An efficient interpolating wavelet-based adaptive-grid numerical method is described for solving systems of bidimensional partial differential equations. The grid is dynamically adapted in both dimensions during the integration procedure so that only the relevant information is stored, saving allocation memory. The spatial derivatives are directly calculated in a nonuniform grid using cubic splines. Numerical results for five typical problems presented illustrate the efficiency and robustness of the method. The adaptive strategy significantly reduces the computational times and the memory requirements, as compared to the fixed-grid approach.