Abstract Salinization of coastal aquifers is a current problem in many regions worldwide. Simulation of this process provides an important tool for the forecast of drinking water resources. The consideration of the unsaturated phreatic zone has an essential influence on the accuracy of the prediction. However, for the large temporal and spatial scales of the aquifers, the presence of the unsaturated subdomains creates challenging difficulties for the numerical methods. In this work, we investigate two approaches for simulating haline density-driven flow in partially saturated aquifers. The first approach, based on the Richards equation, uses a representation of the saturation field and employs an adaptive linearly implicit time discretization scheme. The second approach explicitly represents the water table using a level-set method and relies on weak coupling combined with a standard implicit Euler scheme. Both approaches employ a finite-volume discretization for spatial representation of fluid flow and salt transport. The implementation is based on the UG4 toolkit together with the parallel groundwater flow simulator d3f++. The resulting linear systems are solved using a geometric multigrid method. We compare the aforementioned approaches with respect to theoretical and numerical aspects. The performance of both methods is evaluated in three numerical experiments. In particular we demonstrate robustness of the linearly implicit scheme and introduce a stabilization for the level-set method. The high-performance computing (HPC) potential of the approaches is assessed and demonstrated as well.

Solutions exhibiting weak initial singularities arise in various equations, including diffusion and subdiffusion equations. When employing the well-known L1 scheme to solve subdiffusion equations with weak singularities, numerical simulations reveal that this scheme exhibits varying convergence rates for different choices of model parameters (i.e., domain size, final time $T$, and reaction coefficient $\kappa$). This elusive phenomenon is not unique to the L1 scheme but is also observed in other numerical methods for reaction-diffusion equations such as the backward Euler (IE) scheme, Crank-Nicolson (C-N) scheme, and two-step backward differentiation formula (BDF2) scheme. The existing literature lacks an explanation for the existence of two different convergence regimes, which has puzzled us for a long while and motivated us to study this inconsistency between the standard convergence theory and numerical experiences. In this paper, we provide a general methodology to systematically obtain error estimates that incorporate the exponential decaying feature of the solution. We term this novel error estimate the ‘decay-preserving error estimate’ and apply it to the aforementioned IE, C-N, and BDF2 schemes. Our decay-preserving error estimate consists of a low-order term with an exponential coefficient and a high-order term with an algebraic coefficient, both of which depend on the model parameters. Our estimates reveal that the varying convergence rates are caused by a trade-off between these two components in different model parameter regimes. By considering the model parameters, we capture different states of the convergence rate that traditional error estimates fail to explain. This approach retains more properties of the continuous solution. We validate our analysis with numerical results.

Modeling and analyzing glucose-insulin interactions during diabetes through fractional dynamics in presence of glucagon.

Background: Monkeypox (Mpox) is a re-emerging zoonotic viral disease that poses an increasing threat to global public health. Mathematical modeling is a key tool for understanding transmission dynamics of diseases like Mpox and supporting effective control strategies. Reliable numerical methods are essential for solving the nonlinear differential equations arising from such models. Materials: We developed a deterministic compartmental model to describe Mpox transmission between human and small mammal populations. Human compartments include susceptible, exposed, infected, isolated, and recovered individuals, with corresponding classes for small mammals. We solved the system of nonlinear ordinary differential equations using the fourth-order Runge–Kutta (RK4) method and the Backward Euler method. We validated the model using real outbreak data from the U.S. Clade II Mpox cases reported by the Centers for Disease Control and Prevention (CDC, 2025). Results: Simulation results demonstrate that RK4 provides higher accuracy and faster convergence in non-stiff scenarios, making it suitable for short-term epidemic predictions. The Backward Euler method exhibits superior numerical stability for stiff systems, allowing reliable long-term simulations with larger time steps. Error and computational analyses confirm RK4’s efficiency, while Backward Euler ensures robustness in unstable dynamic regions. Data fitting verifies that RK4 produces closer short-term approximations, whereas Backward Euler yields smoother long-term trends. Conclusion: Both numerical methods are effective for modeling Mpox transmission. RK4 is recommended for accurate short-term analysis, while Backward Euler is preferable for stiff epidemic dynamics requiring high stability. These results highlight the importance of appropriate numerical method selection in computational epidemiology.

