Highly Accurate Difference Methods For Solving The Definite Solution Problems Of The Fourth-order Parabolic Equations With Three Kinds Of Boundary Value Conditions

This paper mainly studies the highly accurate difference methods for solving the definite solution problems of the fourth-order parabolic equations with three different kinds of boundary value conditions.At first,a compact difference scheme for solving the initial-boundary value problem of the fourth-order parabolic equation with the third Neumann boundary value conditions is derived.The values of the first spatial derivative and the third spatial derivative of the unknown function at the boundary are provided.Both the temporal and spatial directions are discretized by weighted average,and the order reduction method is used in the spatial direction.The values of the second spatial derivative at three adjacent points are weighted averaged at the inner points,and the values of the second spatial derivative at two adjacent points are weighted averaged.Then a compact difference scheme is established with the help of the given boundary value conditions and the governing equation.The energy method is used to analyze the unique solvability,convergence and stability of the obtained scheme.It is strictly proved that the convergence order of the scheme in the maximum norm is two in time and four in space.Then a highly accurate computational scheme for the case with the reaction coefficient independent of both time and space is provided.The numerical accuracy and effectiveness of the schemes are verified by two examples.Secondly,a compact difference scheme for solving the first Neumann initial-boundary value problem of the fourth-order parabolic equation is established.The values of the first spatial derivative and the second spatial derivative of the unknown function at the boundary are provided.The Crank-Nicolson type discretization is carried out in the temporal direction,and the spatial second derivative is introduced as an auxiliary function.The original spatial fourth-order problem is equivalent to a second-order system of differential equations.Then a weighted average operator is introduced to achieve the compact approximation in the spatial direction.The energy method is applied to show the unique solvability,convergence and stability of the scheme.Then it is generalized to the case with the reaction term,and a compact difference scheme is derived along with its computational scheme only involving the original unknowns.The accuracy and effectiveness of the difference schemes are verified by numerical examples.It is concluded that the proposed compact difference schemes can achieve the global fourth-order convergence in space.Finally,several compact difference schemes for solving the initial-boundary value problem of the fourth-order parabolic equation with the third Dirichlet boundary value conditions are studied.The values of the unknown function and its third spatial derivative at the boundary.In the spatial direction,several useful numerical differential formulas to approximate the fourth derivative are proposed based on the weighted average and high degree Hermite interpolation techniques.On this basis,three high order compact difference schemes for solving the initial-boundary value problem of the fourth-order parabolic equation with the third Dirichlet boundary conditions are established,and the stability of the developed schemes is briefly analyzed by Fourier method.The numerical accuracy and effectiveness of the schemes are verified by two numerical examples.The global convergence order of the proposed three compact difference schemes can reach two in time and four in space.