Stability Analysis of the Matrix-Free Linearly Implicit Euler Method

Summary

Implicit time stepping methods are useful for the simulation of large scale PDE systems because they avoid the time step limitations imposed by explicit stability conditions. To alleviate the challenges posed by computational and memory constraints, many applications solve the resulting linear systems by iterative methods where the Jacobian-vector products are approximated by finite differences. This paper explains the relation between a linearly implicit Euler method, solved using a Jacobian-free Krylov method, and explicit Runge-Kutta methods. The case with preconditioning is equivalent to a Rosenbrock-W method where the approximate Jacobian, inverted at each stage, corresponds directly to the preconditioner. The accuracy of the resulting Runge-Kutta methods can be controlled by constraining the Krylov solution. Numerical experiments confirm the theoretical findings.