A linearized compact finite difference scheme for Schrödinger- Poisson System

Abstract

In this paper, a novel high accurate and efficient finite difference scheme is proposed for solving the Schr\u00f6dinger-Poisson System. Applying a local extrapolation technique in time to the nonlinear part \u00a0makes \u00a0 the \u00a0proposed \u00a0scheme \u00a0linearized \u00a0in \u00a0the \u00a0implementation. In \u00a0fact, at \u00a0each \u00a0time \u00a0step, only two tri-diagonal linear \u00a0systems \u00a0of \u00a0algebraic \u00a0equations \u00a0are \u00a0solved \u00a0by \u00a0using \u00a0Thomas \u00a0method. \u00a0Another \u00a0feature \u00a0of \u00a0the \u00a0proposed method \u00a0is \u00a0the \u00a0high \u00a0spatial \u00a0accuracy \u00a0on \u00a0account \u00a0of \u00a0adopting \u00a0the \u00a0compact \u00a0finite \u00a0difference \u00a0approximation \u00a0to discrete the system in space. Moreover, the proposed scheme \u00a0preserves \u00a0the \u00a0total \u00a0mass \u00a0in \u00a0discrete \u00a0sense. Under \u00a0certain \u00a0regularity \u00a0assumptions \u00a0of \u00a0the exact \u00a0solution, the \u00a0local \u00a0truncation \u00a0error \u00a0of \u00a0the \u00a0proposed \u00a0 scheme \u00a0is \u00a0analyzed \u00a0in \u00a0detail \u00a0by \u00a0using Taylor\u2019s \u00a0expansion, and \u00a0consequently \u00a0the \u00a0optimal \u00a0error \u00a0estimates \u00a0 of \u00a0the \u00a0numerical \u00a0solutions \u00a0are established by using the standard energy method and a mathematical induction argument. \u00a0The \u00a0convergence \u00a0order \u00a0is \u00a0of \u00a0O(\u03c4 \u00a02 \u00a0+ \u00a0h4) \u00a0in \u00a0the \u00a0discrete \u00a0L2-norm \u00a0and \u00a0L\u221e-norm, \u00a0respectively. Numerical \u00a0results \u00a0are \u00a0reported \u00a0to \u00a0measure \u00a0the \u00a0theoretical \u00a0analysis, which \u00a0shows \u00a0that \u00a0the \u00a0new scheme is accurate and efficient and it preserves well the total mass and energy.