Linear high order finite difference methods with essentially non-oscillatory limiters for hyperbolic conservation laws

A variational mimetic finite difference method for elliptic interface problems on non-matching polytopal meshes with geometric interface inconsistencies

Dynamic response analysis of mooring cables for floating offshore wind turbines under ocean currents using a perturbation-based finite difference method

Valuing operational flexibility in hybrid energy systems: A neural solution to the Hamilton–jacobi–bellman equation

The thermal management of microfluidic devices and renewable energy systems relies heavily on the efficient transport of non-Newtonian fluids. Specifically, these electro-kinetic and transport mechanisms are directly applied in the real-world design of electro-kinetic micropumps and biomedical lab-on-a-chip devices. Fractional calculus extends classical differentiation to non-integer orders, enabling realistic modeling of systems with memory and spatial nonlocality. This study examines the unsteady electro-osmotic flow of a Walter-B viscoelastic fluid past a semi-infinite vertical plate embedded in a Darcy porous medium under a transverse magnetic field. The model incorporates thermal radiation, internal heat generation, Soret-Dufour cross-diffusion, and a first-order chemical reaction, while Caputo fractional derivatives are used to represent memory-dependent heat and mass diffusion mechanism. The governing equations are non-dimensionalized and solved numerically using a fully implicit finite-difference scheme based on the second-order fractional backward-difference formula (FBDF2), ensuring stability and accuracy. To the best of the authors’ knowledge, this is the first study that combines fractional electro-osmotic Walter-B flow with simultaneous Soret–Dufour effects and chemical reaction within an FBDF2-based fully implicit finite-difference framework. The results reveal that smaller fractional orders intensify memory effects and delay thermal and solutal relaxation, reflecting the inherent nonlocality of fractional transport. Quantitatively, the Soret number enhances the heat transfer rate by approximately 5.38% while reducing the mass transfer rate by about 11.02%. The radiation parameter markedly improves thermal transport, producing nearly 33.36% enhancement in the Nusselt number. In contrast, the thermal Grashof number slightly reduces the skin-friction coefficient by 3.20%, whereas the electro-osmotic parameter decreases it by 7.95%. Moreover, the Dufour number yields a modest 2.10% increase in the Sherwood number.

The explicit finite-difference scheme is widely used in seismic wave simulation. Using large time steps can significantly reduce the number of iterations and thus improve computational efficiency, but this approach faces two main challenges: reduced accuracy caused by temporal dispersion and restrictions imposed by the Courant-Friedrichs-Lewy (CFL) stability condition. Based on the wavefield iteration equation in matrix form, an explicit finite-difference method with time step n-tupling and an extended CFL limit is developed for acoustic wave simulation. By combining n successive iteration operators into a single-step operator, two types of n-tupling algorithms are constructed, effectively expanding the CFL stability limit by a factor of n and enabling time steps well beyond conventional thresholds. Both theoretical analysis and numerical experiments demonstrate that simulations with time step n-tupling achieve accuracy equivalent to conventional single-step methods while reducing computation time to approximately 1/n, resulting in a corresponding n-fold increase in overall computational efficiency. Time dispersion is suppressed using two complementary strategies: for relatively small base time steps, n-tupling inherently attains the accuracy of smaller steps; for relatively large base steps, a time-dispersion transform is applied to eliminate errors and maintain high numerical accuracy throughout the simulation.

A reaction–diffusion model coupling bacterial growth, substrate consumption, and biosurfactant production is presented and validated experimentally for microbial enhanced oil recovery. The physical problem addressed is the mobilization of residual oil through the biosurfactant-induced interfacial tension (IFT) reduction. The model employs Monod-type kinetics for biomass growth and a linear production term for biosurfactant formation, coupled to diffusion in a 2D porous domain. Dimensional consistency is verified, and all parameter units are provided. Numerical simulations were conducted using an explicit finite-difference scheme (100 × 100 grid, Δx = Δy = 0.1 cm, Δt = 0.01 h) with Neumann (zero-flux) boundary conditions. A sensitivity analysis with respect to μmax, biosurfactant yield α, and substrate yield YBS identifies these parameters as primary controls of additional oil recovery. Laboratory micromodel experiments using Bacillus subtilis (surfactin producer) and glucose medium (10 g l−1) demonstrate an IFT reduction from 28 mN m−1 to 2.5 mN m−1 over 72 h and an average additional recovery of 18% of the original oil in place (OOIP), in agreement with model predictions 17.5% OOIP, relative error <3%, R2 = 0.96). The novelty of this work lies in the combined analysis of reaction–diffusion induced spatial instabilities (Turing-type patterns) with experimentally measured IFT and quantitative oil recovery within a unified predictive framework. Limitations, including the exclusion of microbial gas production and acidification, are discussed, and directions for extension to heterogeneous reservoir scales are proposed.

