In this thesis, we deliver some of our work on computational fluid dynamics. Firstly we give an introduction on our work in Chapter 1. Some basic concepts and techniques related to our research is reviewed in Chapter 2. In Chapter 3, we introduce a third order adaptive mesh method to arbitrary high order Godunov approach. Our adaptive mesh method consists of two parts, i.e., mesh-redistribution algorithm and solution algorithm. The mesh-redistribution algorithm is derived based on variational approach, while a new solution algorithm is developed to preserve high order numerical accuracy well. The feature of proposed adaptive ADER scheme includes that 1). all simulations in this paper are stable for large CFL number, 2). third order convergence of the numerical solutions is successfully observed with adaptive method, and 3). high resolution and non-oscillatory numerical solutions are obtained successfully when there are shocks in the domain. A variety of numerical examples show the above features well. Finally in Chapter 4, We introduce our on-going work on detonation wave simulation. Governing equations and derivation of initial condition will be recalled, and numerical experiments will be presented.