We establish an original framework for the stability and convergence analysis of implicit and implicit-explicit variable-step BDF3 schemes for linear and nonlinear parabolic equations. By constructing a positive recursion functional, we derive a novel gradient decomposition for the variable-step BDF3 method, and prove its positive definiteness when \(0.5\le \rho _n\le 1.7319\) , where \(\rho _n\) is the adjacent step-size ratio. This significantly improves on existing results. Under this ratio restriction, we obtain robust stability and optimal error estimates utilizing a concise energy method in an abstract setting. The main ingredients in the proofs include the reformulation of the schemes via a class of discrete orthogonal convolution kernels, global properties of the discrete kernels, and novel convolution type inequality tools. The theoretical barrier, originating from the lack of an analytic expression and uniform positivity of the orthogonal kernels and their scaling counterparts, is addressed by using the elliptic matrix norm. This essentially extends the orthogonal kernel based framework used for the analysis of the second-order BDF method. Numerical results are included to support the analysis. The proposed framework is applicable to a wide range of nonlinear parabolic equations and demonstrates a promising approach for higher-order linear multistep methods with variable step-sizes.