We extend the framework of conceptual density functional theory (DFT) to include fourth-order energy derivatives. We present the series of quantum reactivity descriptors at this order for the canonical and the grand canonical ensembles. After introducing a direct computational methodology, we investigate and describe the quartic curvature λ and the third-order Fukui function f(3)(r). These higher-order descriptors capture nonlinear aspects of electronic reactivity that go beyond the conventional concepts of chemical potential, hardness, or Fukui functions. The global descriptor λ, obtained via finite-difference approximations of frontier orbital energies, quantifies the curvature of the chemical hardness and offers insight into the electronic (in)stability of molecular systems under charge perturbations. The local descriptor f(3)(r), derived as the third-order response of the electron density, reveals spatially resolved regions of charge-transfer sensitivity. We apply these descriptors to a series of push-pull organic molecules and open-shell systems, demonstrating their ability to highlight differences in internal charge transfer character and electronic delocalization. The results support the use of fourth-order conceptual DFT tools as chemically meaningful indicators of reactivity beyond the harmonic regime.

Leveraging the latest Sunway supercomputer, we developed a fully optimized earthquake simulation model that accurately captures topographic effects for realistic seismic analysis. Optimizing for the SW26010Pro architecture with DMA/RMA communication mechanisms, data compression schemes, and vectorization, we achieved a speedup exceeding 160×. Our pipeline-based computation and communication overlapping scheme, combined with performance prediction models further minimized computational costs. These optimizations enabled the largest-scale curvilinear grid finite-difference method (CGFDM) earthquake simulations to date, covering 197 trillion grid points and achieving 86.7 PFLOPS on 39 million cores with a weak scaling efficiency of 97.9%. These advancements enabled the successful simulation of the 2008 Wenchuan earthquake, providing high-resolution seismic insights and robust assessments for regional hazard mitigation and disaster preparedness.

We present diffHOD-IA, a fully differentiable implementation of a halo occupation distribution (HOD) model that incorporates galaxy intrinsic alignments (IA). Motivated by the diffHOD framework, we create a new implementation that extends differentiable galaxy population modeling to include orientation-dependent statistics crucial for weak gravitational lensing analyses. Our implementation combines this HOD formulation with an IA model, enabling end-to-end automatic differentiation from HOD and IA parameters through to the galaxy field. We additionally extend this framework to differentiably model two-point correlation functions, including galaxy clustering and IA statistics. We validate diffHOD-IA against the reference halotools-IA implementation using the Bolshoi-Planck simulation, demonstrating excellent agreement across both one-point and two-point statistics. We verify the accuracy of gradients computed via automatic differentiation by comparison with finite-difference estimates for both HOD and IA parameters. We present science use cases leveraging gradients in the simulations to recover the IA parameters of a galaxy field representative of the TNG300 simulation. Finally, we apply diffHOD-IA in a Hamiltonian Monte Carlo analysis and compare its performance with halotools-IA and a neural-network-based emulator, IAEmu. Unlike emulator-based approaches for statistics, diffHOD-IA provides differentiability at the catalog level, enabling integration into field-level inference pipelines and extension to arbitrary summary statistics for next-generation weak-lensing analyses. Our code is publicly available.

This paper presents a numerical method for reconstructing the spatial distribution of sound speed in inhomogeneous media based on the inverse analysis of acoustic wave propagation. The mathematical model relies on the second-order wave equation with variable coefficients. The inverse problem is formulated as an optimization task to minimize the residual functional between simulated and observed pressure data at the domain boundaries. To efficiently calculate the gradient of the functional, an adjoint (auxiliary) problem method is employed, derived via variational calculus. The numerical implementation is performed using an explicit finite-difference scheme. Computational experiments on a one-dimensional model of a heterogeneous medium (soil-metal-soil) demonstrate that the proposed algorithm allows for reliable reconstruction of the velocity profile, particularly in zones of sharp contrast. The study analyzes the sensitivity of the solution and the convergence rate, showing that 500 iterations provide an optimal balance between accuracy and computational cost.

