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Exact splitting methods for kinetic and Schrödinger equations. (English) Zbl 1459.35341
The authors discuss some PDE involving a differential operator \(p^w\) whose symbol (in the Weyl quantization) is a polynomial function of degree 2 or less. It is said that the operator \(p^w\) can be computed by exact splitting if it can be factored into operators having a special basic form (see Definition 1 in the paper).
Having an exact splitting in hand it is possible to discretize the PDE in question, by solving via pseudo-spectral methods or pointwise multiplications, and the discretized solution will converge to an actual solution of the PDE.
The authors discuss transport equations, Focker-Planck equations, magnetic Schrödinger equations and rotating Gross-Pitaevskii equations. Each of these equations involve an operator with a quadratic symbol, for which the authors prove the exact splitting along with numerical results on the solutions.
MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q49 Transport equations
35Q84 Fokker-Planck equations
82C40 Kinetic theory of gases in time-dependent statistical mechanics
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs
Software:
GPELab
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References:
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