Stability and error estimates for the $\theta$-method for strongly monotone and infinitely stiff evolution equations

Summary.

For evolution equations with a strongly monotone operator $F(t,u(t))$ we derive unconditional stability and discretization error estimates valid for all $t>0$. For the $\theta$-method, with $\theta = 1-\frac{1}{2+ \zeta \tau^\nu }, 0<\nu \leq 1, \zeta > 0$, we prove an error estimate $O(\tau^{\frac{4}{3}}), \tau \rightarrow 0$, if $\nu = \frac{1}{3}$, where $\tau$ is the maximal integration step for an arbitrary choice of sequence of steps and with no assumptions about the existence of the Jacobian as well as other derivatives of the operator $F(\cdot,\cdot)$, and an optimal estimate $O(\tau^2)$ under some additional relation between neighboring steps. The first result is an improvement over the implicit midpoint method $(\theta = \frac{1}{2})$, for which an order reduction to $O(\tau)$ sometimes may occur for infinitely stiff problems. Numerical tests illustrate the results.