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.

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.

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.

Stability analysis of the explicit Euler, backward Euler and Crank-Nicholson finite differences methods for solving the generalised Burgers equation using Fourier (von-Neumann) method is studied. The results demonstrated that the forward Euler scheme is conditionally stable, while Crank-Nicholson and implicit schemes are unconditionally stable.

Abstract We consider optimal interpolation of functions analytic in simply connected domains in the complex plane. By choosing a specific structure for the approximant, we show that the resulting first-order optimality conditions can be interpreted as optimal $$\varvec{\mathcal {H}}_{\varvec{2}}$$ H 2 interpolation conditions for discrete-time dynamical systems. Connections to model reduction of discrete-time time-invariant delay systems are also established with particular emphasis on discretized linear systems obtained through the implicit Euler method, the midpoint method, and backward differentiation methods. A data-driven algorithm is developed to compute a (locally) optimal approximant. Our method is tested on three numerical experiments.

ABSTRACT We propose and analyze the numerical approximation for a viscoelastic Euler‐Bernoulli beam model containing a nonlinear strong damping coefficient. The finite difference method is used for spatial discretization, while the backward Euler method and the averaged PI rule are applied for temporal discretization. The stability and error estimate of the numerical solutions are derived for both the semi‐discrete‐in‐space scheme and the fully discrete scheme by the energy argument. Furthermore, the Leray‐Schauder theorem is used to derive the existence and uniqueness of the fully discrete numerical solutions. Finally, the numerical results verify the theoretical analysis.

We consider the numerical simulation of blood flows in a patient-specific kidney including the renal artery, the renal vein, and the kidney tissue using a coupled system of unsteady Stokes-Darcy equations. The Stokes equations and the Darcy equations are implicitly coupled on the interfaces by enforcing three conditions, namely the conservation of mass, the balance of the normal force and the Beavers-Joseph-Saffman condition. To discretize the system we introduce a stabilized P1-P1-P1 finite element method for the spatial variables and an implicit backward Euler method for the temporal variable. A mathematical theory is developed to guarantee the stability and the convergence of the proposed discretization method. To efficiently solve the large, sparse and highly ill-conditioned algebraic systems, we further propose a Krylov subspace method preconditioned by a robust two-scale additive Schwarz method consisting of a mixed-dimensional coarse preconditioner with a 1D central-line preconditioner in the vascular region and a 3D preconditioner for the kidney tissue with some compatibility conditions imposed on the 1D and 3D interfaces. Some numerical experiments for a benchmark problem and a patient-specific kidney with physiologic parameters are presented to verify the accuracy, the robustness, and the effectiveness of the proposed method.

Abstract The propagation of charged particles through a scattering medium in the presence of a magnetic field can be described by a Fokker–Planck equation with Lorentz force. This model is studied both from a theoretical and a numerical point of view. A particular trace estimate is derived for the relevant function spaces to clarify the meaning of boundary values. Existence of a weak solution is then proven by the Rothe method. In the second step of our investigations, a fully practical discretization scheme is proposed based on an implicit Euler method for the energy variable and a spherical-harmonics finite-element discretization with respect to the remaining variables. A complete error analysis of the resulting scheme is given and numerical tests are presented to illustrate the theoretical results and the performance of the proposed method.

By virtue of the novel technique, this paper focuses on the mean-square convergence of the backward Euler method (BEM) for stochastic differential delay equations (SDDEs) without using the moment boundedness of numerical solutions. The convergence rate for SDDE whose drift and diffusion coefficients can both grow polynomially is shown. Furthermore, under fairly general conditions, the novel technique is applied to prove that the BEM can inherit the exponential mean-square stability with a simple proof. At last, some numerical experiments are implemented to illustrate the reliability of the theories.

ABSTRACT In this paper, the ‐Galerkin mixed finite element method (MFEM) is used to solve the time‐fraction‐order damped beam vibration equations with simple support at both ends. Compared with the standard finite element method (FEM), the ‐Galerkin MFEM can calculate the deflection, bending moment and other parameters of the beam more directly, so it is more suitable for solving the beam vibration equations, which are high‐order partial differential equations. The Caputo fractional derivative is approximated using the L1 formula, and a fully discrete numerical scheme is established by linear backward Euler method. The stability of the proposed scheme, as well as the existence, uniqueness, and convergence of the numerical solution, are rigorously analyzed. Numerical examples are provided to verify the theoretical results, with real‐world beam data used to investigate how variations in the damping coefficient and the order of the fractional derivative affect the beam's vibrational behavior. Numerical simulations demonstrate that the amplitude of the beam decays more rapidly with larger damping coefficients. Moreover, as the order of the fractional derivative increases, the decay rate of the beam vibration first increases and then decreases, while the peak of the curve gradually shifts to the right.