A New Approach of Adding Nanoparticles to Micro-Encapsulated Phase Change Material-Water Slurry for Enhancing Melting and Heat Transfer in Buoyancy Driven Enclosure Using Eulerian-Eulerian Model

Bilinear partial differential equations form an important class of nonlinear distributed systems in which the control interacts multiplicatively with the system state. Such models arise in many physical and engineering applications and present significant analytical and computational challenges due to their nonlinear coupling. In this work, we investigate an optimal control problem for a wave system governed by a second-order hyperbolic equation. The objective functional aims to minimise the discrepancy between the spatial gradient of the state and a prescribed target profile while penalising the control energy. We establish the existence of an optimal control and derive first-order necessary optimality conditions via an adjoint system, leading to an explicit characterisation of optimal control in both bounded and unbounded admissible sets. To validate the theoretical results and demonstrate the effectiveness of the proposed method in accurately tracking gradients, a finite-difference numerical scheme, combined with a forward-backward iterative algorithm, was developed.

This article examines the transverse vibration of a rectangular plate structure, with all sides fixed, under normal loads applied to its surface using bimoment plate theory. The problem was solved using the finite difference method. Numerical results for calculating the normal displacements and stresses are presented. Fig.: 2. Tables: 5.

This article introduces a semianalytical method for pricing American options on assets (stocks, ETFs) that pay discrete or continuous dividends. The problem is notoriously complex because discrete dividends create abrupt price drops and affect the optimal exercise timing, making traditional continuous dividend models unsuitable. Our approach uses the Generalized Integral Transform (GIT) method introduced by the author and his co-authors in several previous publications, which transforms the pricing problem from a complex partial differential equation with a free boundary into an integral Volterra equation of the second or first kind. In this article, we illustrate this approach by considering a popular GBM model that accounts for discrete cash and proportional dividends using Dirac delta functions. By reframing the problem as an integral equation, we can sequentially solve for the option price and the early exercise boundary, effectively handling the discontinuities caused by the dividends. Our methodology provides a powerful alternative to standard numerical techniques such as binomial trees or finite difference methods, which can struggle with the jump conditions of discrete dividends by losing accuracy or performance. Several examples demonstrate that the GIT method is highly accurate and computationally efficient, avoiding the need for extensive computational grids or complex backward induction steps.

Accurate prediction of the thermal ablation zone in hepatic radiofrequency ablation (RFA) is critical for preventing the recurrence of local tumor, yet it is complicated by the convective heat sink effect of blood perfusion.Traditional numerical solvers, such as the finite difference method (FDM), are inherently limited by time-step constraints which require greater computational cost and impede real-time clinical applications.This study proposed a mesh-free Physics-Informed Neural Network (PINN) framework to simulate the spatiotemporal dynamics of Pennes bioheat equation.By embedding the governing partial differential equation (PDE) directly into the loss function of the neural network, the model learnt the continuous temperature field without spatial discretization or labeled training data.A comparative analysis against an explicit FDM baseline yielded a relative L2 error norm of 1.9%.Although PINN's continuous functional approximation slightly dampened the theoretical singularity at the tip of the electrode, it accurately resolved the critical 50 C isotherm that defined the boundary of irreversible coagulative necrosis.Furthermore, the framework effectively decoupled computational cost from the time of physical simulation.While offline training required approximately 6 minutes, the optimized network executed online inference in milliseconds.This capability to provide physically consistent and near-instantaneous thermal predictions demonstrates the potential of the PINN framework for intraoperative decision support systems.

Noise pollution has become a significant issue in modern society, with long-term exposure to high-volume environments considerably increasing the risk of hearing damage. Based on Coupled-Mode Theory, this paper proposes an active noise cancellation (ANC) headphone system that integrates acousto-electric coupling modeling and artificial intelligence-based voice enhancement.By establishing a coupled model of acoustic wave propagation and electronic signal processing, and employing the Finite Difference Method and Runge—Kutta algorithm for system simulation, precise noise cancellation is achieved. In terms of hardware, the system uses an ESP32-S3 as the main control unit, integrated with feedforward and feedback microphone structures and an A-29P intelligent voice processing module to achieve active suppression of ambient noise and real-time extraction of human voice. Experimental results demonstrate that the system effectively reduces the risk of hearing impairment even in high-noise environments and significantly improves speech communication clarity.

