An algorithm is presented to model two-dimensional, non-isothermal, low Mach number flows with a local steep density gradient. The algorithm uses an adaptive, locally refined, non-staggered grid and has been developed, especially for modelling laminar flames. The governing equations, based on a stream-function–vorticity formulation, are presented and discretized using hybrid finite differences. A (isothermal) test problem is presented to compare the accuracy of the results of the solver presented in this paper, with the results of algorithms found in the literature. However, this test problem proves to be not well suited for the application of a locally refined grid, since it does not contain a local steep gradient. For this reason an additional test problem is constructed that clearly shows the advantages of the locally refined grid as compared to a uniform grid with respect to both the calculation time as well as the number of grid nodes needed. Furthermore, a laminar premixed flame is modelled with simple chemistry to show that the algorithm, presented in this paper, converges to a stabilized flame when an adaptive grid technique is used.