Stochastic DAEs in Circuit Simulation

Abstract

Stochastic differential-algebraic equations (SDAEs) arise as a mathematical model for electrical network equations that are influenced by additional sources of Gaussian white noise. We sketch the underlying analytical theory for the existence and uniqueness of strong solutions, provided that the systems have noise-free constraints and are uniformly of DAE-index 1. In the main part we analyze discretization methods. Due to the differential-algebraic structure, implicit methods will be necessary. We start with a general p-th mean stability result for drift-implicit one-step methods applied to stochastic differential equations (SDEs). We discuss its application to drift-implicit Euler, trapezoidal and Milstein schemes and show how drift-implicit schemes for SDEs can be adapted to become directly applicable to stochastic DAEs. Test results of a drift-implicit Euler scheme with a mean-square step size control are presented for an oscillator circuit.