Conditional stability of Larkin methods with non-uniform grids

Stability analysis based on the von Neumann method showed that the Larkin methods for two-dimensional heat conduction with non- uniform grids are conditionally stable while they are known to be unconditionally stable with uniform grids. The stability criteria consisting of the dimensionless time step ∆t, the space intervals ∆x, ∆y, and the ratios of neighboring space intervals α, β were derived from the stability analysis. A subsequent numerical experiment demonstrated that solutions derived by the Larkin methods with non-uniform grids lose stability and accuracy when the criteria are not satisfied