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1. Moving interfaces in peridynamic diffusion models and the influence of discontinuous initial conditions: Numerical stability and convergence

   https://doi.org/10.1016/j.camwa.2023.10.016  

   Scabbia, Francesco; Gasparrini, Claudia; Zaccariotto, Mirco; Galvanetto, Ugo; Larios, Adam; Bobaru, Florin (December 2023, Computers & Mathematics with Applications)

   We derive numerical stability conditions and analyze convergence to analytical nonlocal solutions of 1D peridynamic models for transient diffusion with and without a moving interface. In heat transfer or oxidation, for example, one often encounters initial conditions that are discontinuous, as in thermal shock or sudden exposure to oxygen. We study the numerical error in these models with continuous and discontinuous initial conditions and determine that the initial discontinuities lead to lower convergence rates, but this issue is present at early times only. Except for the early times, the convergence rates of models with continuous and discontinuous initial conditions are the same. In problems with moving interfaces, we show that the numerical solution captures the exact interface location well, in time. These results can be used in simulating a variety of reaction-diffusion type problems, such as the oxidation-induced damage in zirconium carbide at high temperatures. 

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2. A direct forcing, immersed boundary method for conjugate heat transport

   https://doi.org/10.1016/j.jcp.2025.114135  

   Koponen, Kimmo; Pal_Singh_Bhalla, Amneet; Sprinkle, Brennan; Wu, Ning; Tilton, Nils (October 2025, Journal of Computational Physics)

   Motivated by applications to fluid flows with conjugate heat transfer and electrokinetic effects, we propose a direct forcing immersed boundary method for simulating general, discontinuous, Dirichlet and Robin conditions at the interface between two materials. In comparison to existing methods, our approach uses smaller stencils and accommodates complex geometries with sharp corners. The method is built on the concept of a “forcing pair,” defined as two grid points that are adjacent to each other, but on opposite sides of an interface. For 2D problems this approach can simultaneously enforce discontinuous Dirichlet and Robin conditions using a six-point stencil at one of the forcing points, and a 12-point stencil at the other. In comparison, prior work requires up to 14-point stencils at both points. We also propose two methods of accommodating surfaces with sharp corners. The first locally reduces stencils in sharp corners. The second uses the signed distance function to globally smooth all corners on a surface. The smoothing is defined to recover the actual corners as the grid is refined. We verify second-order spatial accuracy of our proposed methods by comparing to manufactured solutions to the Poisson equation with challenging dis- continuous fields across immersed surfaces. Next, to explore the performance of our method for simulating fluid flows with conjugate heat transport, we couple our method to the incompressible Navier–Stokes and continuity equations using a finite-volume projection method. We verify the spatial-temporal accuracy of the solver using manufactured solutions and an analytical solution for circular Couette flow with conjugate heat transfer. Finally, to demonstrate that our method can model moving surfaces, we simulate fluid flow and conjugate heat transport between a stationary cylinder and a rotating ellipse or square. 

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3. Convergence of the boundary integral method for interfacial Stokes flow

   https://doi.org/10.1090/mcom/3787  

   Ambrose, David; Siegel, Michael; Zhang, Keyang (March 2023, Mathematics of Computation)

   Boundary integral numerical methods are among the most accurate methods for interfacial Stokes flow, and are widely applied. They have the advantage that only the boundary of the domain must be discretized, which reduces the number of discretization points and allows the treatment of complicated interfaces. Despite their popularity, there is no analysis of the convergence of these methods for interfacial Stokes flow. In practice, the stability of discretizations of the boundary integral formulation can depend sensitively on details of the discretization and on the application of numerical filters. We present a convergence analysis of the boundary integral method for Stokes flow, focusing on a rather general method for computing the evolution of an elastic capsule or viscous drop in 2D strain and shear flows. The analysis clarifies the role of numerical filters in practical computations. 

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4. convergence of a homotopy finite element method for computing steady states of Burgers' equation

   https://doi.org/10.1051/m2an/2018046  

   Yang, Yong; Hao, Wenrui (January 2018, ESAIM: Mathematical Modelling and Numerical Analysis)

   In this paper, the convergence of a homotopy method (1.1) for solving the steady state problem of Burgers’ equation is considered. When ν is fixed, we prove that the solution of (1.1) converges to the unique steady state solution as epsilon → 0, which is independent of the initial conditions. Numerical examples are presented to confirm this conclusion by using the continuous finite element method. In contrast, when ν = epsilon → 0, numerically we show that steady state solutions obtained by (1.1) indeed depend on initial conditions. 

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5. Global Attitude Control via Contraction on Manifolds with Reference Trajectory and Optimization

   https://doi.org/10.1109/CDC42340.2020.9303862  

   Vang, Bee; Tron, Roberto (December 2020, IEEE Conference on Decision and Control)

   null (Ed.)

   In this paper, we present a simple geometric attitude controller that is globally, exponentially stable. To overcome the topological restriction, the controller is designed to follow a reference trajectory that in turn converges to the desired equilibrium (making it discontinuous in the initial conditions, but continuous in time). The system and reference dynamics are studied as a single augmented system that can be analyzed and tuned simultaneously. The controller's stability is proved using contraction analysis (on the manifold), and the bounds on the convergence rate can be found via a semi-definite program with linear matrix inequalities. Additionally, our approach allows the use of the Nelder-Mead algorithm to automatically select controller gains and reference trajectory parameters by optimizing the aforementioned bounds. The resulting controller is verified through simulations. 

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