<p>The wavelet multi-resolution interpolation Galerkin method (WMIGM) is combined with a mixed explicit-implicit time-stepping scheme to solve the one-dimensional Burgers’ equation at high Reynolds numbers, where the solutions exhibit evolving steep local gradients. In the proposed framework, a dynamic sequence of node distributions with local multi-resolution refinement is adaptively constructed according to the gradient information identified by a wavelet transform. The approximate solution at previous time levels, required in the time-stepping procedure, is represented by the same wavelet expansion used in its original construction, thereby eliminating the need for interpolation between different node distributions. Several representative numerical examples are presented to assess the accuracy, convergence, and robustness of the proposed adaptive wavelet method. The results demonstrate that the proposed approach possesses a higher accuracy and a faster convergence rate than many existing numerical methods, and can accurately capture complex shock dynamics without spurious oscillations, including boundary layer formation from smooth initial profiles and shock merging processes.</p>