A stability analysis based on the von Neumann method shows that the Larkin methods for two-dimensional heat conduction using non-uniform grids are conditionally stable, while they are known to be unconditionally stable using uniform grids. The stability criteria consisting of the dimensionless time step Δt, the space intervals Δx, Δy, and the ratios of neighboring space intervals α, β are derived from the stability analysis. A subsequent numerical experiment demonstrates that solutions derived by the Larkin methods using non-uniform grids lose stability and accuracy when the criteria are not satisfied.