A characteristic-spectral-mixed scheme for six-dimensional Wigner-Coulomb dynamics

Abstract

Numerical resolution for 6-D Wigner dynamics under the Coulomb potential faces 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 scheme, 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 weakly singular symbol, CHASM further adopts the truncated kernel method to attain a highly efficient approximation. Several typical experiments including the quantum harmonic oscillator and Hydrogen 1s state demonstrate the accuracy and efficiency of CHASM. The non-equilibrium electron-proton couplings are also clearly displayed and reveal the uncertainty principle and quantum tunneling in phase space. Finally, the scalability of CHASM up to 16000 cores is presented.

Distributed local spline simulator for wave propagation

“…The spline expansion is essentially global as it requires solving global algebraic equations with tridiagonal coefficient matrice. Nonetheless, we will show that the cubic spline can be reconstructed by imposing effective inner boundary conditions on the junctions of local patches [25,29].…”

“…(3.8) This can be realized by imposing effective Hermite boundary conditions on two ends of local splines (see Figure 1(b)) [24,29],…”

“…As a pretreatment in LOSS, the domain was evenly decomposed into 4 × 4 × 4 patches, thereby achieving a perfect balance in overload. Here PMBCs were assembled by truncating the neighborhoods at n nb = 20 [29]. A visualization of the domain decomposition in 3-D space is presented in Figure 9(a), when the whole computational domain is decomposed into 2 × 2 × 2 cells.…”

“…The cubic spline achieves spatial fourth-order convergence [26] and its construction can be realized by the standard sweeping method, where the complexity scales linearly with respect to mesh size [27]. For these reasons, the most recent semi-Lagrangian methods are even capable to resolve the kinetic-type equation in full 6-D phase space, e.g., the Vlasov equation [28] and quantum Wigner equation [29].…”

“…This is based on a key observation that the wavelet basis decay exponentially in the dual space [20,22], so that the vanished off-diagonal elements in the inverse coefficient matrix can be truncated. As a consequence, a perfectly matched boundary condition (PMBC) can be introduced to give a closure of patched spline coefficients and allows local splines to recover the global one as accurately as possible [29]. Since only local communications in adjacent neighbors are needed, LOSS is expected to be suitable for the computational clusters with high-latency network, known as the Beowulf machines [17].…”