Numerical Scheme Research Articles

Mathematical modeling of allelopathic stimulatory phytoplankton species using fractal–fractional derivatives

In the current study, we employ the novel fractal–fractional operator in the Atangana–Baleanu sense to investigate the dynamics of an interacting phytoplankton species model. Initially, we utilize the Picard-Lindelöf theorem to validate the uniqueness and existence of solutions for the model. We then explore equilibrium points within the phytoplankton model and conduct Hyers–Ulam stability analysis. Additionally, we present a numerical scheme utilizing the Newton polynomial to validate our analytical findings. Numerical simulations illustrate the dynamical behavior of the model across various fractal and fractional parameter values, visualized through graphical representations. Our simulations reveal that the stability of equilibrium points is not significantly impacted with the long-term memory effect, which is characterized by fractal–fractional order values. However, an increase in fractal–fractional parameters accelerates the convergence of solutions to their intended equilibrium states.

A nonlinear fractional fishery resource system model with Crowley–Martin functional response under Mittag-Leffler kernel

This article studies the action of predators and the predator-dependent functional response in the fishery resource model. We employ the Atangana–Baleanu–Caputo fractional derivative to study the proposed fractional fishery resource model in the presence of predators with Crowley–Martin functional response. We present a theoretical and numerical analysis of the governing nonlinear differential equations of the model consisting of the biomass density of the fish population inside the unrestricted fishing zone, the biomass density of the fish population inside the reserve or restricted fishing zone, and the predator population. Using the fixed point theory and nonlinear analysis, we establish the existence and uniqueness results of the proposed ABC fractional fishery resource model. We establish stability analysis of the fishery model using the Ulam–Hyers stability approach. The numerical scheme of the fractional Adams–Bashforth method is provided and the approximate solutions for the model under consideration are given and discussed. We observe that an increase in the fish capturing rates increases the size of the predator population and reduces the fish subpopulations. To maintain a high number of fish species, we recommend a control measure to reduce the fish capturing rate by the predators.

A finite volume approach to dosimetric calculations in diffusing alpha-emitters radiation therapy

Objective.In diffusing alpha-emitters radiation therapy ('Alpha DaRT'), the diffusion-leakage (DL) model is used to determine the spatial distributions of the emitters and the corresponding alpha dose, critical for a successful treatment. This work first introduces a finite volume (FV) approach to develop numerical schemes to simulate the DL model in one, two and three dimensions then presents how variations over realistic ranges of the DL model parameters related to desorption, diffusion and leakage processes affect the alpha dose distribution and the position of the clinically significant alpha particle10Gy isodose. This work also presents the effects of three modeling approximations: two source geometry approximations (solid cylinder instead of hollow, pixelized cross section instead of circular), and one dosimetric approximation (single-source dose superposition instead of multiple-sources direct dose calculation).Approach.The introduced FV approach was used to obtain spatial distributions of the emitters, from which the corresponding alpha dose distributions were calculated under the assumption of a local deposition of the alpha particles' energies. Variation ranges of the DL model parameters were based on previously published data. For each modeling approximation studied, the error and relative error on the alpha dose distribution were calculated and the displacement of the10Gy isodose was evaluated.Main results.Over realistic ranges, the desorption probabilities, diffusion lengths, and leakage probabilities affect the position of the alpha particle10Gy isodose by∼0.1mm,∼1.5mm and∼0.5mm, respectively. The three modeling approximations studied have a negligible effect on the alpha particle10Gy isodose position, with displacements⩽0.01mm.Significance.This work quantitatively evaluates the relative importance of different parameters and approximations in Alpha-DaRT alpha dose calculations based on their impact not only on the dose variation at a given distance from the source but also on the displacement of clinically significant isodoses.

SMS: Spiking marching scheme for efficient long time integration of differential equations

We propose a Spiking Neural Network (SNN)-based explicit numerical scheme for long time integration of time-dependent Ordinary and Partial Differential Equations (ODEs, PDEs). The core element of the method is a SNN, trained to use spike-encoded information about the solution at previous timesteps to predict spike-encoded information at the next timestep. After the network has been trained, it operates as an explicit numerical scheme that can be used to compute the solution at future timesteps, given a spike-encoded initial condition. A decoder is used to transform the evolved spiking-encoded solution back to function values. We present results from numerical experiments of using the proposed method for ODEs and PDEs of varying complexity.