This research work describes a new 4-D hyperchaotic system with high complexity. The proposed 4-D system is hyperchaotic as it has a bounded attractor set with two positive Lyapunov exponents. The large positive Lyapunov exponents of the new 4-D hyperchaotic system exhibit the high complexity of the proposed system. We also show that two nearly identical trajectories quickly separate, with significant differences emerging within just 0.6 seconds, which illustrates the high sensitivity of the state trajectories of the proposed hyperchaotic systems to initial conditions and significant unpredictability. Multistability for the newly proposed hyperchaotic system is illustrated by plotting two different coexisting hyperchaotic attractors for the identical values of the system's parameters, but for non-identical initial conditions. We also explore offset boosting, a control strategy that allows us to convert the bipolar signal into a unipolar signal without changing the dynamics, which makes the proposed system more adaptable for a range of applications. The electronic implementation of the new system with high complexity is done herein by applying the well-known Euler's method, which is implemented on a field-programmable gate array (FPGA). In this 4-D system, the nonlinear terms are the multiplication of two state variables that appear in each ordinary differential equation. The FPGA design for the newly developed hyperchaotic system with high complexity is implemented on the Zybo Z7-20 development board with xc7z020clg400-1. In this study, the potential of the new 4D hyperchaotic system in the field of information security was investigated and encryption of voice data was performed using the newly developed hyperchaotic system with high complexity. The variables of the system were evaluated with differential entropy analysis and high randomness levels were verified. The XOR-based encryption algorithm developed using these variables obtained with numerical solutions provides effective protection at time, frequency and statistical levels and allows lossless recovery of data.

On the randomized Euler scheme for stochastic differential equations with integral-form drift

Lake Kivu is distinguished by several unique characteristics that set it apart from other lakes around the world. One of the notable features is a temperature increase with depth, accompanied by unusual staircase-like patterns in the thermodynamic and environmental parameters. The lake also experiences suppressed vertical mixing due to stable density stratification, with its deep water separated from the surface water by chemoclines. Additionally, Lake Kivu contains high concentrations of dissolved methane (CH4) and carbon dioxide (CO2), and there is no standard method for measuring their concentrations. The lake is also recognized as a renewable energy source due to its continuous supply of CH4, and it demonstrates a quadruple-diffusive convection transport mechanism. These factors contribute to the lake's distinctiveness. The occurrence of catastrophic limnic eruptions at Lakes Nyos and Monoun, along with the structural similarities between these lakes and Lake Kivu, raises serious concerns about the likelihood of a similar disaster in Lake Kivu in the future. The scale of threats posed in Lake Kivu can be orders of magnitude greater than the other two lakes, given its 3000 times larger size, two to four orders of magnitude higher content of dissolved CO2, containing substantial quantities of CH4 in addition to CO2 in solution, and holding a far denser population living in its much wider catchment area. The present study aims to assess the probability of a future gas outburst in this giant lake by numerical modeling of its hydrodynamics over the next half a millennium. The turbulent transport is calculated using the extended k-ε model. An implicit Euler method is applied to solve the governing partial differential equations on a vertically staggered grid system, discretized using a finite-volume approach. Since the previously calibrated model successfully reproduces the measured lake profiles, the same tuned parameter values are used in this study, assuming a stable steady-state condition in the future. The results of our simulations effectively address common concerns regarding the risk of a gas burst in the lake due to buoyancy instability-triggered overturn and/or supersaturation of the water column.

