The ${l}^1$-Error Estimates for a Hamiltonian-Preserving Scheme for the Liouville Equation with Piecewise Constant Potentials

Abstract

We study the $l^1$-error of a Hamiltonian-preserving scheme, developed in [S. Jin and X. Wen, Commun. Math. Sci., 3 (2005), pp. 285–315], for the Liouville equation with a piecewise constant potential in one space dimension. This problem has important applications in computations of the semiclassical limit of the linear Schrödinger equation through barriers, and of the high frequency waves through interfaces. We use the $l^1$-error estimates established in [X. Wen and S. Jin, J. Comput. Math., 26 (2008), pp. 1–22; X. Wen, Convergence of an Immersed Interface Upwind Scheme for Linear Advection Equations with Piecewise Constant Coefficients II: Some Related Binomial Coefficient Inequalities, preprint, 2008] for the immersed interface upwind scheme to the linear advection equations with piecewise constant coefficients. We prove that the scheme with the Dirichlet incoming boundary conditions is $l^1$-convergent for a class of bounded initial data and derive the half order $l^1$-error bounds with explicit coefficients. The initial conditions can be satisfied by applying the decomposition technique proposed in [S. Jin, H. L. Liu, S. Osher, and R. Tsai, J. Comput. Phys., 205 (2005), pp. 222–241] for solving the Liouville equation with measure-valued initial data, which arises in the semiclassical limit of the linear Schrödinger equation.