A second-order Magnus-type integrator for quasi-linear parabolic problems

In this paper, we consider an explicit exponential method of classical order two for the time discretisation of quasi-linear parabolic problems. The numerical scheme is based on a Magnus integrator and requires the evaluation of two exponentials per step. Our convergence analysis includes parabolic partial differential equations under a Dirichlet boundary condition and provides error estimates in Sobolev spaces. In an abstract formulation the initial boundary value problem is written as an initial value problem on a Banach space $X$

$u'(t) = A\big(u(t)\big) u(t), \quad 0 < t \leq T, \qquad u(0)$    given$,$

involving the sectorial operator $A(v):D \to X$ with domain $D \subset X$ independent of $v \in V \subset X$. Under reasonable regularity requirements on the problem, we prove the stability of the numerical method and derive error estimates in the norm of certain intermediate spaces between $X$ and $D$. Various applications and a numerical experiment illustrate the theoretical results.