Preface to the Special Issue on “Computational Methods and Applications for Numerical Analysis, 2nd Edition”

1. Introduction

Computational methods and applications for numerical analysis constitute a fundamental pillar underpinning modern science and engineering, playing an indispensable role in complex physical modeling, engineering simulation, and the solution of interdisciplinary problems. With the increasing complexity of engineering systems and the growing demand for computational accuracy, non-standard governing equations, inverse problems, and multiscale multiphysics coupling pose significant challenges to the stability, accuracy, and efficiency of numerical algorithms.

This Special Issue focuses on recent theoretical advances and practical applications in computational methods and numerical analysis. Following a rigorous peer-review process, fifteen research articles and one review paper were selected for publication from a total of twenty submissions. The contributions cover a broad range of topics, including complex mathematical models and non-standard equations, the development of high-accuracy numerical methods, and multiscale and multiphysics applications in engineering and physical systems, such as elastic wave propagation, soliton dynamics, wind turbines and floating platforms, hydrodynamics, and beamforming in communication systems. Collectively, these works highlight recent progress in the design of robust and efficient computational algorithms for addressing a wide range of challenging and strongly coupled computational problems, underscoring the vital role of computational methods and applications for numerical analysis in advancing both scientific research and engineering practice.

2. A Summary of the Papers

The first paper, authored by Borghi (Contribution 1), develops and rigorously analyzes an efficient computational framework for the representation and evaluation of converging factors associated with the superfactorially divergent Stieltjes series. By exploiting inverse factorial expansions, the proposed approach overcomes the intrinsic limitations of classical Padé approximants when applied to strongly divergent series. Moreover, a novel analytical formulation based on an equivalent Cauchy problem is introduced, providing both theoretical insight into the convergence mechanism and a practical pathway for accurate numerical implementation. Numerical experiments demonstrate that the method achieves high accuracy and robustness in regimes where traditional rational approximation techniques fail, thereby offering a powerful tool for the summation and numerical treatment of highly divergent series arising in applied mathematics and theoretical physics.

Hwang et al. (Contribution 2) propose two novel linear beamforming schemes for cooperative non-orthogonal multiple access systems with full duplex amplify and forward relaying, aiming to effectively mitigate residual self-interference and inter user interference while guaranteeing the convergence of the relay transmission. By jointly considering the coupling between beamformer design and full-duplex operation, the proposed schemes achieve a favorable balance between interference suppression and signal enhancement. Theoretical analysis is complemented by extensive numerical simulations, which evaluate the achievable sum-rate performance under various system configurations. The results demonstrate that the proposed beamforming strategies significantly outperform conventional designs, offering improved spectral efficiency and stable convergence behavior, thereby highlighting their potential for practical implementation in next-generation wireless communication networks.

Chang et al. (Contribution 3) present a 2.5D generalized finite difference method for the efficient and accurate simulation of elastic wave propagation in longitudinally invariant structures. By exploiting the structural invariance along one spatial direction, the proposed approach employs Fourier transforms to reduce the original three-dimensional elastodynamic problem into a series of decoupled, sparse two-dimensional formulations. This dimensional reduction significantly lowers computational cost and memory requirements while preserving the essential three-dimensional wave characteristics. The meshless nature of the method further enhances its flexibility in handling complex geometries and irregular domains without the need for mesh generation. The accuracy, stability, and computational efficiency of the proposed scheme are systematically validated through representative numerical examples, demonstrating its strong potential for large-scale structural wave propagation analysis in engineering applications.

Finch et al. (Contribution 4) develop an artificial neural network-based numerical framework for simulating soliton propagation governed by the Rosenau-KdV-RLW equation on unbounded spatial domains. By constructing neural network approximations that inherently satisfy the governing equation, the proposed approach avoids the introduction of artificial boundary conditions, which are commonly required in traditional numerical methods and may induce spurious reflections. The method is supported by a rigorous convergence analysis that establishes its theoretical reliability, while numerical experiments confirm its high accuracy in capturing the evolution and interaction of solitary waves over long-time integration. These results demonstrate the effectiveness of neural network-based solvers as flexible and robust alternatives for nonlinear dispersive wave problems posed on infinite or semi-infinite domains.

Zhang et al. (Contribution 5) present a high-accuracy, meshless method of fundamental solutions for the numerical analysis of antiplane piezoelectricity problems involving multiple inclusions. To address the sensitivity of method of fundamental solution to source-point placement and the associated numerical singularities, an adaptive leave-one-out cross-validation strategy is introduced to automatically determine optimal source locations. This data-driven regularization mechanism significantly improves numerical stability and robustness. Extensive numerical experiments demonstrate that the proposed method achieves exceptional accuracy in evaluating displacement, stress, and electric field concentrations, even in the presence of perturbed or imperfect boundary conditions. The results highlight the effectiveness of combining meshless fundamental solution techniques with cross-validation-based optimization for reliable simulation of complex piezoelectric composite materials and microstructured systems.

