1 Abstract The numerical solution of partial differential equations is often performed on a numerical grid, where the grid points are used for estimating the partial derivatives. The grid can be fully static as in Eulerian type of solution method, or the grid points can move during the solution, which is the case in Lagrangian type of method. In the current article, a numerical solution method is presented , where the grid points are located on iso-contours of the two-dimensional field. The method calculates the local movement of the iso-contours according to an evolution equation described by the PDE, and the solution proceeds by moving the grid points towards the calculated direction. Additional stability is obtained by setting the grid points to move along the iso-contour line. To exemplify the application of the method, numerical examples are calculated for the two-dimensional diffusion equation. 2 Introduction In the previous study [1], the movement of a phase interface was simulated with level-set type method using a physical science based model which takes into account the transformation strains and carbon partitioning and diffusion. In that study the connection to the Allen-Cahn equation [1] was made, which connected it's solution to the level-set approach. As a extension of this idea, the connection between a general partial differential equation (PDE) with first order time derivative and the level set formulation is investigated in the current study. The current approach is closely connected with the level-set method [2], which is often applied for simulations involving phase boundary movement. Level set methods usually perform the solution of a partial differential equation (PDE) in a grid that is not based on the points contained on the isocontours. The level-set method can be used in any types of grids, even in completely Eu-lerian framework [3, 4], where the computational grid does not need to move.