Exponential time difference methods with spatial exponential approximations for solving boundary layer problems

AbstractIn this work, an efficient and stable exponential time difference method is presented for solving boundary layer problems. By combining exponential time difference schemes with spatial direct discontinuous Galerkin discretization based on exponential boundary layer approximations, the proposed algorithm not only may admit large time step sizes but also could provide good spatial approximations even if on rather coarse spatial grids in the boundary layer. Some energy stabilities of the numerical scheme are rigorously derived. Numerical examples illustrate the accuracy, stability and efficiency of the algorithm.

The transmission dynamics of an infectious disease model in fractional derivative with vaccination under real data

The resurgence of monkeypox causes considerable healthcare risks needing efficient immunization programs. This work investigates the monkeypox disease dynamics in the UK, focusing on the impact of vaccination under real data. The key difficulty is to correctly predict the spread of the disease and evaluate the success of immunization efforts. We construct a mathematical model for monkeypox infection and extend it to the fractional case considering the Caputo derivative. The analysis ensures the positivity, boundedness, and uniqueness of the solution for the non-integer system. We conduct a local asymptotical stability analysis (LAS) at the disease-free equilibrium (DFE) D0, showing the result for R0<1. Additionally, we demonstrate the existence of multiple endemic equilibria and provide conditions for backward bifurcation, which are illustrated graphically. Using real case data from the UK, we estimate model parameters via the nonlinear least square method. Our results show that, without vaccination, R2≈0.8, whereas vaccination reduces it to R2v=0.48. We perform sensitivity analysis to identify key parameters influencing disease elimination, presenting the outcomes through graphs. To solve numerically the fractional model, we outline a numerical scheme and provide detailed results under various parameter assumptions. Our findings suggest that high vaccine efficacy, a low waning rate of the vaccines, and increased vaccination of the infected people can significantly reduce the future cases of monkeypox in the UK. The present study offers a comprehensive framework for monkeypox dynamics and informs public health strategies for effective disease control and prevention.

Shifted Legendre Gauss‐Lobatto collocation scheme for solving nonlinear coupled space‐time fractional reaction‐advection diffusion equations

AbstractThis work aims to develop and apply an algorithm for solving space‐time fractional reaction‐advection diffusion (STFRAD) nonlinear coupled equations with specified initial and boundary conditions by employing a Shifted Legendre Gauss‐Lobatto collocation (SLGLC) scheme. The fractional derivatives of time and space are defined in Caputo sense. The approximate solutions have been constructed with the help of shifted Legendre polynomials. The proposed numerical scheme reduces the coupled fractional order differential equations into a system of algebraic equations which can then be easily solved. The error analysis through the two test problems is carried out to validate the efficiency and efficacy of the method. The effect of advection and reaction terms, and various space‐time fractional order derivatives on the solution profiles have been studied and are shown graphically for different particular cases. It has been observed that the species whose transport is simulated using STFRADE shows higher concentration at a specific distance in porous media in case of integer order differential equations as compared to fractional order equations demonstrating overestimation of its impact in case of integer order differential equation. This is extremely important observation with respect to applications in the field of groundwater pollution management. The salient feature of the article is the stability and convergence analyses of the proposed scheme on the concerned model, which demonstrates the effectiveness and capabilities of the developed method.

Error estimate of the conservative difference scheme for the derivative nonlinear Schrödinger equation

In this letter, we propose and rigorously analyze a fully implicit difference scheme for the derivative nonlinear Schrödinger equation. We show that the numerical scheme at least preserves two discrete conserved quantities. Next, to facilitate error estimate, the numerical scheme is converted into an equivalent system, which can be regarded as one-stage Gaussian–Legendre Runge–Kutta method in time. Furthermore, with the help of the cut-off function technique, we prove the convergence of the equivalent system for the first time with the convergence order O(τ2+h2) under discrete L∞-norm without any restriction on step ratio. Finally, the numerical results confirm theoretical findings and capacity in long-time simulations.

