral part will be treated using the trapezoidal method, and the derivative part will be treated using the forward finite difference method. And therefore, the coupled system will be converted into a system of algebraic equation that will be solved together to get the solutions. Finally, we give two examples to illustrate the effectiveness of the suggested approach.

Random vibration and reliability analysis of fluid-conveying pipe under white noise excitations

Due to the random excitation in the external environment, the fluid-conveying pipe would inevitably undergo random vibration. This paper investigates the random vibration and reliability of simply-supported fluid-conveying pipes under white noise excitations. Dynamic equations of the nonlinear pipe conveying fluid under white noise excitations are modeled by the generalized Hamilton principle. Then probability density functions (PDFs) of the amplitude and energy of the system are determined by using the modified stochastic averaging method. The PDFs of displacement and velocity of the system are also discussed. The accuracy of results is verified by the Monte Carlo method. In addition, the first-passage problem of the pipe system is investigated for the first time. In order to obtain the PDFs of reliability and first failure time of the system, the forward finite difference method is adopted to solve the backward Kolmogorov equation of the system. The effects of different parameters on the amplitude, energy, reliability, and first failure time are investigated. The results show that the amplitude of the system becomes large and the first-passage time is advanced with the increase of excitation intensity or fluid speed. The amplitude of the system is decreased and the first-passage time is delayed with the increase of the damping coefficient.

A Composite Algorithm for Numerical Solutions of Two-Dimensional Coupled Burgers’ Equations

In this study, a new composite algorithm with the help of the finite difference and the modified cubic trigonometric B-spline differential quadrature method is developed. The developed method was applied to two-dimensional coupled Burgers’ equation with initial and Dirichlet boundary conditions for computational modeling. The established algorithm is better than the traditional differential quadrature algorithm proposed in literature due to more smoothness of cubic trigonometric B-spline functions. In the development of the algorithm, the first step is semidiscretization in time with the forward finite difference method. Furthermore, the obtained system is fully discretized by the modified cubic trigonometric B-spline differential quadrature method. Finally, we obtain coupled Lyapunov systems of linear equations, which are analyzed by the MATLAB solver for the system. Moreover, comparative study of these solutions with the numerical and exact solutions which are appeared in the literature is also discussed. Finally, it is found that there is good suitability between exact solutions and numerical solutions obtained by the developed composite algorithm. The technique can be extended for various multidimensional Burgers’ equations after some modifications.

Reliability assessment of a planar steel frame subjected to earthquakes in case of an implicit limit-state function

This study determines the probability of failure of a planar steel frame using a proposed framework and Monte Carlo simulation (MCS) considering the uncertainties of the material and cross-section parameters. A new framework (O-FCD method) is proposed by combining the OpenSees software for structural analysis and the first-order reliability method (FORM) implemented using MATLAB for analyzing the structural reliability to determine the failure probability of a case study structure with the implicit limit-state function. The limit-state conditions of this case study structure are defined by the limitations in two scenarios: the total drift defined as the ratio of the maximum top displacement to the building height of 3%, and the maximum interstory drift of 5%. Both static and dynamic loads are applied to compare the results of the O-FCD method and MCS. The investigations show that the O-FCD method can determine well the probability of failure of this case study structure. The absolute errors of the proposed framework and MCS are slightly different under both static and dynamic loads. In addition, the O-FCD method can solve the computationally expensive part of MCS in the case of a lower failure probability. Finally, the O-FCD method uses the finite central difference method to calculate the gradient vector in FORM numerically. The proposed framework can reach the convergence criteria with numerous random variables compared with a framework using the forward finite difference method.

On the exact and numerical solutions to the FitzHugh–Nagumo equation

In this paper, with the help of a computer package program, the auto-Bäcklund transformation method (aBTM) and the finite forward difference method are used for obtaining the wave solutions and the numeric and exact approximations to the FitzHugh–Nagumo (F-N) equation, respectively. We successfully obtain some wave solutions to this equation by using aBTM. We then employ the finite difference method (FDM) in approximating the exact and numerical solutions to this equation by taking one of the obtained wave solutions into consideration. We also present the comparison between exact and numeric approximations and support the comparison with a graphic plot. Moreover, the Fourier von-Neumann stability analysis is used in checking the stability of the numeric scheme. We also present the [Formula: see text] and [Formula: see text] error norms of the solutions to this equation.

