scholarly journals Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes

2018 ◽  
Vol 11 (4) ◽  
pp. 1037-1062 ◽  
Author(s):  
Weizhu Bao ◽  
◽  
Chunmei Su ◽  
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Error Estimates ◽  
Difference Method ◽  
Massless Limit ◽  
Uniform Error ◽  
Klein Gordon ◽  
Schrödinger System ◽  
Uniform Error Estimates

Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrödinger-Poisson System

2013 ◽  
Vol 13 (5) ◽  
pp. 1357-1388 ◽  
Author(s):  
Yong Zhang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Error Estimates ◽  
Difference Method ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Optimal Error Estimates ◽  
Optimal Error ◽  
Compact Finite Difference ◽  
Compact Finite Difference Method

AbstractWe study compact finite difference methods for the Schrodinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave function and external potential V(x). The Crank-Nicolson compact finite difference method and the semi-implicit compact finite difference method are both of order Ҩ(h4 + τ2) in discrete l2,H1 and l∞ norms with mesh size h and time step t. For the errors ofcompact finite difference approximation to the second derivative andPoisson potential are nonlocal, thus besides the standard energy method and mathematical induction method, the key technique in analysisis to estimate the nonlocal approximation errors in discrete l∞ and H1 norm by discrete maximum principle of elliptic equation and properties of some related matrix. Also some useful inequalities are established in this paper. Finally, extensive numerical re-sults are reported to support our error estimates of the numerical methods.

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