Speaker: Prof. Yinhua Xia（夏银华教授）

Time: TBA

Venue: TBA

Abstract

In this talk, we introduce a high-order discontinuous Galerkin (DG) scheme that simultaneously ensures positivity-preserving (PP) properties and maintains the globally divergence-free (GDF) constraint for the ideal magnetohydrodynamics (MHD) equations. Achieving both conditions remains quite challenging in MHD simulations when the PP and GDF reconstructions (theoretically linked) are treated as independent post-processing operations. To overcome this difficulty, we propose the pointwise pressure-preserving correction method, which is easy to implement and highly effective. Specifically, based on the equation of state, we recompute the total energy values at integration nodes using PP hydrodynamic variables and the GDF magnetic field. As long as the GDF magnetic field is highly accurate, the total energy obtained through this correction technique can also maintain high accuracy. The key technique for proving the PP property of the GDF-DG scheme involves the convex decomposition of cell averages. With the geometric quasilinearization framework introduced by Wu et al., our GDF-DG scheme can be theoretically shown to satisfy the PP property. To suppress spurious oscillations, we adopt the jump filter for ideal MHD equations, as a sequel to our recent work. This filter operates locally based on jump information at cell interfaces, and preserves key attributes of the DG method, such as conservation, L2 stability, and high-order accuracy. Moreover, it boasts an impressively low computational cost, given that no local characteristic decomposition is required and all computations are confined to the physical space. The jump filter is applied after each Runge-Kutta stage without altering the DG spatial discretization and maintains both the GDF and PP properties of the scheme. Numerical simulations demonstrate the accuracy, effectiveness, and robustness of the proposed PP-GDF-DG schemes with the jump filter.

About the Speaker

夏银华，中国科学技术大学数学科学学院，教授，博士生导师，安徽省领军人才特聘教授。中国科学技术大学数学系获得博士学位，曾先后到美国布朗大学、香港大学、德国维堡大学等从事博士后研究和访问工作。主要从事高精度数值方法和大规模科学计算的研究，应用于计算流体、天体物理、相场问题、交通流等方面的数值模拟。相关工作发表在包括Math. Comp., J. Comput. Phys., J. Sci. Comput., SIAM J. Num. Anal., SIAM J. Sci. Comput.等杂志。主持国家自然科学基金、教育部、安徽省杰青项目等多项科学基金项目的研究。