This article deals with the analysis of the deformed shape of local stability loss of a reinforced flexible beam, occurring due to constrained expansion during heating. The multiple elastic supports of the beam modeled as an elastic medium, which provides constant resistance to both longitudinal and transverse displacements of the rod. The infinitely long beam divided into a buckling region and an adjacent region under compression. The lengths of these regions are unknown and should be determined during the solution process. A part of the potential energy accumulated during compression is expended on the work of internal forces during bending deformation following the loss of stability. This leads to a reduction in the magnitude of the compressive force in the buckled region. The problem of determining the displacement functions and the critical value of the safe heating temperature is formulated as a system of nonlinear differential equations concerning the deformations in the regions of the flexible beam. The solution obtained using the finite difference method, which transforms a system of differential equations into a system of linear algebraic equations. This system takes the closed form with boundary conditions and transversality conditions. A sufficient number of grid nodes for constructing the difference scheme determined through an iterative procedure that compares two adjacent solutions. The criterion for comparison of solutions is a tuple of areas under the graphs of the sought functions, which are calculated through numerical integration using the trapezoidal rule. The obtained final solution compared with the classical solution to the stability problem of an evenly loaded beam, which does not take into account longitudinal displacements. Additionally, it is contrasted with the known solution in the field of operation of continuously welded railway tracks, which also disregards resistance to longitudinal displacements in the buckled region. The refined results obtained by the proposed modified method for calculation of the parameters of the deformed shape of a flexible beam is important for monitoring the pre-critical state of the modelling system.

ABSTRACT Carbon fiber reinforced polymer (CFRP) encased coal pillar as an innovative support system has attracted growing interest for enhancing stability in underground coal mines. Although CFRP confinement of cylindrical coal samples has been widely studied, the mechanical behavior of square coal pillars commonly found in room‐and‐pillar goafs remains poorly understood and differs significantly. Accordingly, uniaxial compression tests were conducted on CFRP‐encased square coal pillars (CSCP) with 0–4 layers, and a coupled discrete element method–finite difference method (DEM–FDM) numerical model was developed to quantitatively and qualitatively analyze their macroscopic and meso‐mechanical behaviors. The results show that CFRP confinement markedly altered the stress–strain behavior of square coal pillars, inducing double‐peaked curves with post‐peak strengthening behavior that became more pronounced with increasing layers. CFRP wrapping transformed the failure mode from brittle to ductile, with 3‐layer confinement achieving a peak ductility 4.48 times that of unconfined square coal pillars (USCP). Compared to the intense acoustic emission (AE) burst during post‐peak failure in USCP, CSCP exhibited sustained activity before and after the first peak, yet remained stable and gradual during the stress‐maintenance and strengthening stages. This study offers guidance for stable and sustainable mining of residual coal pillars in abandoned mines.

As the boundary between indoor and outdoor spaces, the heat flux of a building envelope is a crucial factor influencing the indoor thermal environment and human thermal comfort, and also an important indicator reflecting the impact of outdoor meteorological factors on the indoor environment. In scenarios involving rapid assessment of existing buildings and engineering projects, the dynamic thermal performance of the building envelope are often affected by factors such as outdoor weather fluctuations, window–wall coupling, wall heat storage, and thermal bridging. To address this issue, this study proposes a dynamic heat flux calculation method that accounts for hysteresis. Simultaneously, the heat conduction equation of the implicit finite difference method (IFDM) and boundary conditions based on wall energy balance are used to optimize the wall surface temperature. An adaptive step size control strategy (Runge–Kutta–Fehlberg) is introduced in the time step setting. Results show that the heat flux R2 of the proposed dynamic heat flux calculation method is 0.9207, and the optimized R2 is 0.9435, both within an acceptable range for engineering applications. Studies have shown that the simplified framework derived from the heat flux analysis of building envelopes retains the characteristics of wall heat storage and delayed heat release, while effectively solving the window–wall coupling problem and significantly reducing the reliance on computationally expensive numerical methods. This method therefore provides an efficient and scalable technical pathway for thermal performance assessment and energy-retrofit decision support for existing building envelopes.