Based On GDQM A Numerical Method Research For One Dimension Hyperbolic Conservation Equations

The exploitation of numerical methods for hyperbolic conservation laws is one of the key research fields in computational mathematics since 1930's, a lot of effective and influential numerical methods have been put forward and the GDQ method was one important and efficient scheme among them. Besides simpleness and legibility, the GDQM is studied and applied widely today since it adds no demand and restriction on the grids.In this paper, we discussed hyperbolic conservation laws, their exact solutions and the background of the discrete GDQ method. For the discontinuous arising in the process of computation about the compressible flows which governed by hyperbolic conservation laws, we pay more attention on the high-order accuracy and high resolution in the construction of adaptive discrete GDQMs and uniform schemes.We analyzed the principle of the adaptive discrete GDQM and uniform GDQM elaborately, including grids-generation, spatial discretization and temporal discretization. The reconstruction of numerical fluxes on the whole field is the key step in spatial discretization because of its crucial impact on efficiency. In temporal discretization, we employed TVD Runge-Kutta sheme.The two GDQMs are both of TVD property.Numerical tests show that our methods could capture shocks and rarefaction wave efficiently and successfully in comparison with the graphs of some typical numerical examples.