Avoiding stability-induced inefficiencies in BDF methods

Abstract

Time-dependent partial differential equations are often treated by semidiscretization and the resulting problem solved using existing ordinary differential equation software restricted to low-order formulas. For certain classes of problems, the Backward Differentiation Formulas (BDF) are often dismissed due to their poor stability behavior near the imaginary axis for orders three and above. We explain why and for what problems this happens, what the appropriate tactic should be, and why this is not the tactic taken by most automatic codes. We present an idea that avoids this inefficiency in one automatic code.