The maximum norm error estimate and Richardson extrapolation methods of a second-order box scheme for a hyperbolic-difference equation with shifts

Abstract

The study aims at the development and theoretical analyses of a box method used to solve a kind of hyperbolic equations with shifts. By using the discrete energy method, it is shown that numerical solutions converge to exact solutions with an order of O(τ ² + h ²) in L ^∞-norm as τ = O(h). Here, τ and h are temporal and spatial meshsizes, respectively. According to local truncation error, the asymptotic expansion formula of the numerical solutions is derived by introducing two auxiliary problems. By applying this asymptotic expansion formula, a class of Richardson extrapolations methods have been derived to achieve extrapolation solutions with an order of O(τ ⁴ + h ⁴) in L ^∞-norm. Numerical results show the high performance of the proposed methods and the exactness of theoretical findings.