An efficient algorithm to find the solution of transient convection–diffusion problems with dominant convection is presented. The main idea is to follow the solution front and activate those subdomains where the solution satisfies a given threshold value. We call this novel method ‘the multiplicative Schwarz method with active subdomains’, and it is motivated by the solution of a problem from activated-carbon filters used in the automotive industry to reduce emissions. Numerical experiments show that this method is more efficient than the preconditioned conjugate gradient method with an incomplete Cholesky factorization.