Quantization-based new integration methods for stiff ordinary differential equations

Abstract

In this paper we introduce new classes of numerical ordinary differential equation (ODE) solvers that base their internal discretization method on state quantization instead of time slicing. These solvers have been coined quantized state system (QSS) simulators. The primary result of the research described in this article is a first-order accurate QSS-based stiff system solver, called the backward QSS (BQSS). The numerical properties of this new algorithm are discussed, and it is shown that this algorithm exhibits properties that make it a potentially attractive alternative to the classical numerical ODE solvers. Some simulation examples illustrate the advantages of this method. As a collateral result, a first-order accurate QSS-based solver designed for solving marginally stable systems is briefly outlined as well. This new method, called the centered QSS (CQSS), is successfully applied to a challenging benchmark problem describing a high-order system that is simultaneously stiff and marginally stable. However, the primary emphasis of this article is on the BQSS method, that is, on a stiff system solver based on state quantization.