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1. Scalar Auxiliary Variable (SAV) Stabilization of Implicit‐Explicit (IMEX) Time Integration Schemes for Non‐Linear Structural Dynamics

   https://doi.org/10.1002/nme.7660  

   Kwon, Sun‐Beom; Prakash, Arun (January 2025, International Journal for Numerical Methods in Engineering)

   Implicit-explicit (IMEX) time integration schemes are well suited for non-linear structural dynamics because of their low computational cost and high accuracy. However, the stability of IMEX schemes cannot be guaranteed for general non-linear problems. In this article, we present a scalar auxiliary variable (SAV) stabilization of high-order IMEX time integration schemes that leads to unconditional stability. The proposed IMEX-BDFk-SAV schemes treat linear terms implicitly using kth-order backward difference formulas (BDFk) and non-linear terms explicitly. This eliminates the need for iterations in non-linear problems and leads to low computational costs. Truncation error analysis of the proposed IMEX-BDFk-SAV schemes confirms that up to kth-order accuracy can be achieved and this is verified through a series of convergence tests. Unlike existing SAV schemes for first-order ordinary differential equations (ODEs), we introduce a novel SAV for the proposed schemes that allows direct solution of the second-order ODEs without transforming them into a system of first-order ODEs. Finally, we demonstrate the performance of the proposed schemes by solving several non-linear problems in structural dynamics and show that the proposed schemes can achieve high accuracy at a low computational cost while maintaining unconditional stability. 

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2. Efficient and linear schemes for anisotropic Cahn–Hilliard model using the Stabilized-Invariant Energy Quadratization (S-IEQ) approach

   Xu, Z; Yang, X; Zhang, H; and Xie, Z. (January 2019, Computer physics communications)

   In this paper, we consider numerical approximations for the anisotropic Cahn–Hilliard equation. We develop two linear and second-order schemes that combine the IEQ approach with the stabilization technique, where several extra linear stabilization terms are added in and they can be shown to be crucial to suppress the non-physical spatial oscillations caused by the strong anisotropy. We show the well-posedness of the resulting linear systems and further prove their corresponding unconditional energy stabilities rigorously. Various 2D and 3D numerical simulations are presented to demonstrate the stability, accuracy, and efficiency of the proposed schemes. 

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3. Sediment Exchange Between the Created Saltmarshes of Living Shorelines and Adjacent Submersed Aquatic Vegetation in the Chesapeake Bay

   https://doi.org/10.3389/fmars.2021.727080  

   Vona, Iacopo; Palinkas, Cindy M.; Nardin, William (October 2021, Frontiers in Marine Science)

   Rising sea levels and the increased frequency of extreme events put coastal communities at serious risk. In response, shoreline armoring for stabilization has been widespread. However, this solution does not take the ecological aspects of the coasts into account. The “living shoreline” technique includes coastal ecology by incorporating natural habitat features, such as saltmarshes, into shoreline stabilization. However, the impacts of living shorelines on adjacent benthic communities, such as submersed aquatic vegetation (SAV), are not yet clear. In particular, while both marshes and SAV trap the sediment necessary for their resilience to environmental change, the synergies between the communities are not well-understood. To help quantify the ecological and protective (shoreline stabilization) aspects of living shorelines, we presented modeling results using the Delft3D-SWAN system on sediment transport between the created saltmarshes of the living shorelines and adjacent SAV in a subestuary of Chesapeake Bay. We used a double numerical approach to primarily validate deposition measurements made in the field and to further quantify the sediment balance between the two vegetation communities using an idealized model. This model used the same numerical domain with different wave heights, periods, and basin slopes and includes the presence of rip-rap, which is often used together with marsh plantings in living shorelines, to look at the influences of artificial structures on the sediment exchange between the plant communities. The results of this study indicated lower shear stress, lower erosion rates, and higher deposition rates within the SAV bed compared with the scenario with the marsh only, which helped stabilize bottom sediments by making the sediment balance positive in case of moderate wave climate (deposition within the two vegetations higher than the sediment loss). The presence of rip-rap resulted in a positive sediment balance, especially in the case of extreme events, where sediment balance was magnified. Overall, this study concluded that SAV helps stabilize bed level and shoreline, and rip-rap works better with extreme conditions, demonstrating how the right combination of natural and built solutions can work well in terms of ecology and coastal protection. 

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4. On a SAV-MAC scheme for the Cahn–Hilliard–Navier–Stokes phase-field model and its error analysis for the corresponding Cahn–Hilliard–Stokes case

   https://doi.org/10.1142/S0218202520500438  

   Li, Xiaoli; Shen, Jie (November 2020, Mathematical Models and Methods in Applied Sciences)

   null (Ed.)

   We construct a numerical scheme based on the scalar auxiliary variable (SAV) approach in time and the MAC discretization in space for the Cahn–Hilliard–Navier–Stokes phase- field model, prove its energy stability, and carry out error analysis for the corresponding Cahn–Hilliard–Stokes model only. The scheme is linear, second-order, unconditionally energy stable and can be implemented very efficiently. We establish second-order error estimates both in time and space for phase-field variable, chemical potential, velocity and pressure in different discrete norms for the Cahn–Hilliard–Stokes phase-field model. We also provide numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of our scheme. 

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5. 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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