Research On High Resolution Numerical Methods For Magneto-hydrodynamics Equations

The main thoughts of this paper is to study high-order accuracy, high resolutionand non-oscillatory numerical methods of magneto-hydrodynamics (MHD) equations.Numerical methods of magneto-hydrodynamics equations have a wide range ofapplications in astrophysics, controlled thermonuclear reaction, the radar systemcommunication, power generation systems, flow control and other fields. The mainwork of this paper includes two aspects: on the one hand, based on the relationshipbetween magneto-hydrodynamics equations and hyperbolic conservation laws, twokinds of high-order accuracy and high resolution numerical methods of hyperbolicconservation laws are extended to solve magneto-hydrodynamics equations. On theother hand, a kind of the existing staggered central difference schemes for solvingmagneto-hydrodynamics equations is improved. Its main content include thefollowing several respects:1. The MmB(Maximum and minimum Bounded) difference scheme forhyperbolic conservation laws is applied to solve the magneto-hydrodynamicsequations. Based on flux splitting and piecewise linear reconstruction of cell-averaged,by properly selecting the numerical derivative and considering Runge-Kutta TVDtime discretization method, a class of two-order accuracy, high resolution andnon-oscillatory MmB schemes for solving magneto-hydrodynamics equations isobtained. Furthermore, the extension to two dimensional magneto-hydrodynamicsequations is implemented by using dimension-by-dimension method. Finally, a seriesof typical numerical examples are given to verify the validation of the resultingschemes.2. The third-order semi-discrete CWENO (Central weighted essentiallynon-oscillatory）method for hyperbolic conservation laws proposed by Kurganov andLevy is applied to solve the magneto-hydrodynamics equations. Based on third-orderaccurate CWENO reconstruction, a class of third-order accurate semi-discreteCWENO methods for magneto-hydrodynamics equations is obtained. Furthermore,the extension to two dimensional magneto-hydrodynamics equations is implementedby using dimension-by-dimension method. Finally, a series of typical numericalexamples are given to verify the validation of the resulting schemes.3. By improving the staggered central difference scheme for solving magneto-hydrodynamics equations proposed by Balbas, Tadmor and Wu, a class ofsecond-order and third-order accuracy, non-staggered, high resolution andnon-oscillatory methods for one and two dimensional magneto-hydrodynamicsequations is obtained. Finally, a series of typical numerical examples are given toverify the validation of the resulting schemes.4. Three classes of schemes in this paper are compared and their advantages anddisadvantages are shown. Finally, the further work in future is present.