Abstract(#br)In many hydrodynamical applications, advection is the most important dynamical process. Advected quantities typically develop greatly disparate spatial scales because of the inherent non-linearity of advection. A method is outlined how to tackle this process numerically. The basic idea of the coordinate transformation equation (CTE) method is to introduce a moving coordinate system in physical space before applying any discretisation. This keeps the problem of discretisation separate from the problem of adaptation. In the new coordinate system, which changes in time, the effects on the numerical representation of the advection terms are twofold. Firstly, the advection is with respect to a relative velocity between coordinates and flow, which is arranged to be less than the... absolute velocity, leading to a less restrictive stability criterion for the explicit time stepping compared to the Courant condition. Since the relative velocity is smaller than the absolute velocity, this also leads to a decrease in the importance of the non-linear advection term in comparison with other terms in the equation being solved, and therefore to a decrease in numerical errors from advection. Secondly, a smoothing of the spatial gradients of the quantities to be advected decreases the diffusion due to the discretisation, especially if monotonic (but therefore diffusive) discretisation schemes are used. Illustrations of the applicability and accuracy of the method are given, using comparisons with the results of other advection techniques and analytical solutions of linear advection problems, i.e., problems in which the velocity field is given, in one and two dimensions, including rarefaction and compression waves. The remaining illustrations are of non-linear problems, in which the velocity field is solved for as part of the problem. In such problems the CTE is coupled non-linearly to the equations of motion. However, the main shortcoming of the method is the large computing time required to solve the CTE system.