BackgroundDeformable image registration (DIR) is a crucial tool in radiotherapy for analyzing anatomical changes and motion patterns. Current DIR implementations rely on discrete volumetric motion representation, which often leads to compromised accuracy and uncertainty when handling significant anatomical changes and sliding boundaries. This limitation affects the reliability of subsequent contour propagation and dose accumulation procedures, particularly in regions with complex anatomical interfaces such as the lung‐chest wall boundary.PurposeGiven that organ motion is inherently a continuous process in both space and time, we aimed to develop a model that preserves these fundamental properties. Drawing inspiration from fluid mechanics, we propose a novel approach using implicit neural representation (INR) for continuous modeling of patient anatomical motion. This approach ensures spatial and temporal continuity while effectively unifying Eulerian and Lagrangian specifications to enable natural continuous motion modeling and frame interpolation. The integration of these specifications provides a more comprehensive understanding of anatomical deformation patterns.MethodsWe propose an INR‐based approach modeling motion continuously in both space and time, named continues‐sPatial‐temporal deformable image registration (CPT‐DIR). This method fits a multilayer perception network to map the 3D coordinate (x,y,z), to its corresponding velocity vector (vx,vy,vz). Displacement vectors (▵x,▵y,▵z) are then calculated by integrating velocity vectors over time using an Euler method numerical scheme. The above spatial and temporal continuous motion design also enables continuous frame interpolation (CPT‐Interp). The DIR's and interpolation's performance were tested on the DIR‐Lab dataset and the Abdominal‐DIR‐QA dataset, using metrics of landmark accuracy (target registration error), contour conformity (Dice), and image similarity (mean absolute error).ResultsCPT‐DIR clearly reduced landmark TRE from 2.79±1.88 to 0.99±1.07 mm over DIRLab and from 8.61±7.92 to 4.79±6.28 mm over the challenging Abdominal‐DIR‐QA dataset, surpassing B‐spline results across all cases. The whole‐body region MAE improved from 35.46±46.99 to 28.99±32.70 HU for DIRLab, and from 37.32±18.69 to 20.65±16.39 HU for Abdominal‐DIR‐QA. In the challenging sliding boundary region, CPT‐DIR demonstrated superior performance compared to B‐spline, reducing ribcage MAE from 75.40±86.70 HU (unregistered) to 42.04±45.60 HU and improving Dice coefficients from 89.30% to 90.56%. The training‐free CPT‐Interp method enhanced previous deep learning‐based approaches, improving upon UVI‐Net with reduced MAE (17.88±3.79 vs. 18.93±3.90) and increased peak signal‐to‐noise ratio (PSNR) (40.26±1.58 vs. 39.76±1.48), while eliminating training dataset dependencies. Both CPT‐DIR and CPT‐Interp achieved substantial computational efficiency, completing operations in under 3 s compared to several minutes required by conventional B‐spline methods.ConclusionBy leveraging the continuous representations, the CPT‐DIR method enhances registration and interpolation accuracy, automation, and speed. The method achieves high accuracy on intra‐fractional thoracic datasets and demonstrates improved performance over conventional methods in more challenging inter‐fractional abdominal registration scenarios, highlighting its potential for robust applications in radiotherapy. The improved efficiency and accuracy of CPT‐DIR make it particularly suitable for real‐time adaptive radiotherapy applications.

We study the enlarged numerical dynamics of time-space discrete p-Laplace equations on the Banach space l p for p>2, where we use the word ‘enlarged’ because l p contains l 2 . We prove the existence and uniqueness of an enlarged numerical attractor as well as numerical l p -solutions for the implicit Euler scheme when the time-size belongs to an existing interval. By establishing an enlarged Taylor formula and estimating the interpolation errors, we prove the upper semicontinuity of enlarged numerical attractors towards to the enlarged global attractor of the original lattice system as the time-size tends to zero. We further show that the lower semicontinuity set of enlarged numerical attractors is an IOD-type (Intersection of countably many Open Dense sets) in the existing interval, moreover, this lower semicontinuity set is dense with the same cardinality as the phase space.