A numerical efficient splitting method for the solution of two dimensional susceptible infected recovered epidemic model of whooping cough dynamics: Applications in bio-medical engineering

Background and Objective The positivity property of the non-linear dynamical systems is one of the essential features in different fields of bio-medical engineering, science and many more. The state variables, involving in the models, describing the natural phenomenon such as concentration, density and population size etc. must be positive. Therefore, the computing techniques used to solve the system of non-linear differential equations must be consisted with the continuous nature of the models. But, unfortunately there are some existing techniques in the literature that do not preserve the positivity property, especially for the multi-space dimensional models. So there is a gap in the literature that should be filled up, by constructing the positivity preserving numerical algorithms. In this study, we consider a susceptible-infected-recovered (SIR) reaction diffusion epidemic model in two space dimensions from biomedical engineering and solved numerically to observe the behavior of the model. Since the state variables involved in this system are population densities therefore we design a novel computational method which is time efficient because of its splitting structure and holds the positivity as well as other important structure of epidemic system. Methods Three different computational techniques are designed to examine the numerical solution of SIR model of infectious disease. Two approaches are well-known existing computing methods named as forward Euler finite difference (FD) method and backward Euler operator splitting finite difference (OS-FD) method. The third approach is operator splitting nonstandard finite difference (OS-NSFD) method which is devised by using the NSFD rules. Results The proposed OS-NSFD technique retains efficiently the stability of equilibria as well as the positivity. Graphical behavior depicts that the existing computing methods can not get success to preserve the structure of the epidemic system of whooping cough dynamics. At the same time OS-NSFD computing method is proven to be reliable and suitable for the system of bio-medical engineering mathematically and graphically. Conclusion A reliable and novel computing technique is developed for the solution of two dimensional reaction diffusion problem. This technique preserves all the imperative characteristics of the model under study. Also the time efficiency of this method makes it easy to find the solution of physical system in two space dimension. The comparison with other techniques shows the efficacy and reliability of the designed technique.

Numerical analysis of diffusive susceptible-infected-recovered epidemic model in three space dimension

Abstract In this article, numerical solution of three dimensional susceptible-infected-recovered (SIR) reaction-diffusion epidemic system is furnished with a time efficient operator splitting nonstandard finite difference (OS-NSFD) method. We perform the comparison of proposed OS-NSFD method with popular forward Euler explicit finite difference (FD) method and time efficient backward Euler operator splitting finite difference (OS-FD) implicit method. The proposed OS-NSFD method is implicit in nature but computationally efficient as compared to forward Euler explicit (FD) scheme. The numerical stability and bifurcation value of transmission coefficient for SIR reaction-diffusion epidemic system is also investigated with the aid of Routh–Hurwitz method. At the end, we give two numerical experiments and simulation. In first experiment, all the numerical schemes are compared with the help of simulations. In second experiment we show the simulations of proposed NSFD technique at different values of parameters. Also we discuss the importance of transmission rate to control the spread of disease with the help of simulations.

Algebra of Fuzzy Random Variable and its Application to Science and Engineering Problem- Radon Transport from Soil into Building

The theme of this paper is to bring out the mathematical details of fuzzy random variable and its application to model radon transport from soil into buildings. The physical processes that influence radon transport are advection and diffusion. Parameters associated with the governing partial differential equation describing radon transport from soil into buildings are radon diffusion coefficient and advective velocity of radon in air. Both the parameters are imprecise. Imprecision of these parameters is addressed as fuzzy random variable due to presence of inherent randomness within their fuzziness. Handling fuzziness within randomness is an innovative concept of computation to quantify the uncertainty of any physical system. Paper presents the mathematical structure (algebra) of fuzzy random variables. The concept of fuzzy randomness is implemented in developing radon transport model. Numerical solution of radon transport model with fuzzy random parameters is obtained by explicit forward time central space finite difference method. Support, uncertainty index, possibility, necessity and credibility of the spatial and temporal behaviour of the random behaviour of radon concentration are also computed to explore the concept and role of the fuzzy randomness in decision making issue

