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Relaxation Crank–Nicolson compact finite difference schemes for Schrödinger–Poisson systems

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  • Cui, Mingrong

Abstract

Relaxation Crank–Nicolson compact finite difference schemes for solving both one- and two dimensional Schrödinger–Poisson systems are given and analyzed in this paper. Using the idea of relaxation scheme, that is, after introducing an auxiliary variable, we get an additional equation that decouples the nonlinear term in the original system. We discretize the time derivative by Crank–Nicolson scheme with the newly added variable approximated on the staggered time mesh, while compact finite differences are employed to approximate the second-order spatial derivatives, then we obtain the fully discrete relaxation Crank–Nicolson compact finite difference schemes. The resulting linear relaxation schemes have the properties of discrete mass conservation and discrete energy conservation. Numerical experiments demonstrate second-order accuracy in time and fourth-order accuracy in space, confirming the accuracy and efficiency of the proposed algorithm.

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  • Cui, Mingrong, 2026. "Relaxation Crank–Nicolson compact finite difference schemes for Schrödinger–Poisson systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 247(C), pages 225-243.
    • Handle: RePEc:eee:matcom:v:247:y:2026:i:c:p:225-243
    DOI: 10.1016/j.matcom.2026.03.014
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    References listed on IDEAS

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    1. Haidong Qu & Xuan Liu, 2015. "A Numerical Method for Solving Fractional Differential Equations by Using Neural Network," Advances in Mathematical Physics, Hindawi, vol. 2015, pages 1-12, October.
    2. Peter A. Markowich & Christian A. Ringhofer & Christian Schmeiser, 1990. "Semiconductor Equations," Springer Books, Springer, number 978-3-7091-6961-2, January.
    3. Haidong Qu & Xuan Liu, 2015. "A Numerical Method for Solving Fractional Differential Equations by Using Neural Network," Advances in Mathematical Physics, John Wiley & Sons, vol. 2015(1).
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