The Stiff Limit of the Exponential Rosenbrock-type Method

In this paper, I explore the exponential Rosenbrock-type method of order two. It is an explicit method that solves stiff ODEs. The method linearizes the equation with each step, then uses a matrix exponential to solve the linearized equation exactly. Previous research has identified stability bounds and convergence results. In this paper, I explore the stiffness capabilities of this second order method as stiffness increases toward infinity. The paper contains estimation-style analysis of the step size and order. I also test the exponential Rosenbrock-type method on two stiff equations: decay of a particle to a circle, and a partial differential equation on a circle with either a viscosity or a hyperviscosity term. Through numerical testing, it is evident that the exponential Rosenbrock-type method exhibits stiffness characteristics in the limit as stiffness tends towards infinity; however, for high stiffness, the method has order reduction to first order.