Koleva et al. (Contribution 6) address both the direct and inverse problems associated with a class of pseudoparabolic equations. For the forward formulation, the authors rigorously establish well-posedness and derive a priori estimates for the solution under given initial and boundary conditions. In the inverse setting, the study focuses on the identification of unknown boundary fluxes on a portion of the domain boundary using integral observation data collected on the same segment. By reformulating the inverse problem as an equivalent direct problem with nonclassical integrodifferential boundary conditions, the authors develop a tailored finite difference scheme capable of efficiently and accurately handling such boundary constraints. Numerical experiments validate the stability, convergence, and reconstruction accuracy of the proposed approach, demonstrating its effectiveness for solving inverse pseudoparabolic problems arising in heat transfer and diffusion-related applications.

Hajjia et al. (Contribution 7) investigate the vibrational behavior of aircraft wing structures within the framework of the Mindlin Reissner plate theory, which incorporates the effects of transverse shear deformation and rotary inertia and is therefore well suited for moderately thick wing components. Finite element simulations are employed to systematically analyze the influence of functionally graded material distributions and geometric configurations, including tapered and interpolated wing meshes, on natural frequencies and mode shapes. The numerical results indicate that interpolated geometries exhibit higher natural frequencies due to increased structural stiffness, whereas tapered wings show lower frequencies associated with enhanced flexibility. Validation against ANSYS simulations confirms the accuracy and reliability of the proposed approach, highlighting the combined effects of geometry and material gradation on the vibrational performance of lightweight and efficient aerospace wing designs.

Yang et al. (Contribution 8) advance the theory and methodology of uniform design by deriving explicit construction formulas for high-dimensional experimental spaces subject to arbitrary linear constraints. By extending the inverse Rosenblatt transformation framework, this study enables the efficient generation of robust and uniformly distributed designs over constrained domains such as hyperplanes and hyperspheres embedded within unit hypercubes. Numerical simulations are presented to verify the feasibility and effectiveness of the proposed constructions, demonstrating their applicability to complex experimental and optimization settings involving general linear constraints.

Zhu et al. (Contribution 9) present a two-dimensional structural topology optimization framework that integrates hole appearance constraints to simultaneously regulate hole shape, number, and minimum spacing. By defining the distance between the evolving structure and a prescribed appearance target image as an inequality constraint, the method enables precise control of geometric features during the optimization process. Hole shapes are guided through adaptable equivalent shape templates, minimum inter-hole scales are enforced via a hole shrinkage strategy, and hole numbers are regulated using a dedicated calculation and filling procedure. Embedded within the SIMP interpolation-based topology optimization model, the proposed approach is validated through numerical examples, demonstrating its effectiveness in achieving target structural performance while maintaining geometric consistency.

Zhao et al. (Contribution 10) develop a path-conservative discontinuous Galerkin framework for non-conservative hyperbolic partial differential equations, incorporating the one-stage ADER approach for temporal discretization. By employing a differential transformation procedure in place of the classical Cauchy-Kowalewski expansion, the method avoids the need to solve generalized Riemann problems at cell interfaces. Compared with traditional Runge–Kutta DG schemes, the proposed formulation significantly reduces memory requirements by eliminating intermediate stages, while retaining arbitrary high-order accuracy in both space and time. Numerical experiments on one- and two-dimensional shallow water equations confirm the method’s high resolution for discontinuous solutions and its overall computational efficiency.

Koleva et al. (Contribution 11) address both direct and inverse heat conduction problems in two-dimensional domains composed of disjoint rectangles coupled through Robin-type interface conditions. The well-posedness of the forward and inverse formulations is rigorously established. By exploiting integral observations of the solution and the specific interface conditions, the inverse problem of reconstructing unknown Dirichlet boundary data is reduced to a one-dimensional forward heat equation. The resulting problem is solved using the explicit Saul’yev finite difference scheme, and numerical examples demonstrate the proposed approach’s accuracy and efficiency in handling nonclassical interface conditions and recovering unknown boundary information.

