一类非线性抛物型方程的紧差分格式

A COMPACT DIFFERENCE SCHEME FOR NON-LINEAR PARABOLIC EQUATION

本文研究了一维非线性抛物型方程的紧差分格式.首先将非线性项线性化,并参照线性抛物型方程的紧差分格式的推导思路导出了非线性抛物型方程的紧差分格式,并给出了截断误差表达式.其次用能量方法分析了紧差分格式,导出了先验估计式,证明了差分格式的可解性、稳定性和收敛性,确定收敛阶为O(τ2+h4).然后将Richardson外推法应用于紧差分格式,外推一次得到具有O(τ4+τ2h4+h6)阶精度的近似解.最后通过数值算例,表明非线性抛物型方程的紧差分格式及其外推格式具有较高的收敛精度.

A compact difference scheme for non-linear parabolic equation is proposed. Firstly, the non-linear term in the parabolic equation is linearized, and a compact difference scheme for non-linear parabolic equation is derived referred to the derivation of the compact difference scheme for linear parabolic equation. The truncation error of the scheme is analyzed. Secondly, a priori estimation of the solution of the compact difference scheme is given using the energy analysis method, which proves the solvability, stability and convergence of the scheme. The convergence order is O (r2 +/i4). Thirdly, the Richardson * s extrapolation method is applied to the compact difference scheme and the solution with accuracy O (r4 + r 2/i4 H-/i6) is obtained through once extrapolation. Finally, the high accuracy of the compact difference scheme for non-linear parabolic equation and its extrapolation scheme is proved through a concrete numerical example.