Schrödinger-Poisson方程守恒DDG方法研究

Schrödinger-Poisson equations are nonlinear equations of a wide range of applications. Direct discontinuous Galerkin (DDG) method is one of Discontinuous Galerkin methods (DG). The DG discretization results in an extremely local, element based approximation, well suited for hp-adaptivity, maintaining high order of accuracy. They also have excellent provable structure preserving property. A key ingredient of this method is the suitable design of the interelement boundary treatments (the so-called numerical fluxes) to obtain high order accurate and stable schemes. In this project, we mainly study the mass conservation and energy conservation by direct discontinuous Galerkin methods for Schrödinger -Poisson equations. We search the numerical fluxes such that mass conservation and energy conservation, and prove the conservation for both semi discrete scheme and fully discrete scheme. For space discretization, we use DDG method, and for time discretization, we use the Strang splitting method. We study the error estimate for the one dimensional discrete scheme, and extend the results to higher dimension.

Schrödinger-Poisson方程是一种应用广泛的非线性方程。直接间断有限元(DDG)方法是间断有限元(DG)方法的一种。使用DDG方法离散后的结果是一个非常局部的基于有限元的逼近，有非常好的自适应性，并能保持高阶精度，也有出色的可证明的保结构特性。DDG方法的关键在于设计合适的数值流来获得高精度的稳定格式。本项目主要研究Schrödinger-Poisson方程的保守恒性的DDG解法。我们寻求使得质量守恒和能量守恒的数值流，并证明其半离散格式和全离散格式的守恒性，在空间上使用DDG离散，在时间上采用Strang splitting离散，对一维半离散格式进行误差分析，并将结果推广至高维。