Pan et al. (Contribution 12) propose an intelligent low-consumption optimization strategy for the economic operation of hydropower stations by integrating machine learning techniques. An improved long short-term memory neural network, optimized via an enhanced particle swarm algorithm, is employed to accurately model the flow characteristic curves of hydraulic turbines under varying operating conditions. On this basis, a random forest model, further refined using K-means clustering, is constructed to perform optimal load distribution. A case study of a hydropower station in China demonstrates that the proposed framework significantly improves prediction accuracy, enhances unit operation efficiency, and reduces total water consumption, thereby supporting economically efficient and resource-saving hydropower management.

Chen et al. (Contribution 13) investigate the dynamic response and structural safety of a 10 MW concrete semi-submersible floating wind turbine platform under harsh marine environments through fluid–structure interaction analysis. The study evaluates stress distribution, deformation, safety factors, and fatigue life under combined wind, wave, and current loads. The results reveal that environmental load incident angles have a pronounced influence on surge, sway, pitch, and yaw motions, as well as on stress concentrations. While the maximum stress remains within strength requirements, localized regions exhibit reduced fatigue life, highlighting critical areas susceptible to fatigue damage and providing valuable guidance for the design and long-term stability of floating wind turbine platforms.

Yang et al. (Contribution 14) examine the influence of suction flow control on the hydrodynamic performance of a NACA0009 blunt trailing-edge hydrofoil using the γ-transition turbulence model. Three-dimensional numerical simulations demonstrate that suction control increases the velocity gradient within the boundary layer, delays transition, and suppresses vortex shedding in the wake. By systematically analyzing the effects of suction coefficient and suction slot location, the study shows that appropriately tuned suction parameters significantly reduce wake velocity fluctuations and enhance the lift-to-drag ratio. The results provide practical insights into the effective application of active flow control strategies for improving hydrofoil performance.

Guo et al. (Contribution 15) introduce an efficient two-step numerical scheme for solving large systems of absolute value equations, combining a generalized Newton method as a predictor with a three-point Newton–Cotes formula as a corrector. The convergence properties of the proposed method are rigorously analyzed. As an application, a beam equation is transformed into a system of absolute value equations and solved using the proposed approach. Numerical experiments demonstrate that the method achieves high accuracy and superior computational efficiency compared with existing techniques, making it particularly suitable for large-scale nonlinear systems.

Lastly, Leon et al. (Contribution 16) provide a comprehensive systematic review of mathematical and computational approaches for geomechanically informed rock–blast design, guided by the PRISMA protocol. Ninety-seven representative studies are categorized into high-fidelity finite-element and finite-discrete element simulations, geomechanics-enhanced empirical models, and machine-learning-based surrogate models coupled with multi-objective optimization techniques. The review highlights significant advances in damage prediction accuracy, uncertainty quantification, and surrogate-assisted PDE-constrained optimization, while also identifying persistent challenges related to scalable uncertainty analysis, coupled fracture modeling, and the rigorous integration of physics-informed and data-driven methodologies.

3. Conclusions

As Guest Editors of this Special Issue, we are highly satisfied with the diversity and scientific merit of the published contributions. We sincerely thank all authors for sharing their high-quality research and for their constructive cooperation throughout the review and revision process. Our appreciation also extends to the reviewers, whose time, expertise, and insightful comments were essential in ensuring the rigor and overall quality of this Special Issue. Furthermore, we gratefully acknowledge the editorial staff of Mathematics (MDPI) for their professional support and efficient coordination of the peer-review process, which played a crucial role in the successful completion of this Special Issue.

Looking ahead, future research in computational science and engineering should increasingly focus on the integration of high-fidelity numerical methods, machine-learning-assisted modeling, and robust optimization techniques within a unified computational framework. Building on the advances presented in this Special Issue, several key methodological directions can be identified: the development of meshless, high-order, and dimension-reduction techniques for the efficient and stable simulation of complex physical systems; the incorporation of intelligent, data-driven strategies into numerical modeling and optimization; the advancement of topology optimization and structure-aware design methods under multiple constraints; and the enhancement of numerical solvers for complex governing equations, including inverse, non-conservative, and strongly coupled formulations, through innovative discretization and adaptive algorithms. Such efforts are expected to promote both methodological innovation and the reliable, efficient, and scalable application of advanced numerical algorithms to challenging problems in science and engineering.

The Guest Editors extend their sincere appreciation to all of the authors for their valuable contributions to this Special Issue. We are also deeply grateful to the anonymous reviewers for their insightful and professional evaluation reports, which have significantly enhanced the quality of the submitted manuscripts. Furthermore, we acknowledge the excellent collaboration with the publisher, the constant assistance provided by the MDPI associate editors in bringing this project to the end, and the support of the Managing Editor of this Special Issue, Dr. Nemo Guan.