Research On High-Resolution Moving Mesh Rotated Flux Method For Two Kinds Of Multi-Wave MHD Equations

The numerical solution of the magnetohydrodynamics(MHD)equations is of great significance in the fields of plasma physics,flow control,and high-energy density physics,where the flow complexity of governing equations directly leads to the stringent requirements for numerical schemes.Given that the MHD equations is essentially a nonstrict hyperbolic conservation laws,the entropy-related theory was extended to the field of magnetohydrodynamics many years ago.Now there are still two contradictory hot issues regarding the spurious oscillation eliminated and the smearing phenomenon supressed simultaneously.In this paper,based on the classical ES schemes,a new numerical scheme for multi-wave MHD problems is proposed to resolve multi-waves more accurately by using the moving mesh strategy,flux limiter and rotated flux method.The main work of this paper is as follows:(1).An entropy stable(ES)scheme based on the moving mesh is constructed for the ideal MHD and the Shallow Water MHD(SWMHD)equations.The ES scheme is used to discretize the governing equations in space,and the mesh evolution equation is constructed by the variational method,which is solved iteratively by Gauss-Seidel method.Furthermore,the conservative interpolation formula is used for the quantity transmission from the old nodes to the new ones.The 3-order Runge-Kutta method is utilized in time.Numerical experiments show that the new algorithm can effectively capture the structure of solutions(especially shock waves and rarefaction waves),with good versatility and strong robustness.(2).A new moving-mesh high-resolution scheme based on the S-M flux limiter is proposed for the ideal MHD and the SWMHD equations.Two improvements are proposed based on the numerical results of the scheme in(1).Firstly,a new monitor function is defined to identify multiple structural features of solutions at each particular time level,to assign appropriate weights to all regions containing waves,and to construct a mesh evolution equation for increasing the mesh density of these regions.Secondly,a S-M flux limiter is introduced to weight the ES flux and the anti-diffusive flux for a new flux with high resolution and entropy stability properties.By adjusting the amplitude of anti-diffusive flux adaptively,the new flux can reduce its dissipation in smooth fields and increase its dissipation at discontinuous ones.Numerical experiments show that the new algorithm is more responsive to multi-wave recognition with higher resolution,and still retains strong robustness.(3).A high-resolution rotated flux scheme is constructed to solve those two kinds of MHD systems based on the moving mesh.The introduction of the moving mesh strategy might cause a large deformation of the mesh,leading to a lack of physical meaning in the numerical scheme.Fortunately,the rotated flux method could be used to ensure the isotropic motion of each boundary of the control volume and thus suppress the generation of nonphysical phenomenon if the equations meet the rotational invariance.For the two 2D systems,the rotation-like matrices of flux functions are constructed,and the corresponding rotational invariance theorems are proved.According to this property,the semi-discrete rotated flux scheme is derived by dealing with the governing equations in quasi-1D form.Combining the numerical flux in(2),a high-resolution moving mesh rotated flux was obtained,which can adaptively adjust the dissipation term.Numerical results show that the new scheme is more physically meaningful,and can be easily extended to higher dimensions.