Numerical Solution of High-Dimensional Shockwave Equations by Bivariate Multi-Quadric Quasi-Interpolation

Radial basis function-based quasi-interpolation performs efficiently in high-dimensional approximation and its applications, which can attain the approximant and its derivatives directly without solving any large-scale linear system. In this paper, the bivariate multi-quadrics (MQ) quasi-interpolation is used to simulate two-dimensional (2-D) Burgers’ equation. Specifically, the spatial derivatives are approximated by using the quasi-interpolation, and the time derivatives are approximated by forward finite difference method. One advantage of the proposed scheme is its simplicity and easy implementation. More importantly, the proposed scheme opens the gate to meshless adaptive moving knots methods for the high-dimensional partial differential equations (PDEs) with shock or soliton waves. The scheme is also applicable to other non-linear high-dimensional PDEs. Two numerical examples of Burgers’ equation (shock wave equation) and one example of the Sine–Gordon equation (soliton wave equation) are presented to verify the high accuracy and efficiency of this method.

Modeling of soft tissue thermal damage based on GPU acceleration

Hyperthermia treatments require precise control of thermal energy to form the coagulation zones which sufficiently cover the tumor without affecting surrounding healthy tissues. This has led modeling of soft tissue thermal damage to become important in hyperthermia treatments to completely eradicate tumors without inducing tissue damage to surrounding healthy tissues. This paper presents a methodology based on GPU acceleration for modeling and analysis of bio-heat conduction and associated thermal-induced tissue damage for prediction of soft tissue damage in thermal ablation, which is a typical hyperthermia therapy. The proposed methodology combines the Arrhenius Burn integration with Pennes’ bio-heat transfer for prediction of temperature field and thermal damage in soft tissues. The problem domain is spatially discretized on 3-D linear tetrahedral meshes by the Galerkin finite element method and temporally discretized by the explicit forward finite difference method. To address the expensive computation load involved in the finite element method, GPU acceleration is implemented using the High-Level Shader Language and achieved via a sequential execution of compute shaders in the GPU rendering pipeline. Simulations on a cube-shape specimen and comparison analysis with standalone CPU execution were conducted, demonstrating the proposed GPU-accelerated finite element method can effectively predict the temperature distribution and associated thermal damage in real time. Results show that the peak temperature is achieved at the heat source point and the variation of temperature is mainly dominated in its direct neighbourhood. It is also found that by the continuous application of point-source heat energy, the tissue at the heat source point is quickly necrotized in a matter of seconds, while the entire neighbouring tissues are fully necrotized in several minutes. Further, the proposed GPU acceleration significantly improves the computational performance for soft tissue thermal damage prediction, leading to a maximum reduction of 55.3 times in computation time comparing to standalone CPU execution.

Using a shifted Grünwald‐Letnikov scheme for the Caputo derivative to study anomalous solute transport in porous medium

SummaryWe propose an extension of the shifted Grünwald‐Letnikov method to solve fractional partial differential equations in the Caputo sense with arbitrary fractional order derivative α and with an advective term. The method uses the relation between Caputo and Riemann‐Liouville definitions, the shifted Grünwald‐Letnikov, and the traditional backward and forward finite difference method. The stability of the method is investigated for the implicit and explicit scheme with homogeneous boundary conditions, and a stability criterion is found for the advective‐dispersive equation. An application of the method is used to solve contaminant diffusion and advective‐dispersive problems. The numerical solution for the fractional diffusion and fractional advection‐dispersion is compared with their respective analytical solutions for different time and space grid refinements. The diffusion simulation exhibited a good fit between the analytical and numerical solutions, with the explicit scheme going from stable to unstable as the time and space refinement changes. The fractional advection‐dispersion application produced small deviations from the analytical solution. These deviations, however, are analogous to the numerical dispersions encountered in conventional finite difference solutions of the advection‐dispersion equation. The new method is also compared with the traditional L2 method. Notably, an example that involves asymmetrical fractional conditions, a fractional diffusivity that depends on time, and a source term show how the methods compare. Overall, this study assesses the quality and easiness of use of the numerical method.

Simplified algorithm for reliability sensitivity analysis of structures: A spreadsheet implementation.

