An Adaptable Discontinuous Galerkin Scheme for the Wigner-fokker-planck Equation
نویسندگان
چکیده
The right hand side Q ~,FP (w) models the averaged environmental intera tions with the system and is referred to as the Quantum Fokker-Plan k operator. The operator Θ[V ] is a pseudo-di erential operator and takes into a ount the nonlo al a tion of the potential V . In this paper we propose a Dis ontinuous Galerkin approximation for the above problem. The omputation applies to a wide range of approximation spa es and does not rely on a basis of polynomials. We present also estimates showing onvergen e and stability of the s heme. The Dis ontinuous Galerkin (DG) approa h proposed here provides several opportunities to optimize the approximation spa e. In parti ular, the use of non-polynomial basis fun tions, as proposed by Yuan and Shu in [28℄, allows for improvement over mesh re nement, in reased polynomial order, and global or lo al basis set adjustments. The method is suitable to be adjusted to unstru tured grids in spa e and time. The basis set may be a priori or adaptively optimized, depending on the spe i ir umstan es of the al ulation. Taken to the extreme, this allows the method to transition from a traditional DG solver to an essentially spe tral solver. For example, to study perturbations of the harmoni potential one ould use the known eigenfun tions of the harmoni ase.
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