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.