An important segment of the reliability-based optimization problems is to get access to the sensitivity derivatives. However, since the failure probability is not a closed-form function of the input variables, the derivatives are not explicitly computable and rather require a full reliability analysis which is computationally expensive. In this paper, a step-by-step algorithm has been presented to calculate the derivatives of the probability of failure and safety index with respect to the input parameters based on the advanced first-order second-moment (AFOSM) reliability method. The proposed algorithm is then implemented in a spreadsheet using Visual Basic for Application (VBA) programming language. Two geotechnical and structural examples are then presented to examine the program and describe the modeling procedure. The robustness of the proposed method is examined using a Gaussian random perturbation. The capability of the proposed method in the calculation of the sensitivity derivatives of the model uncertainty is explained in a separate section. Finally, the proposed model has been compared to the forward finite difference (FFD) method and the results are validated.

Transient Mixed Convection Boundary Layer Flow of an Incompressible Fluid Past a Wedge in Presence of Magnetic Field

In this paper, an analysis is performed to explorethe transient, laminar two-dimensional, mixed convection boundary layer flow of a viscous and incompressible fluid past a vertical wedge taking into account the effect of magnetic field. With appropriate transformations the boundary layer equations are reduced to a local nonsimilarity equations and the solutions are obtained employing three distinct methods, namely, (i) perturbation method for small time; (ii) asymptotic solution method for large time; (iii) straight forward finite difference method for any time. The agreement between the solutions obtained from prescribed methods is found to be excellent. In this study the evaluation of skin-friction coefficient and the local Nusselt number with the effects of different governing parameters such as different time, τ, the exponent, m (= 0.4, 0.5, 1.0), mixed convection parameter, λ (= 0.0, 0.2, 0.4) and magnetic field parameter, M (=0.0, 1.0) for fluids having Prandtl number, Pr= 0.72, 1.0 and 7.0have been discussed. It is observed that both the local skin friction and local Nusseltnumber decreases due to an increase in the value of M. It is also found that an increase in the value of Prandtl number, Pr, leads to a decrease in the value of local skin friction coefficient and the value of local Nusselt number coefficient increases with the increasing values of Prandtl number.

Comparison of Caputo and conformable derivatives for time-fractional Korteweg–de Vries equation via the finite difference method

In this study, we investigate the nonlinear time-fractional Korteweg–de Vries (KdV) equation by using the (1/G[Formula: see text])-expansion method and the finite forward difference method. We first obtain the exact wave solutions of the nonlinear time-fractional KdV equation. In addition, we used the finite-forward difference method to obtain numerical solutions in this equations. When these solutions are obtained, the indexed forms of both Caputo and conformable derivatives are used. By using indexing technique, it is shown that the numerical results of the nonlinear time-fractional KdV equation approaches the exact solution. The two- and three-dimensional surfaces of the obtained analytical solutions are plotted. The von Neumann stability analysis of the used numerical scheme with the studied equation is carried out. The L2and L[Formula: see text] error norms are computed. The exact solutions and numerical approximations are compared by supporting with graphical plots and tables.

On the exact and numerical solutions to the coupled Boussinesq equation arising in ocean engineering

The studies of the dynamic behaviors of nonlinear models arising in ocean engineering play a significant role in our daily activities. In this study, we investigate the coupled Boussinesq equation which arises in the shallow water waves for two-layered fluid flow. The modified exp $$(-\varphi (\zeta ))$$ -expansion function method is utilized in reaching the solutions to this equation such as the topological kink-type soliton and singular soliton solutions. The interesting 2D and 3D graphics of the obtained analytical solutions in this study are presented. Via one of the reported analytical solutions, the finite forward difference method is used in obtaining the approximate numerical and exact solutions to this equation. The Fourier–Von Neumann analysis is used in checking the stability of the used numerical method with the studied model. The $$L_{2}$$ and $$L_{\infty }$$ error norms are computed. We finally present a comprehensive conclusion to this study.