A filtered Rankine–Hugoniot method for consistent stochastic particle LES-FDF modelling of combustion in high-speed flows

A new model consistency scheme based on the filtered form of the Rankine–Hugoniot (R–H) relation is developed for hybrid Eulerian/Lagrangian Large Eddy Simulation-Filtered Density Function (LES-FDF) methods to model combustion in compressible high-speed flows. The approach particularly pertains to the evolution of the FDF stochastic Lagrangian particles in enthalpy space, accounting for the subgrid effects in the filtered R–H relation to ensure stable solutions and model consistency with the total energy field solved by an Eulerian compressible finite volume scheme. An analytical form of the subgrid term appearing in the filtered R–H relation is derived and it is shown to have a significant effect on the ability to accurately produce the post-shock thermochemical state. Subsequently, two novel numerical schemes are proposed based on either a modelled subgrid scale kinetic energy or a relaxation over a physical timescale. In testing for 1D normal shock cases, the former approach is found to be strongly dependent on the filter width and Mach number, while the latter approach produces consistent results. The relaxation method is further tested for a turbulent 3D shock tube case and a 1D detonation case and found to produce numerically consistent and accurate temperature and species fields. It is shown that the subgrid term is critical for accurately and consistently predicting the temperature fields in both cases and for achieving the correct detonation structure involving a leading shock and autoignition in the second case.Novelty and significance statementLES-FDF methods have not been widely used for high-speed combusting flows with strong shocks. Such flows are common in aerospace combustors and accurate modelling of these is essential for their optimisation, especially with rapid transition to alternative fuels. The first novelty is the development of a filtered Rankine–Hugoniot model to ensure energy consistency between the Eulerian LES and the FDF modelled through Lagrangian particles. The inclusion of a subgrid kinetic energy term in the Lagrangian particle energy equation is shown to be essential for producing consistent solutions and accurate prediction of thermodynamics states through discontinuities. This subgrid term is not closed and, therefore, the second novelty lies in proposing a practical numerical scheme which accounts for it. The consistent model and numerical scheme is validated in canonical shock tube and detonation cases.

Efficient energy stable schemes for incompressible flows with variable density

We present novel numerical schemes for the incompressible Navier-Stokes equations with variable density and address two critical concerns, i.e., the preservation of lower density bounds and the preservation of energy inequality under the gravitational force. Spatial discretization is performed using upwind discontinuous Galerkin methods for density, and continuous finite element methods are used for velocity and pressure. Additionally, velocity fields are projected onto divergence-free Raviart-Thomas finite element space in the transport equation. This relaxes using divergence-free finite elements and therefore simplifies the implementation. The proposed schemes are tested on benchmark examples to demonstrate the excellent performance in accurately capturing the complex dynamics of these problems. Overall, the proposed numerical schemes exhibit enhanced stability, efficiency, and accuracy, making them suitable for modeling and simulating incompressible flows with variable density in practical applications.

Domain preserving and strongly converging explicit scheme for the stochastic SIS epidemic model

In this article, we construct a numerical method for a stochastic version of the Susceptible–Infected–Susceptible (SIS) epidemic model, expressed by a suitable stochastic differential equation (SDE), by using the semi-discrete method to a suitable transformed process. We prove the strong convergence of the proposed method, with order 1, and examine its stability properties. Since SDEs generally lack analytical solutions, numerical techniques are commonly employed. Hence, the research will seek numerical solutions for existing stochastic models by constructing suitable numerical schemes and comparing them with other schemes. The objective is to achieve a qualitative and efficient approach to solving the equations. Additionally, for models that have not yet been proposed for stochastic modeling using SDEs, the research will formulate them appropriately, conduct theoretical analysis of the model properties, and subsequently solve the corresponding SDEs.

