Compact Difference Method For A Class Of Fourth-order Nonlinear Parabolic Equation

Fourth-order nonlinear parabolic equations,which have received more and more attention in recent years,are widely used in the fields of thin film theory,flame propagation,bistable phase transition and high-order diffusion and so on.Due to the high-order and nonlinear properties of such equations,it is difficult to obtain their exact solutions.Therefore,it has important theoretical and practical significance to study the efficient numerical methods for this kind of equations.In this thesis,we mainly study the initial boundary value problem of the following fourth-order nonlinear parabolic equation:(?)where ? is a bounded domain with piecewise smooth boundary in R~N(N=1,2),? is the boundary of ?.p is an integer greater than ?_?(t)(?=1,2,3) are sufficiently smooth functions with positive upper and lower bounds.(?) is a given function.This thesis mainly includes two parts,in which compact difference method for the above problem in one and two-dimensional cases are considered respectively.Firstly,we adopt the idea of order reduction to treat the fourth-order spatial derivative term,that is,the original fourth-order equation is transformed into a second-order coupled system by introducing an intermediate variable.Then,we adopt the fourth-order compact difference method to discretize the spatial derivative term.At the same time,the Crank-Nicolson method is used to discretize the time derivative term in the one-dimensional case,and the implicit central difference method is used in the two-dimensional case.Thus,two compact difference schemes are established.Next,we prove the existence,uniqueness and boundedness of the numerical solutions.Futhermore,the unconditional stability and convergence rate of (?) of the schemes are proved by energy analysis method.Finally,the effectiveness of the schemes are verified by numerical examples.