A Provable Positivity-Preserving Local Discontinuous Galerkin Method for the Viscous and Resistive MHD Equations

In this paper,we construct a high-order discontinuous Galerkin(DG)method which can preserve the positivity of the density and the pressure for the viscous and resistive magnetohydrodynamics(VRMHD).To control the divergence error in the magnetic field,both the local divergence-free basis and the Godunov source term would be employed for the multi-dimensional VRMHD.Rigorous theoretical analyses are presented for one-dimensional and multi-dimensional DG schemes,respectively,showing that the scheme can maintain the positivity-preserving(PP)property under some CFL conditions when combined with the strong-stability-preserving time discretization.Then,general frameworks are established to construct the PP limiter for arbitrary order of accuracy DG schemes.Numerical tests demonstrate the effectiveness of the proposed schemes.

On Arbitrary-Lagrangian-Eulerian One-Step WENO Schemes for Stiff Hyperbolic Balance Laws

In this article we present a new family of high order accurate Arbitrary Lagrangian-Eulerian one-step WENO finite volume schemes for the solution of stiff hyperbolic balance laws.High order accuracy in space is obtained with a standard WENO reconstruction algorithm and high order in time is obtained using the local space-time discontinuous Galerkinmethod recently proposed in[20].In the Lagrangian framework considered here,the local space-time DG predictor is based on a weak formulation of the governing PDE on a moving space-time element.For the spacetime basis and test functions we use Lagrange interpolation polynomials defined by tensor-product Gauss-Legendre quadrature points.The moving space-time elements are mapped to a reference element using an isoparametric approach,i.e.the spacetime mapping is defined by the same basis functions as the weak solution of the PDE.We show some computational examples in one space-dimension for non-stiff and for stiff balance laws,in particular for the Euler equations of compressible gas dynamics,for the resistive relativistic MHD equations,and for the relativistic radiation hydrodynamics equations.Numerical convergence results are presented for the stiff case up to sixth order of accuracy in space and time and for the non-stiff case up to eighth order of accuracy in space and time.