高阶曲面重构是用于无CAD、基于网格的几何与物理建模，以及工程应用中求解偏微分方程（PDE）的高阶数值方法的重要技术。在本文中，我们引入了一种新方法，用于从三角化曲面中准确且稳健地重构分段光滑曲面。我们提出的方法在两个主要方面扩展了Continuous Moving Frames (CMF)和Weighted Averaging of Local Fittings (WALF)方法（Engrg. Comput. 28 (2012)）。首先，它利用Hermite式最小二乘逼近，即使输入网格相对粗糙，也能实现具有紧支集的四阶及更高阶精度。其次，它引入了一种迭代的特征感知参数化，以确保在尖锐特征附近实现高阶精度且G0连续的重构。我们给出了该方法的理论框架，并在精度和稳定性方面将其与CMF和WALF进行了比较。我们还展示了在有限元方法（FEM）求解PDE时使用Hermite式重构的情况，并表明使用高阶重构曲面的四次和六次FEM产生的结果与使用精确几何几乎相同，同时提供了额外的灵活性。
英文摘要：
High-order surface reconstruction is an important technique for CAD-free, mesh-based geometric and physical modeling, and for high-order numerical methods for solving partial differential equations (PDEs) in engineering applications. In this paper, we introduce a novel method for accurate and robust reconstructions of piecewise smooth surfaces from a triangulated surface. Our proposed method extends the Continuous Moving Frames (CMF) and the Weighted Averaging of Local Fittings (WALF) methods (Engrg. Comput. 28 (2012)) in two main aspects. First, it utilizes a Hermite-style least squares approximation to achieve fourth and higher-order accuracy with compact support, even if the input mesh is relatively coarse. Second, it introduces an iterative feature-aware parameterization to ensure high-order accurate, G0 continuous reconstructions near sharp features. We present the theoretical framework of the method and compare it against CMF and WALF in terms of accuracy and stability. We also demonstrate that the use of the Hermite-style reconstruction in the solutions of PDEs using finite element methods (FEM), and show that quartic and sextic FEMs using the high-order reconstructed surfaces produce nearly identical results as using exact geometry while providing additional flexibility.
A Hermite WENO reconstruction for fourth order temporal accurate schemes based on the GRP solver for hyperbolic conservation laws
School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）；Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics（应用物理与计算力学研究所计算物理实验室）
In this paper, we propose a hybrid finite volume Hermite weighted essentially non-oscillatory (HWENO) scheme for solving one and two dimensional hyperbolic conservation laws. The zeroth-order and the first-order moments are used in the spatial reconstruction, with total variation diminishing Runge-Kutta time discretization. The main idea of the hybrid HWENO scheme is that we first use a shock-detection technique to identify the troubled cell, then, if the cell is identified as a troubled cell, we would modify the first order moment in the troubled cell and employ HWENO reconstruction in spatia…
Reconstructing sparse exponential polynomials from samples: Stirling numbers and Hermite interpolation
Lehrstuhl für Mathematik mit Schwerpunkt Digitale Bildverarbeitung & FORWISS, University of Passau（帕绍大学数字图像处理重点数学研究所与FORWISS）；University of Passau（帕绍大学）
We present fourth-order conservative non-splitting semi-Lagrangian (SL) Hermite essentially non-oscillatory (HWENO) schemes for linear transport equations with applications for nonlinear problems including the Vlasov-Poisson system, the guiding center Vlasov model, and the incompressible Euler equations in the vorticity-stream function formulation. The proposed SL HWENO schemes combine a weak formulation of the characteristic Galerkin method with two newly constructed HWENO reconstruction methods. Fourth-order accuracy is accomplished in both space and time under a non-splitting setting. Mass …
In this paper, we propose a well-balanced fifth-order finite difference Hermite WENO (HWENO) scheme for the shallow water equations with non-flat bottom topography in pre-balanced form. For achieving the well-balance property, we adopt the similar idea of WENO-XS scheme [Xing and Shu, J. Comput. Phys., 208 (2005), 206-227.] to balance the flux gradients and the source terms. The fluxes in the original equation are reconstructed by the nonlinear HWENO reconstructions while other fluxes in the derivative equations are approximated by the high-degree polynomials directly. And an HWENO limiter is …
In this paper, a fifth-order Hermite weighted essentially non-oscillatory (HWENO) scheme with artificial linear weights is proposed for one and two dimensional hyperbolic conservation laws, where the zeroth-order and the first-order moments are used in the spatial reconstruction. We construct the HWENO methodology using a nonlinear convex combination of a high degree polynomial with several low degree polynomials, and the associated linear weights can be any artificial positive numbers with only requirement that their summation equals one. The one advantage of the HWENO scheme is its simplicit…
In this paper, we propose to combine the fifth order Hermite weighted essentially non-oscillatory (HWENO) scheme and fast sweeping method (FSM) for the solution of the steady-state $S_{N}$ transport equation in the finite volume framework. It is well-known that the $S_{N}$ transport equation asymptotically converges to a macroscopic diffusion equation in the limit of optically thick systems with small absorption and sources. Numerical methods which can preserve the asymptotic limit are referred to as asymptotic preserving methods. In the one-dimensional case, we provide the analysis to demonst…
In this paper, we propose a novel Hermite weighted essentially non-oscillatory (HWENO) fast sweeping method to solve the static Hamilton-Jacobi equations efficiently. During the HWENO reconstruction procedure, the proposed method is built upon a new finite difference fifth order HWENO scheme involving one big stencil and two small stencils. However, one major novelty and difference from the traditional HWENO framework lies in the fact that, we do not need to introduce and solve any additional equations to update the derivatives of the unknown function $ϕ$. Instead, we use the current $ϕ$ and t…
In this paper, we combine the nonlinear HWENO reconstruction in \cite{newhwenozq} and the fixed-point iteration with Gauss-Seidel fast sweeping strategy, to solve the static Hamilton-Jacobi equations in a novel HWENO framework recently developed in \cite{mehweno1}. The proposed HWENO frameworks enjoys several advantages. First, compared with the traditional HWENO framework, the proposed methods do not need to introduce additional auxiliary equations to update the derivatives of the unknown function $ϕ$. They are now computed from the current value of $ϕ$ and the previous spatial derivatives of…