This study analyzes the efficiency of Repeated Richardson Extrapolation (RRE) as an alternative for reducing and estimating discretization error ( E h ) in the numerical resolution of the one- and two-dimensional two-phase flow problem in porous media. The numerical solution was obtained using the finite volume method (FVM) in space and the implicit Euler method for time discretization. For linearization, we employed the modified Picard method, and to solve the linear equations system, we used the Gauss-Seidel solver coupled with the multigrid method to accelerate the convergence of the iterative process. The variables of interest analyzed were the wetting and non-wetting pressures located at the central point of the domain. The results indicate that the employed methodology was suitable, with an increase in the accuracy level of numerical solutions from 10 − 3 to 10 − 14 , and additionally, accurate estimates for E h were obtained.

This study presents a unified constitutive model capable of simulating monotonic behaviour of clay and sand, incorporating a non-associated flow rule and critical state concept. The bounding surface approach has been used to anticipate a smooth transition from the elastic phase to the plastic phase of the soil. A novel dilatancy relationship is introduced to represent the volumetric behaviour of both sand and clay in a unified way. The model is implemented using the implicit Euler method. To capture anisotropic soil behaviour, the model is extended within a multilaminate framework comprising 13 elastic-plastic planes. The overall response is derived by integrating the behaviour of these individual planes, each governed by unconventional constitutive equations. The model realistically reproduces strain softening and induced anisotropy through a non-classical plasticity approach. Simulations of six soil samples under monotonic drained and undrained loading show good agreement with experimental results, demonstrating the model’s effectiveness.

In this paper, C1-conforming element methods are analyzed for the stream function formulation of a single layer non-stationary quasi-geostrophic equation in the ocean circulation model. In its first part, some new regularity results are derived, which show an exponential decay property when the wind shear stress is zero or exponentially decaying. Moreover, when the wind shear stress is independent of time, the existence of an attractor is established. In its second part, finite element methods are applied in the spatial direction, and for the resulting semi-discrete scheme, the exponential decay property and the existence of a discrete attractor are proved. By introducing an intermediate solution of a discrete linearized problem, optimal error estimates are derived both for smooth and non-smooth solutions. Based on the backward Euler method, a completely discrete scheme is obtained and uniform in time a priori estimates are established. Moreover, the existence of a discrete solution is proved by appealing to a variant of the Brouwer fixed point theorem, and then, an optimal error estimate is derived. Finally, several computational experiments with benchmark problems are conducted to confirm our theoretical findings.

This paper addresses the numerical approximations of solutions for one dimensional parabolic singularly perturbed problems of reaction-diffusion type. The solution of this class of problems exhibit boundary layers on both sides of the domain. The proposed numerical method involves combining the backward Euler method on a uniform mesh for temporal discretization and an upwind finite difference scheme for spatial discretization on a modified graded mesh. The numerical solutions presented here are calculated using a modified graded mesh and the error bounds are rigorously assessed within the discrete maximum norm. The primary focus of this study is to underscore the crucial importance of utilizing a modified graded mesh to enhance the order of convergence in numerical solutions. The method demonstrates uniform convergence, with first-order accuracy in time and nearly second-order accuracy in space concerning the perturbation parameter. Theoretical findings are supported by numerical results presented in the paper.

This study investigates the numerical modeling of two-dimensional anisotropic diffusion processes involving a spatially localized and temporally limited energy or thermal source. The governing model is formulated as a parabolic partial differential equation, discretized in space using the Finite Element Method (FEM) with linear triangular elements, and in time using both explicit and implicit Euler integration schemes. To ensure spatial accuracy, a dense mesh configuration is employed, which has been shown to produce smooth and representative solution distributions. Simulation results demonstrate that the implicit Euler method exhibits superior numerical stability across various time step sizes, whereas the explicit method requires significantly smaller time steps to remain stable. Analysis of the transient regime reveals that the numerical solution gradually converges toward a steady-state configuration once the source is deactivated. These findings confirm that the combination of FEM with implicit time integration and dense meshing is effective in capturing the spatiotemporal dynamics of anisotropic diffusion processes with localized sources, a phenomenon relevant to thermal analysis, anisotropic materials, and environmental modeling.

