High Accuracy GDQ Method For One Dimension Hyperbolic Conservation Equation

As one of the most important partial differencial equations, hyperbolic conservation equation is widely used in fluid dynamics, aeronautics and paceflight, aerody and shipbuilding fields. It has an important characteristic: even if the initial value is continuous, the solution may become discontinuous. So the key of numerical solution is to find a method which can catch discontinuity (shock wave) effectively and accurately. In this thesis, combining traditional GDQ method with upwind idea and Lax-wendroff time discretization respective, we get two new high accuracy difference schemes, High Order TVD Upwind Scheme and Based on Lax-Wendroff Time Discretization GDQ Method, which are used to solve hyperbolic conservation equations.The thesis combines traditional GDQ method which is used to solve ordinary differencial equation with upwind idea, modifies flux by analyzing the character of computational domain, constructs a new semi-discretization scheme for hyperbolic conservation equations, High Order TVD Upwind Scheme. The thesis gives the construction process of new scheme for quasilinear hyperbolic conservation equation and system of quasilinear hyperbolic conservation equations, and proves the TVD character of quasilinear hyperbolic conservation equation strictly. Aplaying the new scheme to three typical equations, the numerical experiments verify that it can efficiently simulate shock wave, compress wave, rarefaction wave and contactdiscontinuity.Combining traditional GDQ method with Lax-Wendroff time discretization, using flux modifing, we get anew entirety discretization method for hyperbolic conservation equations, Based on Lax-Wendroff Time Discretization GDQ Method. It is simple and high efficiency, uses less memory. The numerical experiments verify that it can simulate shock wave, compress wave and rarefaction wave well.