标题： 一种新的离散散度自由、保正的高阶有限体积法求解理想磁流体力学方程 显示英文标题

标题： A new discretely divergence-free positivity-preserving high-order finite volume method for ideal MHD equations

摘要： 本文提出并分析了一种新颖的高效高阶有限体积法，用于理想磁流体力学（MHD）。 作为一种独特的特性，该方法同时保持了磁场的离散散度自由（DDF）约束和保正性（PP）性质，确保密度、压力和内能的非负性。 为了强制实施DDF条件，我们设计了一种新的离散投影方法，该方法将单元界面处重构的点值投影到一个DDF空间上，而无需使用任何近似多项式。 此投影方法非常高效，易于实现，并且特别适合于标准的高阶有限体积WENO方法，后者通常仅返回重构中的点值。 此外，我们还开发了一个新的有限体积框架，用于构建理想MHD系统的可证明PP方案。 该框架包括离散投影技术、对Godunov-Powell源项的适当逼近以及一个简单的PP限制器。 我们对所提出的有限体积方法的PP性质进行了严格的分析，表明DDF条件和对源项的适当逼近消除了磁通量散度项对PP性质的影响。 由于内能函数的非线性和DDF与PP性质之间的复杂关系，这种分析具有挑战性。 为了解决这些挑战，采用了最近发展起来的几何拟线性化方法，该方法将非线性约束转化为一组线性约束。 最后，我们通过几个基准和苛刻的数值例子验证了所提出方法的有效性。 结果表明，所提出的方法稳健、精确且非常有效，证实了所提出的DDF投影和PP技术的重要性。 显示英文摘要

摘要： This paper proposes and analyzes a novel efficient high-order finite volume method for the ideal magnetohydrodynamics (MHD). As a distinctive feature, the method simultaneously preserves a discretely divergence-free (DDF) constraint on the magnetic field and the positivity-preserving (PP) property, which ensures the positivity of density, pressure, and internal energy. To enforce the DDF condition, we design a new discrete projection approach that projects the reconstructed point values at the cell interface into a DDF space, without using any approximation polynomials. This projection method is highly efficient, easy to implement, and particularly suitable for standard high-order finite volume WENO methods, which typically return only the point values in the reconstruction. Moreover, we also develop a new finite volume framework for constructing provably PP schemes for the ideal MHD system. The framework comprises the discrete projection technique, a suitable approximation to the Godunov--Powell source terms, and a simple PP limiter. We provide rigorous analysis of the PP property of the proposed finite volume method, demonstrating that the DDF condition and the proper approximation to the source terms eliminate the impact of magnetic divergence terms on the PP property. The analysis is challenging due to the internal energy function's nonlinearity and the intricate relationship between the DDF and PP properties. To address these challenges, the recently developed geometric quasilinearization approach is adopted, which transforms a nonlinear constraint into a family of linear constraints. Finally, we validate the effectiveness of the proposed method through several benchmark and demanding numerical examples. The results demonstrate that the proposed method is robust, accurate, and highly effective, confirming the significance of the proposed DDF projection and PP techniques.