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1. Unconditionally Stable Central Difference Dissipative Algorithm for Multi-Directional Real-Time Hybrid Simulations of Large Nonlinear Structural Systems

   https://doi.org/10.1080/13632469.2024.2333822  

   Al-Subaihawi, Safwan; Ricles, James M; Quiel, Spencer E; Marullo, Thomas (April 2024, Journal of Earthquake Engineering)

   The central difference is a popular algorithm used to integrate the equations of motion, yet suffers from two drawbacks: (1) it is only conditionally stable and requires a small-time step to maintain numerical stability; (2) it is nondissipative, and high-frequency spurious oscillations may appear and compromise the accuracy of the solution. These drawbacks are detrimental to applying the algorithm to the real-time hybrid simulation of large, complex nonlinear structural systems. In this paper, the conventional central difference algorithm is modified to overcome these drawbacks, and the modified algorithm is applied to the real-time hybrid simulation of complex structural systems. 

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2. A semi-adaptive finite difference method for simulating two-sided fractional convection–diffusion quenching problems

   https://doi.org/10.1016/j.cam.2025.116796  

   Dong, Rumin; Zhu, Lin; Sheng, Qin; Zhao, Bingxin (July 2025, Journal of Computational and Applied Mathematics)

   na (Ed.)

   This paper investigates quenching solutions of an one-dimensional, two-sided Riemann–Liouville fractional order convection–diffusion problem. Fractional order spatial derivatives are discretized using weighted averaging approximations in conjunction with standard and shifted Grünwald formulas. The advective term is handled utilizing a straightforward Euler formula, resulting in a semi-discretized system of nonlinear ordinary differential equations. The conservativeness of the proposed scheme is rigorously proved and validated through simulation experiments. The study is further advanced to a fully discretized, semi-adaptive finite difference method. Detailed analysis is implemented for the monotonicity, positivity and stability of the scheme. Investigations are carried out to assess the potential impacts of the fractional order on quenching location, quenching time, and critical length. The computational results are thoroughly discussed and analyzed, providing a more comprehensive understanding of the quenching phenomena modeled through two-sided fractional order convection-diffusion problems. 

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3. Efficient numerical scheme for a dendritic solidification phase field model with melt convection

   Chen, C; Yang, X. (March 2019, Journal of computational physics)

   In this paper, we consider numerical approximations for a dendritic solidification phase field model with melt convection in the liquid phase, which is a highly nonlinear system that couples the anisotropic Allen-Cahn type equation, the heat equation, and the weighted Navier-Stokes equations together. We first reformulate the model into a form which is suitable for numerical approximations and establish the energy dissipative law. Then, we develop a linear, decoupled, and unconditionally energy stable numerical scheme by combining the modified projection scheme for the Navier-Stokes equations, the Invariant Energy Quadratization approach for the nonlinear anisotropic potential, and some subtle explicit-implicit treatments for nonlinear coupling terms. Stability analysis and various numerical simulations are presented. 

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4. Adaptive Control of a Two-Link Robot Using Batch Least-Square Identifier

   https://doi.org/10.1109/JAS.2020.1003459  

   Bagheri, Mostafa; Karafyllis, Iasson; Naseradinmousavi, Peiman; Krstic, Miroslav. (January 2021, IEEECAA journal of automatica sinica)

   Liu, Tengfei; Ou, Yan. (Ed.)

   We design a regulation-triggered adaptive controller for robot manipulators to efficiently estimate unknown parameters and to achieve asymptotic stability in the presence of coupled uncertainties. Robot manipulators are widely used in telemanipulation systems where they are subject to model and environmental uncertainties. Using conventional control algorithms on such systems can cause not only poor control performance, but also expensive computational costs and catastrophic instabilities. Therefore, system uncertainties need to be estimated through designing a computationally efficient adaptive control law. We focus on robot manipulators as an example of a highly nonlinear system. As a case study, a 2-DOF manipulator subject to four parametric uncertainties is investigated. First, the dynamic equations of the manipulator are derived, and the corresponding regressor matrix is constructed for the unknown parameters. For a general nonlinear system, a theorem is presented to guarantee the asymptotic stability of the system and the convergence of parameters’ estimations. Finally, simulation results are discussed for a two-link manipulator, and the performance of the proposed scheme is thoroughly evaluated. 

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5. A local discontinuous Galerkin method for nonlinear parabolic SPDEs

   https://doi.org/10.1051/m2an/2020026  

   Li, Yunzhang; Shu, Chi-Wang; Tang, Shanjian (January 2021, ESAIM: Mathematical Modelling and Numerical Analysis)

   null (Ed.)

   In this paper, we propose a local discontinuous Galerkin (LDG) method for nonlinear and possibly degenerate parabolic stochastic partial differential equations, which is a high-order numerical scheme. It extends the discontinuous Galerkin (DG) method for purely hyperbolic equations to parabolic equations and shares with the DG method its advantage and flexibility. We prove the L 2 -stability of the numerical scheme for fully nonlinear equations. Optimal error estimates ( O ( h (k+1) )) for smooth solutions of semi-linear stochastic equations is shown if polynomials of degree k are used. We use an explicit derivative-free order 1.5 time discretization scheme to solve the matrix-valued stochastic ordinary differential equations derived from the spatial discretization. Numerical examples are given to display the performance of the LDG method. 

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