ABSTRACT This study investigates the Pennes bioheat transfer model within the framework of the Virtual Element Method, presenting for the first time its application as a computational approach to analyze the thermal behavior of biological tissues. The virtual element method, conceived as a natural extension of the finite element approach, is particularly advantageous for handling any type of complex meshes within the framework of Galerkin approximation. To ensure the robustness of this discretization technique, we establish the theoretical foundations of the model, including the derivation of optimal convergence rates, error estimates in the ‐norm, and proofs of existence and uniqueness of the discrete solution. A numerical investigation of the proposed scheme is conducted by applying it to the generalized bioheat transfer equation, discretized in time using Euler's method. A comparative analysis against the classical finite element method is carried out to rigorously evaluate the effectiveness of the proposed virtual element scheme. The results underscore the computational advantages of the virtual element method, demonstrating its ability to preserve accuracy and stability over complex mesh structures while providing deeper insights into the thermo‐regulatory mechanisms of biological tissues.

Binary alloy solidification involves the transition of a liquid mixture of two metals into a solid phase and presents several complex challenges that researchers aim to address. These problems can be categorized into issues related to thermodynamics, diffusion, and macro- and microstructural evolution during the cooling process. The Sivashinsky equation is a fourth-order nonlinear partial differential equation that arises in the mathematical modeling of binary alloy solidification problems. In this article, we apply the Fourier spectral method combined with the Euler method to numerically solve the 2D Sivashinsky equation with periodic boundary conditions. A numerical study of the Sivashinsky equation is important because its analytical solution does not exist, except for trivial solutions. The error estimation of the approximate solution is provided. Furthermore, we show, both theoretically and numerically, that the proposed method preserves the decreasing mass condition of the obtained numerical solutions. Finally, to validate the theoretical results, three examples with different initial conditions are investigated.

This paper addresses the trajectory tracking problem for a six-degree-of-freedom (6-DOF) underwater manipulator subject to complex disturbances and input saturation. It proposes a fixed-time preset performance sliding mode control method considering input saturation (FT-PP-SMC-IS), aiming to achieve rapid and stable tracking performance under these constraints. Firstly, to improve modeling accuracy, the Newton–Euler method and Morison’s equation are integrated to establish a more precise dynamic model of the underwater manipulator. Secondly, to balance dynamic and steady-state performance, a preset performance function is designed to constrain the tracking error boundaries. Based on dual-limit homogeneous theory, a fixed-time sliding mode surface is constructed, significantly enhancing the convergence speed and fixed-time stability. Furthermore, to suppress the effects of input saturation, a fixed-time auxiliary system is designed to compensate in real-time for deviations caused by actuator saturation. By separately constructing the sliding mode reaching law and equivalent control law, global fixed-time convergence of the system states is ensured. Based on Lyapunov stability theory, the fixed-time stability of the closed-loop system is rigorously proven. Finally, comparative simulation experiments verify the effectiveness and superiority of the proposed method.

Based on the modified Hamilton–Ostrogradsky principle, a mathematical model of a distributed-parameter high-voltage HVAC line that includes lightning shield wires is proposed. A partial differential equation of a five-wire power line is produced as a result. Therefore, a methodology for looking for boundary conditions of a long line equation in the five-wire version is proposed here. A mathematical model is introduced as an example of a section of a power line that consists of a high-voltage long line that includes shield wires operating in an equivalent concentrated-parameter power system presented in its circuit version. The system is described with both partial and ordinary derivative differential equations. Poincaré boundary conditions of the third type are applied to solve the state equations of the object discussed. A discrete line model is thus presented, described with ordinary differential equations based on the well-known straight-line method. Transient processes across the system are analysed exactly at the moment of a lightning strike against a shield wire in the middle section of the line. To this end, a mathematical lightning strike model is developed by means of cubic spline interpolation. The original system of differential equations is integrated into the implicit Euler method, considering the Seidel method. The end results of the computer simulation are presented graphically and analysed. The results show the effectiveness of the proposed method of analysing transients across ultra-high-voltage lines that include lightning protection wires and can serve as accurate calculations of power supply lightning protection at the stages of design and production.

