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- ItemAdditive splitting methods for parallel solution of evolution problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Amiranashvili, Shalva; Radziunas, Mindaugas; Bandelow, Uwe; Busch, Kurt; Čiegis, RaimondasWe demonstrate how a multiplicative splitting method of order P can be used to construct an additive splitting method of order P + 3. The weight coefficients of the additive method depend only on P, which must be an odd number. Specifically we discuss a fourth-order additive method, which is yielded by the Lie-Trotter splitting. We provide error estimates, stability analysis, and numerical examples with the special discussion of the parallelization properties and applications to nonlinear optics.
- ItemUnusual ways of four-wave mixing instability(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Amiranashvili, Shalva; Bandelow, UweA pump carrier wave in a dispersive system may decay by giving birth to blue- and red-shifted satellite waves due to modulation or four-wave mixing instability. We analyse situations where the satellites are so different from the carrier wave, that the red-shifted satellite either changes its propagation direction (k < 0, ω > 0) or even gets a negative frequency (k, ω < 0). Both situations are beyond the envelope approach and require application of Maxwell equations.