This paper is concerned with the asymptotical behavior of the impulsive linearly implicit Euler method for the SIR epidemic model with nonlinear incidence rates and proportional impulsive vaccination. We point out the solution of the impulsive linearly implicit Euler method for the impulsive SIR system is positive for arbitrary step size when the initial values are positive. By applying discrete Floquet’s theorem and small-amplitude perturbation skills, we proved that the disease-free periodic solution of the impulsive system is locally stable. Additionally, in conjunction with the discrete impulsive comparison theorem, we show that the impulsive linearly implicit Euler method maintains the global asymptotical stability of the exact solution of the impulsive system. Two numerical examples are provided to illustrate the correctness of the results.

ABSTRACTIn this paper, we present a stable numerical scheme for solving two‐dimensional ‐component reaction–diffusion systems. The proposed approach utilizes the backward Euler method for temporal discretization and the hybridized discontinuous Galerkin (HDG) method for spatial discretization. We analyze the stability of the proposed HDG method for problems with Dirichlet and Neumann boundary conditions, demonstrating that this method is stable, in the sense of the energy method, under certain mild conditions on the stabilization parameters. Several numerical experiments are provided to validate the proposed scheme, with applications to two reaction–diffusion models: the Brusselator and glycolysis systems. Numerical results confirm that the proposed method achieves the expected optimal convergence rate for both the approximate solutions and their first derivatives. To exhibit relevant physical concepts, we demonstrate the convergence behavior of approximate solutions at the stable equilibrium points of the selected reaction–diffusion system with small diffusion coefficients. Furthermore, the nonconvergence behavior is given at unstable equilibrium points.

Boundary preservation and convergence of the linearly implicit Euler method with truncated Wiener process for the Wright-Fisher model

Background: The aging population is increasing rapidly, with individuals aged 65 and older now representing more than 15% of the global population. This demographic shift is associated with a rising incidence of age-related cardiovascular diseases (CVDs). Early prediction and prevention of cardiovascular aging are essential to improve health outcomes among elderly patients. Objective: This study aimed to develop and externally validate a mathematical model for predicting cardiovascular aging in individuals aged 65 and older, based on general clinical and behavioral data. Methods: The model was built using data from 800 individuals aged 65+ from Almaty, Kazakhstan. Predictors included sex, marital status, education, smoking, alcohol use, disability, physical activity, total cholesterol, hypertension, BMI, coronary artery disease (CAD), myocardial infarction, diabetes mellitus, and chronic heart failure. A system of ordinary differential equations was used to simulate the dynamic interactions of these factors. Numerical integration was performed using the Runge–Kutta, Adams–Bashforth, and backward Euler methods. The model was verified statistically using Pearson correlation analysis and externally validated on independent age cohorts. In addition, we applied k-means clustering to identify hidden patterns and risk profiles within the dataset. A Random Forest classifier was trained to distinguish between high-risk and low-risk individuals using the same feature set. These machine learning approaches were used as complementary tools to enhance the robustness and interpretability of the modeling results. Results: The model trained on the 65–74 age group achieved an external validation accuracy of 98.8% and an AUC of 0.989 when applied to the 75–89 group. Risk modeling showed that in the 65–74 group, smoking and alcohol increased the risk of myocardial infarction, hypertension, and obesity by up to 53%. In the 75–89 group, these factors increased the likelihood of hypertension by 21%, chronic heart failure by 16%, and CAD by 14%. Among individuals aged 90+, hypercholesterolemia increased the risk of chronic heart failure by 17%, while hypertension increased myocardial infarction risk by 16%. Conclusions: The proposed model demonstrated high accuracy in predicting cardiovascular aging and identifying high-risk individuals across elderly subgroups. The integration of clustering and classification methods (k-means and Random Forest) provided additional insights and confirmed the consistency of the findings. This multi-method approach may serve as a valuable tool for developing personalized prevention strategies in geriatric care and improving healthy life expectancy.

In this paper, we propose a linear, fully decoupled and unconditionally energy-stable discontinuous Galerkin (DG) method for solving the tumor growth model, which is derived from the variation of the free energy. The fully discrete scheme is constructed by the scalar auxiliary variable (SAV) for handling the nonlinear term and backward Euler method for the time discretization. We rigorously prove the unconditional energy stability and optimal error estimates of the scheme. Finally, several numerical experiments are performed to verify the energy stability and validity of the proposed scheme.