Stability of Variable-Step BDF2 and BDF3 Methods
Zhaoyi Li, Hong-lin Liao
Abstract
We prove that the two-step backward differentiation formula (BDF) method is stable on arbitrary time grids; while the variable-step three-step backward differentiation formula scheme is stable if almost all adjacent step ratios are less than 2.553. These results relax the severe mesh restrictions in the literature and provide a new understanding of variable-step BDF methods. Our main tools include the discrete orthogonal convolution kernels and an elliptic type matrix norm.
Topics & Concepts
MathematicsVariable (mathematics)Backward differentiation formulaConvolution (computer science)Norm (philosophy)Stability (learning theory)Applied mathematicsScheme (mathematics)Mathematical analysisDifferential equationOrdinary differential equationComputer scienceArtificial neural networkCollocation methodPolitical scienceMachine learningLawNumerical methods for differential equationsAdvanced Numerical Methods in Computational MathematicsMatrix Theory and Algorithms