The morphological linear stability and nonlinear stability of the solid-liquid interface of an Al-0.3 wt.% Si-0.1 wt.% Cu ternary alloy were investigated via thermodynamics methods. Typical parameters of the phase diagram and different stable regions between the linear stability and nonlinear stability of the solid-liquid interface were discussed via two parameters, which were calculated via the thermodynamic method and the linear parameter method, respectively. The stable regions, unstable regions, and subcritical bifurcation regions of the solid-liquid interface are also presented. The results reveal that the typical parameters of the phase diagram obtained via thermodynamic calculations vary with temperature rather than being linear parameters with constant values. The stable regions calculated by coupling with the thermodynamic method are less than those calculated by linear parameters under the conditions of either linear stability or nonlinear stability theory, and the judgment of solid-liquid interface instability is more stringent. When the perturbation frequency is large enough or small enough, the parameters obtained via thermodynamic calculation and linearized parameters can be substituted for each other, with little impact on the final judgment of solid-liquid interface stability.