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 mesh method, and 3). high resolution and non-oscillatory numerical solutions are obtained successfully when there are shocks in the solution. A variety of numerical examples show the feature well.
