Improved error estimates for splitting methods applied to highly-oscillatory nonlinear Schrödinger equations
TLDR
The main result states that, compared to established error estimates, the Strang splitting method is more accurate by a factor e, provided that the time stepsize is chosen as an integer fraction of the period.Abstract:
In this work, the error behavior of operator splitting methods is analyzed for highly-oscillatory differential equations. The scope of applications includes time-dependent nonlinear Schrodinger equations, where the evolution operator associated with the principal linear part is highly-oscillatory and periodic in time. In a first step, a known convergence result for the second-order Strang splitting method applied to the cubic Schrodinger equation is adapted to a wider class of nonlinearities. In a second step, the dependence of the global error on the decisive parameter 0 < e < < 1, defining the length of the period, is examined. The main result states that, compared to established error estimates, the Strang splitting method is more accurate by a factor e, provided that the time stepsize is chosen as an integer fraction of the period. This improved error behavior over a time interval of fixed length, which is independent of the period, is due to an averaging effect. The extension of the convergence result to higher-order splitting methods and numerical illustrations complement the investigations.read more
Citations
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Error estimates of some splitting schemes for charged-particle dynamics under strong magnetic field
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TL;DR: Under the maximal ordering scaling case, a novel energy-preserving splitting scheme with computational cost per step independent from the strength of the magnetic field is proposed and in fact for a class of Lie-Trotter type splitting schemes, a uniform and optimal error bound is established.
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Error Estimates of Some Splitting Schemes for Charged-Particle Dynamics under Strong Magnetic Field
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TL;DR: In this paper, the error estimates of some splitting schemes for the charged-particle dynamics under a strong magnetic field were considered. And a novel energy-preserving splitting scheme was proposed.
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