A numerical technique with simultaneous time integration of a meshed grid system is proposed, in which the fine-mesh region is able to move within the coarse-mesh grid. The interface boundary conditions employed are shown analytically to be the only stable specification of those tested for a simple linear case. Numerical experiments with linear and nonlinear systems in one dimension are used to demonstrate the method by which the fine-mesh region is kept centered over a specified disturbance. Forecast results using the meshed system are compared with those from uniform coarse and fine grids. One important criterion is that the solution within the fine-mesh region of the meshed grid must have nearly the same accuracy as in a system which uses a fine mesh everywhere. The technique is applied to a two-dimensional (y, p), ten-level, primitive equation model. Behavior of the meshed model is examined in experiments in which a small-scale heat source is imbedded within an undisturbed zonal flow pattern.... Abstract A numerical technique with simultaneous time integration of a meshed grid system is proposed, in which the fine-mesh region is able to move within the coarse-mesh grid. The interface boundary conditions employed are shown analytically to be the only stable specification of those tested for a simple linear case. Numerical experiments with linear and nonlinear systems in one dimension are used to demonstrate the method by which the fine-mesh region is kept centered over a specified disturbance. Forecast results using the meshed system are compared with those from uniform coarse and fine grids. One important criterion is that the solution within the fine-mesh region of the meshed grid must have nearly the same accuracy as in a system which uses a fine mesh everywhere. The technique is applied to a two-dimensional (y, p), ten-level, primitive equation model. Behavior of the meshed model is examined in experiments in which a small-scale heat source is imbedded within an undisturbed zonal flow pattern....