Research On Limiters Of Discontinuous Galerkin Methods For A Class Of Nonlinear Hyperbolic Equations

In this dissertation,we study limiters for the discontinuous Galerkin methods for a class of hyperbolic equations.As a very important type of partial differential equations in mathematics,hyperbolic equations are extensively used in scientific research and engineering.In general,the analytical solutions of hyperbolic equations are not easy to find.Even if the initial data is sufficiently smooth,discontinuities may appear at a later time.The discontinuous Galerkin methods is a high-order accuracy and high-resolution numerical method.The high-order accuracy is illustrated by that the smooth solution can be numerically approximated with any order,and the high resolution is reflected by that the discontinuous solution can be sharply captured by the limiter.First,based on the existence of the boundary of the initial value,we define bound-preserving principle for the numerical solution of the hyperbolic equation.For the feasibility of the algorithm,this bound-preserving property is equivalently applied to the cell average of the discontinuous Galerkin methods.In order to meet the sufficient conditions of bound-preserving,we utilize the scaling limiter on the Legendre-GaussLobatto points.Finally,we obtain a high-order bound-preserving discontinuous Galerkin methods that is easy to realize numerically.Then,according to the nonnegative initial value boundary,the positive-preserving property of the compressible Euler equations is defined.We can also construct the high-order positive-preserving discontinuous Galerkin methods.The numerical flux we used in this dissertation,is the Lax-Friedrichs flux and the generalized local Lax-Friedrichs flux.The latter is a general numerical flux,which has greater flexibility compared with the traditional upwind numerical flux.Using the generalized local Lax-Friedrichs flux,we adjust the numerical viscosity of the scheme adaptively.In the numerical experiments,we verify the high-order accuracy of boundpreserving discontinuous Galerkin methods for linear hyperbolic equations,and the high-order accuracy and high resolution of bound-preserving discontinuous Galerkin methods for nonlinear hyperbolic equations.In addition,by adjusting the values of the weights,we found the generalized local Lax-Friedrichs numerical flux has potential advantages for simulating oscillations.