14 research outputs found
Positivity-Preserving Finite Difference WENO Schemes with Constrained Transport for Ideal Magnetohydrodynamic Equations
In this paper, we utilize the maximum-principle-preserving flux limiting
technique, originally designed for high order weighted essentially
non-oscillatory (WENO) methods for scalar hyperbolic conservation laws, to
develop a class of high order positivity-preserving finite difference WENO
methods for the ideal magnetohydrodynamic (MHD) equations. Our schemes, under
the constrained transport (CT) framework, can achieve high order accuracy, a
discrete divergence-free condition and positivity of the numerical solution
simultaneously. Numerical examples in 1D, 2D and 3D are provided to demonstrate
the performance of the proposed method.Comment: 21 pages, 28 figure
A new discretely divergence-free positivity-preserving high-order finite volume method for ideal MHD equations
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.Comment: 26 page