An efficient adaptive multiresolution numerical method is described for solving systems of partial differential equations. The grid is dynamically adapted during the integration procedure so that only the relevant information is stored. The convection terms are discretised with high-resolution methods, thus ensuring boundedness. The proposed method is general, but is particularly useful for highly convective problems involving sharp moving fronts, a situation that frequently occurs in many chemical engineering problems, and where standard procedures may lead to unphysical oscillations in the computed solution. Numerical results for five test problems are presented to illustrate the efficiency and robustness of the method. The adaptive strategy is found to significantly reduce the computation time and memory requirements, as compared to the fixed grid approach. (C) 2003 Elsevier Science Ltd. All rights reserved.