  We describe an implementation of compressible inviscid fluid solvers with
block-structured adaptive mesh refinement on Graphics Processing Units using
NVIDIA's CUDA. We show that a class of high resolution shock capturing schemes
can be mapped naturally on this architecture. Using the method of lines
approach with the second order total variation diminishing Runge-Kutta time
integration scheme, piecewise linear reconstruction, and a Harten-Lax-van Leer
Riemann solver, we achieve an overall speedup of approximately 10 times faster
execution on one graphics card as compared to a single core on the host
computer. We attain this speedup in uniform grid runs as well as in problems
with deep AMR hierarchies. Our framework can readily be applied to more general
systems of conservation laws and extended to higher order shock capturing
schemes. This is shown directly by an implementation of a magneto-hydrodynamic
solver and comparing its performance to the pure hydrodynamic case. Finally, we
also combined our CUDA parallel scheme with MPI to make the code run on GPU
clusters. Close to ideal speedup is observed on up to four GPUs.
