When modeling a turbulent fluid flow, an Approximate Deconvolution Model (ADM) is sometimes chosen - in particular, due to the high order spatial accuracy. A method has been presented in , that demonstrates an approach to increase the temporal accuracy of the ADM, by combining it with the Deferred Correction method (DCM). The resulting model, DC-ADM, is at least second order accurate in both space and time, and theoretically an arbitrarily high order of accuracy is achievable, provided that enough computational resources are available. However, in some applications, especially when random input data are present, the efficiency of the ADM becomes at least as important, as its accuracy. In order to keep the second order accuracy, but reduce the computational time, we propose a modification of the DC-ADM: in the first, defect step, we replace the turbulence model with the computationally cheaper artificial viscosity approximation. Obviously, this defect step approximation loses accuracy, compared with that obtained in the DC-ADM method, but we will show that the accuracy is recovered in the correction step. The numerical tests demonstrate that the new approach is more efficient, if the compared algorithms are run sequentially: the results from the proposed method are obtained in 35% less time, than those obtained by DC-ADM. The tests also demonstrate that there is no loss of quality of the solutions, when switching from the DC-ADM to the new model. When parallelization is introduced, the methods are comparable in both accuracy and computational time.