Maximum-Principle-Satisfying And Positivity-Preserving CDG Method

Hyperbolic conservation laws are in a class of essential partial differential equations. They play significant roles in various area of applications such as hydromechanics, aerodynamics, aerospace and shipbuilding and so on. Because of the complexity of these equations, in general, it is not possible to derive their analytic solutions and leads us to use some appropriate numerical methods for their approximate solution in practical applications. Central discontinuous Galerkin method is a powerful computational method for solving hyperbolic conservation laws, and it has a broad application prospect. Therefore, the improvement and development of this method have very important scientific significances.In particular, when using numerical method to solve hyperbolic conservation laws, it is quite common to get numerical solutions violating some physical properties, for example, the maximum principle, nonnegative pressure and so on. Like other high order numerical schemes for solving conservation laws, Runge-Kutta central discontinuous Galerkin method with a TVB(total variation bounded) limiter satisfies neither a strict maximum principle for scalar conservation laws nor the positivity preserving property of pressure and density for compressible Euler equations. In this thesis, we first develop a maximum-principle-satisfying high order central discontinuous Galerkin method for the one-dimensional and two-dimensional scalar conservation laws. We prove a sufficient condition for the cell averages of the numerical solutions in a central discontinuous Galerkin method with Euler forward time discretization, to satisfy the maximum principle. This sufficient condition can be achieved by a maximum-principle-satisfying limiter proposed by Zhang and Shu without destroying accuracy and conservation under a suitable CFL condition. Next, we use this framework to construct positivity-preserving central discontinuous Galerkin methods for compressible Euler equations in both one and two dimensional spaces, employing the positivity preserving techniques advanced by Zhang and Shu.For the above two methods, essential theoretical results will be discussed and their performance will be demonstrated through a set of numerical experiments. In our examples, we use the high order TVD(total variation diminishing) time discretization which will not destroy the maximum principle and positivity-preserving property.