TY - JOUR
T1 - $ℓ^1$-Error Estimates on the Hamiltonian-Preserving Scheme for the Liouville Equation with Piecewise Constant Potentials: A Simple Proof
AU - Li , Xinchun
JO - Journal of Computational Mathematics
VL - 6
SP - 814
EP - 827
PY - 2017
DA - 2017/12
SN - 35
DO - http://doi.org/10.4208/jcm.1701-m2016-0717
UR - https://global-sci.org/intro/article_detail/jcm/10496.html
KW - Liouville equations, Hamiltonian-preserving schemes, Piecewise constant potentials, $ℓ^1$-error estimate, Half-order error bound, Semiclassical limit.
AB - <p style="text-align: justify;">This work is concerned with $ℓ^1$-error estimates on a Hamiltonian-preserving scheme for the Liouville equation with piecewise constant potentials in one space dimension. We provide an analysis much simpler than these in literature and obtain the same half-order convergence rate. We formulate the Liouville equation with discretized velocities into a series of linear convection equations with piecewise constant coefficients, and rewrite the numerical scheme into some immersed interface upwind schemes. The $ℓ^1$-error estimates are then evaluated by comparing the derived equations and schemes.</p>