A design optimization method for rarefied and continuum gas flows

A design optimization method applicable to both rarefied and continuum gas flows with good efficiency is proposed in the framework of gas-kinetic theory. The method follows the methodology of topology optimization where the areas of gas and solid are marked by the material density, based on which a fictitious porosity model is used to reflect the effect of the solid on the gas and mimic the diffuse boundary condition on the gas-solid interface. The formula of this fictitious porosity model is modified to make the model work well from the free-molecular flow regime to the continuum flow regime. To find the optimized material density, a gradient-based optimizer is adopted and the design sensitivity is obtained by the continuous adjoint method. To solve the primal kinetic equation and the corresponding adjoint equation, efficient multiscale numerical schemes are constructed. Several airfoil optimization problems are solved to demonstrate the performance of the present design optimization method in wide ranges of flow speed and gas rarefaction. It is found that the thickness of the optimized airfoil first increases and then decreases as the Knudsen number increases, and on the supersonic condition the optimized airfoil has quite different shapes of leading edge for different degrees of gas rarefaction.

Structure preserving finite element schemes for the Navier-Stokes-Cahn-Hilliard system with degenerate mobility

In this work we present two new numerical schemes to approximate the Navier-Stokes-Cahn-Hilliard system with degenerate mobility using finite differences in time and finite elements in space. The proposed schemes are conservative, energy-stable and preserve the maximum principle approximately (the amount of the phase variable being outside of the interval [0,1] goes to zero in terms of a truncation parameter). Additionally, we present several numerical results to illustrate the accuracy and the well behavior of the proposed schemes, as well as a comparison with the behavior of the Navier-Stokes-Cahn-Hilliard model with constant mobility.

A robust optimisation approach for the 2D-VO fractional optimal control problems

This paper provides a new numerical scheme with the help of discrete Krawtchouk polynomials (DKPs), the Ritz method, and the two-dimensional (2D) Gauss-Legendre quadrature rule for dealing with two-dimensional variable-order (VO) fractional optimal control problems. For this aim, we construct the derivative operational matrices of DKPs. Also, we impose the conditions into the approximation of state and control functions. Then, we expand the functions in the problem given DKPs and operational matrices. Meanwhile, DKPs and the 2D Gauss-Legendre quadrature rule evaluate the performance index function. According to the numerical algorithm process, the proposed problem is reduced to a system of algebraic equations. The chosen polynomial as basis functions and the Ritz method provide a powerful and flexible scheme. Also, we discuss the error of the VO-fractional derivative of the approximate solution and performance index. At last, we include several illustrative examples to indicate the validity and applicability of the present technique.

The Repulsion Property in Nonlinear Elasticity and a Numerical Scheme to Circumvent It

The Repulsion Property in Nonlinear Elasticity and a Numerical Scheme to Circumvent It

NUMERICAL SOLUTION OF NONLINEAR SCHRODINGER EQUATION USING COLLOCATION METHOD WITH CHEBYSHEV POLYNOMIALS

The nonlinear Schrodinger equation (NSE) is a partial differential equation (PDE) with numerous applications in quantum mechanics. Analytical methods for the solution of NSE are almost impossible due to their complexity. Thus, the need for a numerical scheme to seek the approximate solution of the NSE. Hence, this paper considered the numerical solution of the NSE via the spectral collocation method (SCM) with Chebyshev orthogonal polynomials of the first kind. The scheme was efficiently constructed to solve the NSE equation with the aid of MAPLE 18, and numerical evidence compared with Adomian Decomposition Method (ADM), Homotopy Analysis Transform Method (HATM) and Residual Power Series Method (RPSM) as available in the literature. The findings show that the SCM is an effective solver for NSE with rapid convergence to the exact solution as the parameter �varies.

Estimation of equilibration time scales from nested fraction approximations.

