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__Exponential type integrators for abstract quasilinear parabolic equations with variable domains__

C. González

**Abstract**

In this talk, I propose an exponential explicit integrator for the time discretization of quasilinear parabolic problems. My numerical scheme is based on Runge-Kutta methods. 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)+ b(t), 0 < t \leq T, u(0) {\rm given},
$
involving the sectorial operator $A(v):D(v) \to X$ with variable domains $D(v) \subset X$ with regard to $v \in V \subset X$. Under reasonable regularity requirements on the problem, I analyze the stability and the convergence behaviour of the numerical methods.

**Bibliography**

[1] C. González and M. Thalhammer, A second-order Magnus type integrator for quasilinear parabolic problems, Math. Comp., 76 (2007),
pp. 205-231.

[2] M. Hochbruck and Ch. Lubich, On Magnus integrators for
time-dependent Schrödinger equations, SIAM J. Numer. Anal. 41 (2003), pp. 945-963.

[3] M. Hochbruck and A. Ostermann, Explicit exponential Runge-Kutta methods for semilinear parabolic problems, SIAM J. Numer. Anal. 43 (2005), pp. 1069-1090.

[4] M. Hochbruck and A. Ostermann, Exponential multistep methods of Adams-type, BIT Numer. Math., 51 (2011), pp. 889-908.