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

González, C.; Thalhammer, M.

doi:10.1090/S0025-5718-06-01883-7

A second-order Magnus-type integrator for quasi-linear parabolic problems
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by C. González and M. Thalhammer;

Math. Comp. 76 (2007), 205-231

DOI: https://doi.org/10.1090/S0025-5718-06-01883-7

Published electronically: August 15, 2006

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Abstract:

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 $X$ \begin{equation*} u’(t) = A\big (u(t)\big ) u(t), \quad 0 < t \leq T, \qquad u(0) \text { given}, \end{equation*} 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.

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