# Magnus expansion for linear operators

I want to learn about Magnus expansion for the time dependent pdes of the form $u_t(t,x)=A(t)u(t,x)$. According to the wikipedia explanation http://en.wikipedia.org/wiki/Magnus_expansion $A$ has to be a matrix. However, I don't want to descritize in $x$ dimension to get a matrix which approximates $A$, I want to keep it as an operator. So, does the theory still holds by replacing a matrices by operators or I might get in trouble? I don't know much about operator theory but I want to make sure I can apply Magnus expansion to the case of operators as well.

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## 1 Answer

You can take $A$ to be an operator. However, you are likely to run into problems regarding the convergence of the series given by the Magnus expansion. As the Wikipedia article states, the series converges if $\int_0^T \|A(t)\| \,\mathrm{d}t < \pi$. If $A$ is an operator containing spatial derivatives, then it is unbounded, and the series will not converge.

However, the Magnus expansion may still be useful. Even in cases where the series is not proven to converge, the first few terms can still be a good approximation. This is shown in a particular case by Hochbruck and Lubich, On Magnus Integrators for Time-Dependent Schrödinger Equations, http://dx.doi.org/10.1137/S0036142902403875 (the details of that paper do depend on operator theory, but you should be able to understand the numerical experiments without any knowledge of operator theory).

Generally, the last of the four references in the Wikipedia article is a good review of the Magnus expansion and its application (disclosure: it cites a paper of mine).

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thanks. Assuming $A(t):=a(t)u_x(t,x)+b(t)u_{xx}(t,x)$ and I want to apply Lie splitting to that. In order to calculate the splitting error I do the Taylor expansion and compare the two(for the solution itself and the result of splitting). However, how do boundary conditions come in play? Assume I solve it on $[0,T]\times[x_{m}, x_M]$ and I have $$u(t,x_m)=v(t,x),u(t,x_M)=w(t,x), u(0,x)=f(x)$$This is what I don't find for the Magnus expansion and splitting methods. They have to affect the Taylor expansion in some way, don't they? – Medan Sep 30 '12 at 13:30