Symplectic numerical methods have become a widely-used choice for the accurate simulation of Hamiltonian systems in various fields, including celestial mechanics, molecular dynamics and robotics. Even though their characteristics are well-understood mathematically, relatively little attention has been paid in general to the practical aspect of how the choice of coordinates affects the accuracy of the numerical results, even though the consequences can be computationally significant.The present article aims to fill this gap by giving a systematic overview of how coordinate transformations can influence the results of simulations performed using symplectic methods. We give a derivation for the non-invariance of the modified Hamiltonian of symplectic methods under coordinate transformations, as well as a sufficient condition for the non-preservation of a first integral corresponding to a cyclic coordinate for the symplectic Euler method. We also consider the possibility of finding order-compensating coordinate transformations that improve the order of accuracy of a numerical method. Various numerical examples are presented throughout.

The development of derivative instruments in modern financial markets has created a growing need for option pricing methods that are both accurate and easy to implement. This study aims to calculate the price of European call options using the Black-Scholes model through a semidiscretization numerical approach. The method used involves time transformation, discretization of space variables, and explicit Euler scheme iteration to obtain numerical solutions. This method is applied to real stock price data, and the numerical results are compared with the Black-Scholes analytical solution at various grid numbers. The results show that accuracy increases with the number of grids, and the relative error is very small when M is large enough, so that this method is capable of producing a numerical approximation that is consistent with the analytical solution. These findings also confirm the trade-off between efficiency and accuracy,but still show that semidiscretization can be a practical, fast, and flexible alternative when analytical solutions are difficult to use or when parameter changes need to be evaluated dynamically. This research contributes by showing that a simple numerical approach can still work effectively in real market conditions, making it a practical and efficient alternative for analysts who need fast and flexible calculations without the complexity of advanced numerical methods.

The paper considered a moving mesh finite element method (MMFEM) in solving malaria transmission model that incorporates control strategies. The SEIR-styled reaction-diffusion governing PDE equations were cast into weak formulations to track the features of interest, specifically the disease dynamics. The numerical method involves discretising the spatial domain into finite elements (FEM) guided by appropriate basis function and time discretisation using the implicit (backward Euler) method which ensures unconditionally stable requirements and achieved through the geometric conservation approach via a suitable monitor function (M) and equidistributed so that the integral between M consecutive nodes are equal. The theoretical model was numerically validated to confirm the practical utility of the MMFEM numerical scheme under varying epidemiological scenarios. The validation confirms that the MMFEM approach provides a reliable, accurate, and stable framework for modelling vector-borne disease dynamics. A further research could explore the utility of the Physics Informed Neural Networks (PINNs) technique in solving the reaction-diffusion malaria transmission model which was not considered in this study.

The HIV/AIDS epidemic remains a major public health issue, particularly in regions such as West Nusa Tenggara, Indonesia, where the number of reported cases continues to rise. To better understand and predict the spread of the disease, mathematical modeling provides a valuable analytical tool. This study employs the SIA (Susceptible-Infected-AIDS) compartmental model to describe the dynamics of HIV/AIDS transmission within the population. The model incorporates a latent phase, distinguishing it from simpler models and making it more suitable for diseases with long incubation periods such as HIV/AIDS. Two numerical methods, that is Euler and the 4th Order Runge–Kutta (RK4), are applied to solve the system of nonlinear differential equations derived from the model. Using official epidemiological data from 2023 in West Nusa Tenggara, the simulation tracks the evolution of the population across the three compartments over a 12-month period. The results indicate a consistent decline in the number of susceptible individuals and an increase in both infected and AIDS-diagnosed individuals. The basic reproduction number, , suggesting that the disease is endemic in the region. Stability analysis further confirms that the disease-free equilibrium is unstable, while the endemic equilibrium is locally asymptotically stable. These findings highlight the urgency of targeted interventions and demonstrate the importance of mathematical models in guiding public health strategies for disease control.

