The approximation of transient detonation waves requires numerical methods that are able to 
resolve a wide range of different scales. Especially the accurate consideration of detailed 
chemical kinetics is extremely demanding. This thesis describes an efficient solution strategy 
for the Euler equations of gas dynamics for mixtures of thermally perfect species with detailed, 
non-equilibrium reaction that tackles the problem of source term stiffness by temporal and 
spatial dynamic mesh adaptation. All gas dynamically relevant scales are sufficiently resolved.

The blockstructured adaptive mesh refinement technique of Berger and Colella is utilized to 
supply the required resolution locally on the basis of hydrodynamic refinement criteria. This 
adaptive method is tailored especially for time-explicit finite volume schemes and uses a 
hierarchy of spatially refined subgrids which are integrated recursively with reduced time steps.
A parallelization strategy for distributed memory machines is developed and implemented. It 
follows a rigorous domain decomposition approach and partitions the entire grid hierarchy.

A time-operator splitting technique is employed to decouple hydrodynamic transport and chemical 
reaction. It allows the separate numerical integration of the homogeneous Euler equations with 
time-explicit finite volume methods and the usage of an time-implicit discretization only for 
the stiff reaction terms. High-resolution shock capturing schemes are constructed for the 
homogeneous Euler equations with complex equation of state. In particular, a reliable hybrid 
Roe-solver-based method is derived. The scheme avoids unphysical values due to the Roe 
linearization and utilizes additional numerical viscosity to stabilize the approximation of 
strong shocks that inherently appear at the head of detonation waves. In different test 
configurations it is shown that this hybrid Roe-type method is superior for detonation 
simulation to any other method considered.

Large-scale simulations of unstable detonation structures of hydrogen-oxygen detonations run on 
recent Beowulf clusters demonstrate the efficiency of the entire approach. In particular, 
computations of regular cellular structures in two and three space dimensions and their 
development under transient conditions, e.g. Mach reflection and diffraction, are presented. The 
achieved resolutions go far beyond previously published results and provide new reference 
solutions.