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1. An Energy-Based Summation-by-Parts Finite Difference Method For the Wave Equation in Second Order Form

   https://doi.org/10.1007/s10915-022-01829-4  

   Wang, Siyang; Appelö, Daniel; Kreiss, Gunilla (April 2022, Journal of Scientific Computing)

   Abstract We develop a new finite difference method for the wave equation in second order form. The finite difference operators satisfy a summation-by-parts (SBP) property. With boundary conditions and material interface conditions imposed weakly by the simultaneous-approximation-term (SAT) method, we derive energy estimates for the semi-discretization. In addition, error estimates are derived by the normal mode analysis. The proposed method is termed as energy-based because of its similarity with the energy-based discontinuous Galerkin method. When imposing the Dirichlet boundary condition and material interface conditions, the traditional SBP-SAT discretization uses a penalty term with a mesh-dependent parameter, which is not needed in our method. Furthermore, numerical dissipation can be added to the discretization through the boundary and interface conditions. We present numerical experiments that verify convergence and robustness of the proposed method. 

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2. Interpolation-based immersogeometric analysis methods for multi-material and multi-physics problems

   https://doi.org/10.1007/s00466-024-02506-z  

   Fromm, Jennifer_E; Wunsch, Nils; Maute, Kurt; Evans, John_A; Chen, Jiun-Shyan (June 2024, Computational Mechanics)

   Abstract Immersed boundary methods are high-order accurate computational tools used to model geometrically complex problems in computational mechanics. While traditional finite element methods require the construction of high-quality boundary-fitted meshes, immersed boundary methods instead embed the computational domain in a structured background grid. Interpolation-based immersed boundary methods augment existing finite element software to non-invasively implement immersed boundary capabilities through extraction. Extraction interpolates the structured background basis as a linear combination of Lagrange polynomials defined on a foreground mesh, creating an interpolated basis that can be easily integrated by existing methods. This work extends the interpolation-based immersed isogeometric method to multi-material and multi-physics problems. Beginning from level-set descriptions of domain geometries, Heaviside enrichment is implemented to accommodate discontinuities in state variable fields across material interfaces. Adaptive refinement with truncated hierarchically refined B-splines (THB-splines) is used to both improve interface geometry representations and to resolve large solution gradients near interfaces. Multi-physics problems typically involve coupled fields where each field has unique discretization requirements. This work presents a novel discretization method for coupled problems through the application of extraction, using a single foreground mesh for all fields. Numerical examples illustrate optimal convergence rates for this method in both 2D and 3D, for partial differential equations representing heat conduction, linear elasticity, and a coupled thermo-mechanical problem. The utility of this method is demonstrated through image-based analysis of a composite sample, where in addition to circumventing typical meshing difficulties, this method reduces the required degrees of freedom when compared to classical boundary-fitted finite element methods. 

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3. Discontinuous Galerkin methods for stochastic Maxwell equations with multiplicative noise

   https://doi.org/10.1051/m2an/2022084  

   Sun, Jiawei; Shu, Chi-Wang; Xing, Yulong (March 2023, ESAIM: Mathematical Modelling and Numerical Analysis)

   In this paper we propose and analyze finite element discontinuous Galerkin methods for the one- and two-dimensional stochastic Maxwell equations with multiplicative noise. The discrete energy law of the semi-discrete DG methods were studied. Optimal error estimate of the semi-discrete method is obtained for the one-dimensional case, and the two-dimensional case on both rectangular meshes and triangular meshes under certain mesh assumptions. Strong Taylor 2.0 scheme is used as the temporal discretization. Both one- and two-dimensional numerical results are presented to validate the theoretical analysis results. 

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4. A Universal Predictor‐Corrector Approach for Minimizing Artifacts Due To Mesh Refinement

   https://doi.org/10.1029/2023MS003688  

   Du, Shukai; Stechmann, Samuel N (November 2023, Journal of Advances in Modeling Earth Systems)

   Abstract With nested grids or related approaches, it is known that numerical artifacts can be generated at the interface of mesh refinement. Most of the existing methods of minimizing these artifacts are either problem‐dependent or numerical methods‐dependent. In this paper, we propose a universal predictor‐corrector approach to minimize these artifacts. By its construction, the approach can be applied to a wide class of models and numerical methods without modifying the existing methods but instead incorporating an additional step. The idea is to use an additional grid setup with a refinement interface at a different location, and then to correct the predicted state near the refinement interface by using information from the other grid setup. We give some analysis for our method in the setting of a one‐dimensional advection equation, showing that the key to the success of the method depends on an optimized way of choosing the weight functions, which determine the strength of the corrector at a certain location. Furthermore, the method is also tested in more general settings by numerical experiments, including shallow water equations, multi‐dimensional problems, and a variety of underlying numerical methods including finite difference/finite volume and spectral element. Numerical tests suggest the effectiveness of the method on reducing numerical artifacts due to mesh refinement. 

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5. A high order immersed finite element method for parabolic interface problems

   https://doi.org/10.1051/itmconf/20192901007  

   Jones, Derrick; Zhang, Xu; Herisanu, N.; Razzaghi, M.; Vasiu, R. (January 2019, ITM Web of Conferences)

   We present a high order immersed finite element (IFE) method for solving 1D parabolic interface problems. These methods allow the solution mesh to be independent of the interface. Time marching schemes including Backward-Eulerand Crank-Nicolson methods are implemented to fully discretize the system. Numerical examples are provided to test the performance of our numerical schemes. 

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