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We consider the numerical computation of one-dimensional detonation waves. Detonation waves are travelling wave solutions of the reactive Euler equations. An essential difficulty in the numerical computation of detonation waves is the occurrence of nonphysical solu-tions. In order to study this problem we consider a 2 x 2 model problem. For this model problem it is illustrated that nonphysical solutions are weak detonation waves. This is used to obtain a simple criterion which ensures that, also for relatively large mesh sizes, the nu-merical solution approximates the physically correct weak solution. Finally, we extend this criterion to the reactive Euler equations. Numerical results support the use of this cri-terion to exclude nonphysical weak detonation waves. A.M.S. Classifications