Numerical Analysis for the Oxidation of Phenol with TiO2 in Wastewater Photocatalytic Reactors

Phenolic compounds in wastewater (even at low levels) are found to be quite toxic to humans due to their carcinogenic effects. Photocatalysis has been widely studied for the removal of phenol from industrial wastewater. In this study, photocatalytic oxidation of phenol, under UV irradiation in the presence of TiO2, has been numerically investigated. Phenol mass balance and forward finite difference method (explicit) along with various assumed/calculated parameters, from previous works, were used to numerically plot phenol concertation profiles in water with different initial phenol concentrations. Phenol compounds were observed to be totally oxidized at the bottom of the reactor and the maximum conversion rates occur near the reactor walls. It was found that higher irradiation times increase phenol oxidation rates due to higher water hydrolysis. Oxidation rate of phenol (consumption) increases with the increase in initial phenol concentration.

Mathematical Modeling of Ethane Cracking Furnace of Olefin Plant with Coke Formation Approach

Abstract In this study, ethylene furnace modeling is carried out by ethane pyrolysis (thermal cracking or hydrocracking) method in Arya Sasol Petrochemical Company (ninth olefin unit, Assaluyeh, Iran), which includes the solution of kinetic equations and transfer phenomena, by the forward finite difference method in the MATLAB. Due to study and compare coke formation, a specific time period has been selected in equal segments and equations have been solved. It means that in a length segment of coil (Δz), momentum, energy as well as mass equations are solved, then the amount of precipitated coke in each length segment is achieved. With new efficient coil diameter calculating all mentioned approach will be repeated for next time segment. The results of this model have been compared with actual data and deviation has been reported. It was found that modeling approach is more capable to define the parameters of coke formation equations. The model has a good agreement between the values of prediction and experimental of in most cases.

Finite element response sensitivity analysis of three-dimensional soil-foundation-structure interaction (SFSI) systems

The nonlinear finite element (FE) analysis has been widely used in the design and analysis of structural or geotechnical systems. The response sensitivities (or gradients) to the model parameters are of significant importance in these realistic engineering problems. However the sensitivity calculation has lagged behind, leaving a gap between advanced FE response analysis and other research hotspots using the response gradient. The response sensitivity analysis is crucial for any gradient-based algorithms, such as reliability analysis, system identification and structural optimization. Among various sensitivity analysis methods, the direct differential method (DDM) has advantages of computing efficiency and accuracy, providing an ideal tool for the response gradient calculation. This paper extended the DDM framework to realistic complicated soil-foundation-structure interaction (SFSI) models by developing the response gradients for various constraints, element and materials involved. The enhanced framework is applied to three-dimensional SFSI system prototypes for a pile-supported bridge pier and a pile-supported reinforced concrete building frame structure, subjected to earthquake loading conditions. The DDM results are verified by forward finite difference method (FFD). The relative importance (RI) of the various material parameters on the responses of SFSI system are investigated based on the DDM response sensitivity results. The FFD converges asymptotically toward the DDM results, demonstrating the advantages of DDM (e.g., accurate, efficient, insensitive to numerical noise). Furthermore, the RI and effects of the model parameters of structure, foundation and soil materials on the responses of SFSI systems are investigated by taking advantage of the sensitivity analysis results. The extension of DDM to SFSI systems greatly broaden the application areas of the d gradient-based algorithms, e.g. FE model updating and nonlinear system identification of complicated SFSI systems.

Numerical Study on Density Gradient Carbon–Carbon Composite for Vertical Launching System

This study presents new carbon–carbon (C/C) composite that has a density gradient within single material, and estimates its heat conduction performance by a numerical method. To address the high heat conduction of a high-density C/C, which can cause adhesion separation in the steel structures of vertical launching systems, density gradient carbon–carbon (DGCC) composite is proposed due to its exhibiting low thermal conductivity as well as excellent ablative resistance. DGCC is manufactured by hybridizing two different carbonization processes into a single carbon preform. One part exhibits a low density using phenolic resin carbonization to reduce heat conduction, and the other exhibits a high density using thermal gradient-chemical vapor infiltration for excellent ablative resistance. Numerical analysis for DGCC is performed with a heat conduction problem, and internal temperature distributions are estimated by the forward finite difference method. Material properties of the transition density layer, which is inevitably formed during DGCC manufacturing, are assumed to a combination of two density layers for numerical analysis. By comparing numerical results with experimental data, we validate that DGCC exhibits a low thermal conductivity, and it can serve as highly effective ablative material for vertical launching systems.