一类四阶非线性抛物方程的紧致差分格式

A COMPACT DIFFERENCE SCHEME FOR A CLASS OF FOURTH ORDER NONLINEAR PARABOLIC EQUATIONS

针对一类四阶非线性抛物方程的初边值问题建立紧致差分格式,利用降阶的思想,通过引入中间变量将原四阶问题转化成二阶非线性方程组.对方程中的时间导数项和空间导数项分别采用Crank-Nicolson格式和四阶紧致差分格式进行离散,对非线性项采用外插的方法进行处理,从而得到原问题的三层线性紧致差分格式,其局部截断误差为Ο(τ^2+h^4).数值算例表明该格式具有良好的计算效果.基于四阶非线性抛物方程在薄膜理论等问题中的重要作用,对此类方程构造高精度的紧致差分格式,可以使该方程在有关工程计算方面得到更好的应用,因此该研究成果具有重要的理论意义和广泛的应用前景.

A compact difference scheme is established for the initial boundary value problem of a class of fourth-order nonlinear parabolic equations.The original fourth-order problem is transformed into a system of second-order nonlinear equations by introducing an intermediate variable with the idea of order reduction.The Crank-Nicolson scheme and the fourth-order compact difference scheme are used to discretize the time and space derivatives respectively and the nonlinear term is processed by extrapolation method.The three-level linear compact difference scheme for the original problem is obtained,of which the local truncation error is O(τ^2+h^4).Finally,the numerical experiment shows the performance of this scheme is satisfying.Based on the important role of the fourth-order nonlinear parabolic equation in the application of thin film theory and so on,the construction of high-precision compact difference scheme for this kind of equation can make the equation get better application in relevant engineering calculations.Therefore,the research results have important theoretical significance and wide application prospect.