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1. Finite element-based invariant-domain preserving approximation of hyperbolic systems: Beyond second-order accuracy in space

   https://doi.org/10.1016/j.cma.2023.116470  

   Guermond, Jean-Luc; Nazarov, Murtazo; Popov, Bojan (January 2024, Computer Methods in Applied Mechanics and Engineering)

   This paper proposes an invariant-domain preserving approximation technique for nonlinear conservation systems that is high-order accurate in space and time. The algorithm mixes a high order finite element method with an invariant-domain preserving low-order method that uses the closest neighbor stencil. The construction of the flux of the low-order method is based on an idea from Abgrall et al. (2017). The mass flux of the low-order and the high-order methods are identical on each finite element cell. This allows for mass preserving and invariant-domain preserving limiting. 

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2. An Arbitrarily High-order Spectral Difference Method with Divergence Cleaning (SDDC) for Compressible Magnetohydrodynamic Simulations on Unstructured Grids

   https://doi.org/10.3847/1538-4357/ac6e61  

   Chen, Kuangxu; Liang, Chunlei (June 2022, The Astrophysical Journal)

   Abstract This paper reports a recent development of the high-order spectral difference method with divergence cleaning (SDDC) for accurate simulations of both ideal and resistive magnetohydrodynamics (MHD) on curved unstructured grids consisting of high-order isoparametric quadrilateral elements. The divergence cleaning approach is based on the improved generalized Lagrange multiplier, which is thermodynamically consistent. The SDDC method can achieve an arbitrarily high order of accuracy in spatial discretization, as demonstrated in the test problems with smooth solutions. The high-order SDDC method combined with the artificial dissipation method can sharply capture shock interfaces with the oscillation-free property and resolve small-scale vortex structures and density fluctuations on relatively sparse grids. The robustness of the codes is demonstrated through long time simulations of ideal MHD problems with progressively interacting shock structures, resistive MHD problems with high Lundquist numbers, and viscous resistive MHD problems on complex curved domains. 

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3. An Angular Approach to Untangling High-Order Curvilinear Triangular Meshes

   https://doi.org/10.1007/978-3-030-13992-6_18  

   Stees, Mike; Shontz, Suzanne M. (July 2019, Lecture notes in computational science and engineering)

   To achieve the full potential of high-order numerical methods for solving partial differential equations, the generation of a high-order mesh is required. One particular challenge in the generation of high-order meshes is avoiding invalid (tangled) elements that can occur as a result of moving the nodes from the low-order mesh that lie along the boundary to conform to the true curved boundary. In this paper, we propose a heuristic for correcting tangled second- and third-order meshes. For each interior edge, our method minimizes an objective function based on the unsigned angles of the pair of triangles that share the edge. We present several numerical examples in two dimensions with second- and third-order elements that demonstrate the capabilities of our method for untangling invalid meshes. 

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4. Untangling high-order meshes based on signed angles

   https://doi.org/10.5281/zenodo.3653412  

   Stees, Mike; Dotzel, Myra; Shontz, Suzanne M. (February 2020, Proceedings of the 28th International Meshing Roundtable)

   One challenge in the generation of high-order meshes is that mesh tangling can occur as a consequence of moving the new boundary nodes to the true curved boundary. In this paper, we propose a new optimization-based method that uses signed angles to untangle invalid second- and third-order triangular meshes. Our proposed method consists of two passes. In the first pass, we loop over each high-order interior edge node and minimize an objective function based on the signed angles of the pair of triangles that share the node. In the second pass, we loop over face nodes and move them to the mean of the high-order nodes of the triangle to which the face node belongs. We present several numerical examples in two dimensions with second- and third-order elements that demonstrate the capabilities of our method for untangling invalid meshes. 

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5. System Reliability Analysis With Second Order Saddlepoint Approximation

   https://doi.org/10.1115/DETC2019-97561  

   Wu, Hao (January 2019, The ASME 2019 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference)

   The second order saddlepoint approximation (SPA) has been used for component reliability analysis for higher accuracy than the traditional second order reliability method. This work extends the second order SPA to system reliability analysis. The joint distribution of all the component responses is approximated by a multivariate normal distribution. To maintain high accuracy of the approximation, the proposed method employs the second order SPA to accurately generate the marginal distributions of component responses; to simplify computations and achieve high efficiency, the proposed method estimates the covariance matrix of the multivariate normal distribution with the first order approximation to component responses. Examples demonstrate the high effectiveness of the second order SPA method for system reliability analysis. 

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