量子Liouville-BGK方程的分裂格式
A splitting scheme for the quantum Liouville-BGK equation
摘要：
在这项工作中，我们介绍了一个有效的数值方法的量子Liouville-BGK方程，它模拟量子粒子的扩散输运的模拟。该模型的基石是BGK碰撞算子，通过在碰撞过程中粒子的局部密度守恒的约束下最小化量子自由能获得。这导致了一个大的系统耦合非线性非局部偏微分方程的分辨率是具有挑战性的。然后，我们定义了一个分裂方案，分离的运输和碰撞部分，利用粒子的局部守恒，导致一个完全线性的碰撞步骤。后者涉及到一个约束最优化问题的解决方案，这是处理与非线性共轭梯度算法。我们证明了时间半离散格式是收敛的，并作为我们数值格式的应用，验证了作为量子Liouville-BGK方程扩散极限的量子漂移扩散模型.
英文摘要：
We introduce in this work an efficient numerical method for the simulation of the quantum Liouville-BGK equation, which models the diffusive transport of quantum particles. The corner stone to the model is the BGK collision operator, obtained by minimizing the quantum free energy under the constraint that the local density of particles is conserved during collisions. This leads to a large system of coupled nonlinear nonlocal PDEs whose resolution is challenging. We then define a splitting scheme that separates the transport and the collision parts, which, exploiting the local conservation of particles, leads to a fully linear collision step. The latter involves the resolution of a constrained optimization problem that is is handled with the nonlinear conjugate gradient algorithm. We prove that the time semi-discrete scheme is convergent, and as an application of our numerical scheme, we validate the quantum drift-diffusion model that is obtained as the diffusive limit of the quantum Liouville-BGK equation.
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