Abstract: In this work we use an alternative form for the state transition matrix from a differential equation system, the {\it Non-modal Matrix}. This matrix form is derived and applied to time integration of a barotropic limited area model, the DYNAMO model. We start from a matrix formulation for DYNAMO model, which is described as a non-linear matrix differential equation. It can be solved by an analytical non-modal matrix solution. The Helmholtz wind field reconstruction is made by using the singular value decomposition in conection with Moore-Penrose generalized inverse.