This paper contributes to the topic of unphysical one-cell-per-time-step travelling combustion wave solutions in numerical computations of detonation waves in the presence of stiff chemical source terms. These false weak detonation solutions appear when a gas-dynamics-chemistry operator-splitting technique is used in conjunction with modern shock-capturing schemes for compressible flow simulations. A detailed analysis of piecewise constant three-state weak solutions of the Fickett-Majda detonation model equations is carried out. These structures are idealized analogues of the fake numerical solutions observed in computations. The analysis suggests that the problem can be cured by introducing a suitable ignition temperature below which the chemistry is frozen. It is found that the threshold temperatures needed to effectively suppress the undesired numerical artefacts are considerably lower than any temperature actually found in the reaction zone of a resolved detonation. This is in contrast to earlier suggestions along the same lines in the literature and it allows us to propose the introduction of such a low and otherwise irrelevant ignition temperature threshold as a routine measure for overcoming the problem of artificial weak detonations. The criterion for choosing the ignition temperature is then extended to the reactive Euler equations and extensive computational tests for both the model and the full equations demonstrate the effectiveness of our strategy. We consider the behaviour of a first-order Godunov-type scheme as well as its second-order extension in space and time using van Leer's MUSCL approach and Strang splitting.