On exponential splitting methods for semilinear abstract Cauchy problems. (English) Zbl 1520.47120
Summary: Due to the seminal works of M. Hochbruck and A. Ostermann [Appl. Numer. Math. 53, No. 2–4, 323–339 (2005; Zbl 1070.65099); Acta Numerica 19, 209–286 (2010; Zbl 1242.65109)], exponential splittings are well established numerical methods utilizing operator semigroup theory for the treatment of semilinear evolution equations whose principal linear part involves a sectorial operator with angle greater than \(\frac{\pi}{2}\) (meaning essentially the holomorphy of the underlying semigroup). The present paper contributes to this subject by relaxing the sectoriality condition, but in turn requiring that the semigroup operators act consistently on an interpolation couple (or on a scale of Banach spaces). Our conditions (on the semigroup and on the semilinearity) are inspired by the approach of T. Kato [Math. Z. 187, 471–480 (1984; Zbl 0545.35073)] to the local solvability of the Navier-Stokes equation, where the \(\mathrm{L}^p\)-\(\mathrm{L}^r\)-smoothing of the Stokes semigroup was fundamental. The present abstract operator theoretic result is applicable for this latter problem (as was already the result of Hochbruck and Ostermann), or more generally in the setting of Hochbruck and Ostermann [2005, loc.cit.], but also allows the consideration of examples, such as non-analytic Ornstein-Uhlenbeck semigroups or the Navier-Stokes flow around rotating bodies.
MSC:
| 47N40 | Applications of operator theory in numerical analysis |
| 47J35 | Nonlinear evolution equations |
| 65J08 | Numerical solutions to abstract evolution equations |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
semilinear Cauchy problems; exponential splitting methods; convergence order; \(C_0\)-semigroups; scales of Banach spacesReferences:
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