## Section: New Results

### Numerical analysis and simulation of heterogeneous systems

Participants : Xavier Antoine, Qinglin Tang.

In [1], we propose a simple accelerated pseudo-spectral algorithm to compute the stationary states of the Gross-Piteavskii Equation (GPE) with possibly multiple components. The method is based on the adaptation of new optimization algorithms under constraints coming from mathematical imaging to the imaginary time (gradient-like) method for the GPE arising in Bose-Einstein Condensation.

In [3] we propose original efficient preconditioned conjugate gradient methods coming from molecular physics to the GPE for spectrally computing the stationary states of the GPE. The method allows a gain of a factor 100 for 3D problems with extremely large nonlinearities and fast rotations. The HPC solver is being developed.

In [17], we develop new robust and efficient algorithms for computing the dynamics of 2-components GPEs with dipolar interaction. The main particularity of the method is that high accuracy is obtained by a new FFT based evaluation of nonlocal kernels applied to the nonlinear part of the operator.

In [4], we propose an asymptotic mathematical analysis of domain decomposition techniques for solving the 1D nonlinear SchrÃ¶dinger equation and GPE. The analysis uses advanced techniques related to fractional microlocal analysis for PDEs. Simulations confirm the mathematical analysis.

In [2], we extend, by some very technical mathematical analysis, approaches for the results stated in [4]. Again, numerical simulations validate the theoretical analysis.

In [5], we develop and implement in parallel simple new solvers for computing the dynamics of solutions to the Dirac equation arising in quantum physics. Numerical examples are developed to analyze the capacity of these algorithms for a parallel implementation.

In [6], we introduce the concept of Absorbing Boundary Conditions and Perfectly Matched Layers for the dynamics of nonlinear problems related to classical and relativistic quantum wave problems (Wave equation, SchrÃ¶dinger equation, Dirac eqution). In particular, we show application examples and detail the methods so that they can be implemented by researchers coming for quantum physics.