On the boundedness stepsizes-coefficients of A-BDF methods

Abstract

Physical constraints must be taken into account in solving partial differential equations (PDEs) in modeling physical phenomenon time evolution of chemical or biological species. In other words, numerical schemes ought to be devised in a way that numerical results may have the same qualitative properties as those of the theoretical results. Methods with monotonicity preserving property possess a qualitative feature that renders them practically proper for solving hyperbolic systems. The need for monotonicity signifies the essential boundedness properties necessary for the numerical methods. That said, for many linear multistep methods (LMMs), the monotonicity demands are violated. Therefore, it cannot be concluded that the total variations of those methods are bounded. This paper investigates monotonicity, especially emphasizing the stepsize restrictions for boundedness of A-BDF methods as a subclass of LMMs. A-stable methods can often be effectively used for stiff ODEs, but may prove inefficient in hyperbolic equations with stiff source terms. Numerical experiments show that if we apply the A-BDF method to Sod's problem, the numerical solution for the density is sharp without spurious oscillations. Also, application of the A-BDF method to the discontinuous diffusion problem is free of temporal oscillations and negative values near the discontinuous points while the SSP RK2 method does not have such properties.

Keywords

• monotonicity
• linear multistep method
• total-variation-diminishing
• total-variation-bounded
• method of lines
• a-bdf method