We consider an autocorrelation function of a quantum mechanical system through the lens of the so-called recursive method, by iteratively evaluating Lanczos coefficients or solving a system of coupled differential equationsin the Mori formalism. We first show that both methods are mathematically equivalent, each offering certain practical advantages. We then propose an approximation scheme to evaluate the autocorrelation function and use it to estimate the equilibration time τ for the observable in question. With only a handful of Lanczos coefficients as the input, this scheme yields an accurate order of magnitude estimate of τ, matching state-of-the-art numerical approaches. We develop a simple numerical scheme to estimate the precision of our method. We test our approach using several numerical examples exhibiting different relaxation dynamics. Our findings provide a practical way to quantify the equilibration time of isolated quantum systems, a question which is both crucial and notoriously difficult.

COMPARING SCIENTIFIC COMPUTING ENVIRONMENTS FOR SIMULATING 2D NON-BUOYANT FLUID PARCEL TRAJECTORY UNDER INERTIAL OSCILLATION: A PRELIMINARY EDUCATIONAL STUDY

This study presents a preliminary numerical investigation of the two-dimensional trajectory of a non-buoyant fluid parcel subjected to inertial oscillations and abrupt external forcing events. The simulations were implemented using Python, GNU Octave, R, Julia, and Fortran open-source scientific computing environments. By running 1,000 iterations in each environment, we evaluated the computational performance of these languages in tackling this idealized problem. The results, visualized through static plots and animation, validate the numerical model's ability to represent the fundamental physics governing fluid motion. Statistical analysis using the Kruskal-Wallis test and Dunn's post-hoc test with Bonferroni correction revealed that Fortran exhibits significantly faster execution times than other environments. However, the choice of programming language should also consider factors such as coding expertise, library availability, and scalability requirements. This study focuses on the performance of scientific computing environments within each language rather than the languages themselves. The observed execution times should be interpreted in the context of the specific environments used, as they often leverage optimized libraries written in lower-level languages. Despite the limitations of this work, such as the simplified 2D model and the use of a single hardware configuration, this study provides valuable insights into selecting appropriate computational tools. It contributes to educational resources for teaching idealized fluid dynamics models. Future studies could explore more complex scenarios, a more comprehensive range of programming environments, and the impact of different numerical schemes and physical parameterizations.

The experimental and numerical studies of formation and collapse processes of ventilated supercavitating flow

The aim of this paper is to investigate the ventilated cavitating flow behavior, through experimental and numerical methods. The experiments are performed for a test model placed in a water tunnel and a high-speed camera is used to image the formation and collapse of the supercavity. For the numerical simulations, Ansys CFX commercial software is employed to solve the Reynolds-averaged Navier–Stokes (RANS) equations and additional transport equation for the liquid volume fraction in the unsteady condition. Both the experimental observation and numerical prediction show a hysteresis behavior for ventilated supercavity in which the value of the air entrainment coefficient required to maintain the supercavity in the formation process is less than the amount required to form it. The good agreement observed between the numerical prediction and experimental data, revealed the accuracy and capability of the numerical scheme. Also, the numerical simulation shows that the variations in the supercavity length, in addition to the air entrainment, are a function of the air leakage regime from the closure region of supercavity.

Impact of north-west Bay of Bengal bathymetry along the off-coast of Digha on the propagation of shallow water waves using finite difference method

Numerical simulation of shallow water equations has been carried out using the Crank-Nicolson scheme of Finite Difference Method (FDM) to study the interaction of waves with different forms of bathymetry especially that of the off-coast of Digha, a tourist spot situated in the north-west coast of Bay of Bengal. MATLAB code has been developed for the simulation. One-dimensional (1-D) linear shallow water model has been considered here. Three different bathymetry profile are chosen for the investigation: (i) flat bathymetry (with or without seamount/basin); (ii) piecewise linear approximation to the real bathymetry of the off-coast of Digha; (iii) real bathymetry of the off-coast of Digha. Real bathymetry data has been obtained from the gridded data set of GEBCO (General Bathymetric Chart of the Oceans) published in 2022. Validation of the present numerical scheme has been performed by considering test problems with results existing in the literature. Comparative study of the effect of the bathymetries, on the propagation of shallow water waves, has been performed and demonstrated with figures and tables. Results of wave-bathymetry interaction obtained here are believed to be new with respect to the present study area and will definitely give some insight into the phenomena in this region.