Numerical schemes provably preserving the positivity of density and pressure are highly desirable for ideal magnetohydrodynamics (MHD), but the rigorous positivity-preserving (PP) analysis remains challenging. The difficulties mainly arise from the intrinsic complexity of the MHD equations as well as the indeterminate relation between the PP property and the divergence-free condition on the magnetic field. This paper presents the first rigorous PP analysis of conservative schemes with the Lax--Friedrichs (LF) flux for 1D and multidimensional ideal MHD. The significant innovation is the discovery of the theoretical connection between the PP property and a discrete divergence-free (DDF) condition. This connection is established through the generalized LF splitting properties, which are alternatives to the usually expected LF splitting property that does not hold for ideal MHD. The generalized LF splitting properties involve a number of admissible states strongly coupled by the DDF condition, making their derivation very difficult. We derive these properties via a novel equivalent form of the admissible state set and an important inequality, which is skillfully constructed by technical estimates. Rigorous PP analysis is then presented for finite volume and discontinuous Galerkin schemes with the LF flux on uniform Cartesian meshes. In the 1D case, the PP property is proved for the first-order scheme with proper numerical viscosity, and also for arbitrarily high-order schemes under conditions accessible by a PP limiter. In the 2D case, we show that the DDF condition is necessary and crucial for achieving the PP property. It is observed that even slightly violating the proposed DDF condition may cause failure to preserve the positivity of pressure. We prove that the 2D LF type scheme with proper numerical viscosity preserves both the positivity and the DDF condition. Sufficient conditions are derived for 2D PP high-order schemes, and extension to 3D is discussed. Numerical examples provided in the supplementary material further confirm the theoretical findings.

We construct a local Lax-Friedrichs type positivity-preserving flux for compressible Navier-Stokes equations, which can be easily extended to high dimensions for generic forms of equations of state, shear stress tensor and heat flux. With this positivity-preserving flux, any finite volume type schemes including discontinuous Galerkin (DG) schemes with strong stability preserving Runge-Kutta time discretizations satisfy a weak positivity property. With a simple and effi- cient positivity-preserving limiter, high order explicit Runge-Kutta DG schemes are rendered preserving the positivity of density and internal energy without los- ing local conservation or high order accuracy. Numerical tests suggest that the positivity-preserving flux and the positivity-preserving limiter do not induce excessive artificial viscosity, and the high order positivity-preserving DG schemes without other limiters can produce satisfying non-oscillatory solutions when the nonlinear diffusion in compressible Navier-Stokes equations is accurately resolved.

In this paper, we propose new spectral viscosity methods based on the generalized Hermite functions for the solution of nonlinear scalar conservation laws in the whole line. It is shown rigorously that these schemes converge to the unique entropy solution by using compensated compactness arguments, under some conditions. The numerical experiments of the inviscid Burger's equation support our result, and it verifies the reasonableness of the conditions.

In this paper, we propose and analyze a novel stabilized finite element method (FEM) for the system of generalized Stokes equations arising from the time-discretization of transient Stokes problem. The system involves a small viscosity, which is proportional to the inverse of large Reynolds number, and a large reaction coefficient, which is the inverse of small time step. The proposed stabilized FEM employs the $C^0$ piecewise linear elements for both velocity field and pressure on the same mesh and uses the residuals of the momentum equation and the divergence-free equation to define the stabilization terms. The stabilization parameters are fixed and element-independent, without a comparison of the viscosity, the reaction coefficient and the mesh size. Using the finite element solution of an auxiliary boundary value problem as the interpolating function for velocity and the $H^1$-seminorm projection for pressure, instead of the usual nodal interpolants, we derive error estimates for the stabilized finite element approximations to velocity and pressure in the $L^2$ and $H^1$ norms and most importantly, we explicitly establish the dependence of error bounds on the viscosity, the reaction coefficient and the mesh size. Our analysis reveals that this stabilized FEM is particularly suitable for the generalized Stokes system with a small viscosity and a large reaction coefficient, which has never been achieved before in the error analysis of other stabilization methods in the literature. We then numerically confirm the effectiveness of the proposed stabilized FEM. Comparisons made with other existing stabilization methods show that the newly proposed method can attain better accuracy and stability.

In this paper, we apply discontinuous Galerkin (DG) methods to solve two model equations: Krause's consensus models and pressureless Euler equations. These two models are used to describe the collisions of particles, and the distributions can be identified as density functions. If the particles are placed at a single point, then the density function turns out to be a $\delta$-function and is difficult to be well approximated numerically. In this paper, we use DG method to approximate such a singularity and demonstrate the good performance of the scheme. Since the density functions are always positive, we apply a positivity-preserving limiter to them. Moreover, for pressureless Euler equations, the velocity satisfies the maximum principle. We also construct special limiters to fulfill this requirement.