A dynamic grid adaptation (DGA) scheme is developed using various combinations of the multidimensional positive definite advection transport algorithm (MPDATA) to show the applicability of DGA with the MPDATA scheme to solve advection problems. A one-dimensional model is used to show the effects of varying the number of grid points and the parameters that control the grid redistribution scheme, to determine a stability criteria for the scheme and to investigate the effect of several MPDATA options. A two-dimensional model is used to show the applicability of the scheme in multiple dimensions and to illustrate the effects of DGA in combination with MPDATA options. Diffusion errors are reduced by more than 90% using DGA when compared to static, uniformly spaced grid computations. Phase errors are reduced using certain MPDATA options by more than 25%.