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.

翻译：本文提出并分析了一种新颖高效的理想磁流体动力学（MHD）高阶有限体积方法。该方法的一个显著特征是同时保持了磁场的离散无散（DDF）约束和保正（PP）性质，从而确保密度、压力和内能的正定性。为强制执行DDF条件，我们设计了一种新的离散投影方法，无需使用近似多项式，即可将单元界面处的重构点值投影到DDF空间。该投影方法高效、易于实现，特别适用于通常仅返回重构中点值的标准高阶有限体积WENO方法。此外，我们还发展了一种新的有限体积框架，用于为理想MHD系统构造可证明保正的格式。该框架包括离散投影技术、对Godunov-Powell源项的适当近似以及一个简单的保正限制器。我们对所提有限体积方法的保正性质进行了严格分析，证明DDF条件和源项的恰当近似消除了磁散度项对保正性质的影响。由于内能函数的非线性和DDF与PP性质之间的复杂关系，该分析具有挑战性。为应对这些挑战，采用了近期发展的几何拟线性化方法，将非线性约束转化为一系列线性约束。最后，我们通过多个基准和具有挑战性的数值算例验证了所提方法的有效性。结果表明，该方法稳健、精确且高效，证实了所提出的DDF投影和PP技术的重要价值。