@article {35164,
	title = {A characteristic-spectral-mixed scheme for six-dimensional Wigner-Coulomb dynamics},
	journal = {SIAM Journal on Scientific Computing},
	volume = {45},
	year = {2023},
	pages = {B906-B931},
	abstract = {



Numerical resolution for 6-D Wigner dynamics under the Coulomb potential is faced with the combined challenges of high dimensionality, nonlocality, oscillation, and singularity. In particular, the extremely huge memory storage of 6-D grids hinders the usage of all existing deterministic numerical schemes, which is well known as the curse of dimensionality. To surmount these difficulties, we propose a massively parallel solver, termed the characteristic-spectral-mixed (CHASM) scheme, by fully exploiting two distinct features of the Wigner equation: locality of spatial advection and nonlocality of quantum interaction. Our scheme utilizes the local cubic B-spline basis to interpolate the local spatial advection. The key is to use a perfectly matched boundary condition to give a closure of spline coefficients, so that distributed pieces can recover the global one as accurately as possible owing to the rapid decay of wavelet basis in the dual space, and communication costs are significantly reduced. To resolve the nonlocal pseudodifferential operator with a singular symbol, CHASM further adopts the truncated kernel method to attain a highly efficient approximation. Several typical experiments including the quantum harmonic oscillator and the 1s state of hydrogen demonstrate the accuracy and efficiency of CHASM. Nonequilibrium electron-proton couplings are also clearly displayed and illuminate the uncertainty principle and quantum tunneling in phase space. Finally, the scalability of CHASM up to 16000 cores is presented.



},
	url = {https://doi.org/10.1137/22M1494294},
	author = {Yunfeng Xiong and Yong Zhang and Sihong Shao}
}