As a multi-switching power electronic circuit with complex variable topology, the three-level active neutral point clamped (ANPC) converter is a complex system with strong coupling and low linearity. It has numerous high-speed switching devices, a large number of switch states, and a high matrix dimension. Modeling each switch will undoubtedly further increase the circuit size. While in real-time simulation, updating all states of the model to produce outputs within a single time step results in a significant computational load, causing an increasing consumption of FPGA hardware resources as the number of switches and circuit size grow. In order to solve this problem, the current common practice is to decompose the entire complex power electronic system into smaller serial subsystems for modeling. The overall modeling approach for small circuits can be achieved, but when the size of the circuit increases, the overall modeling complexity and difficulty are increased or even impossible to achieve. Decoupling power electronic circuits with this decomposition into subsystem modeling not only reduces the matrix dimension and simplifies the modeling process, but also improves the computational efficiency of the real-time simulator. However, this inevitably generates simulation delays between different subsystems, leading to numerical oscillations. In an effort to overcome this challenge, this paper adopts the method of parallel computation after subsystem partitioning. There is no one-beat delay between different subsystems, and there is no loss of accuracy, which can improve the numerical stability of the modeling and can effectively reduce the step length of real-time simulation and alleviate the problem of real-time simulation resource consumption. In addition, to address the problems of low accuracy due to the traditional forward Euler method as a solver and the possibility of significant errors at some moments, this paper uses a modified prediction correction method to solve the discrete mathematical model, which provides higher accuracy as well as higher stability. And, different from the traditional control method, this paper uses an improved FCS-MPC strategy to control the switching transients of the ANPC model, which achieves a very good control effect. Finally, a simulation step size of less than 60 ns is successfully realized by empirical demonstration on the Speedgoat test platform. Meanwhile, the accuracy of our model can be objectively evaluated by comparing it with the simulation results of the Matlab Simpower system.

A nonstationary mixed hemivariational inequality is studied for an incompressible fluid flow described by the Stokes equations subject to a nonsmooth boundary condition of friction type described by the Clarke subdifferential. The solution existence is shown through a limiting procedure based on temporally semi-discrete approximations. Uniqueness of the solution and its continuous dependence on data are also established. Fully discrete numerical methods are introduced to solve the nonstationary mixed hemivariational inequality. The backward Euler scheme is applied to discretize the time derivative, and mixed finite element methods are used for the spatial discretization. An error bound is derived for the numerical solution of the unknown velocity. Numerical results are reported on computer simulations of some examples.

We give a comprehensive description of Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals $\mathcal F_\nu \coloneqq \text{MMD}_K^2(\cdot, \nu)$ towards given target measures $\nu$ on the real line, where we focus on the negative distance kernel $K(x,y) \coloneqq -|x-y|$. In one dimension, the Wasserstein-2 space can be isometrically embedded into the cone $\mathcal C(0,1) \subset L_2(0,1)$ of quantile functions leading to a characterization of Wasserstein gradient flows via the solution of an associated Cauchy problem on $L_2(0,1)$. Based on the construction of an appropriate counterpart of $\mathcal F_\nu$ on $L_2(0,1)$ and its subdifferential, we provide a solution of the Cauchy problem. For discrete target measures $\nu$, this results in a piecewise linear solution formula. We prove invariance and smoothing properties of the flow on subsets of $\mathcal C(0,1)$. For certain $\mathcal F_\nu$-flows this implies that initial point measures instantly become absolutely continuous, and stay so over time. Finally, we illustrate the behavior of the flow by various numerical examples using an implicit Euler scheme, which is easily computable by a bisection algorithm. For continuous targets $\nu$, also the explicit Euler scheme can be employed, although with limited convergence guarantees.