Efficient Numerical Methods For Fourth Order Parabolic Equations

The problem of beam transverse vibration can be reduced mathematically to the initial-boundary value problem of the fourth-order parabolic equation.In order to numerically solve the fourth-order parabolic equation,this paper proposes the finite difference method and the quintic spline element approximation method.The main research contents are as follows:First,based on Taylor expansion,the explicit and implicit three-level difference schemes for solving the fourth-order parabolic equation are obtained,and the order of the local truncation error is given,that is,the convergence speed of the difference scheme.The artificial boundary is introduced to discretize the boundary condition-s,discrete Fourier analysis is used to prove that explicit schemes are conditionally stable and implicit schemes are absolutely stable.Numerical experiments verify the theoretical results.Secondly,based on the variational form of the fourth-order parabolic equation,using the quintic spline function as the shape function,a polynomial spline function space is constructed and solved in this function space.In the space direction,the original problem is transformed into a sparse linear algebraic equation set by using finite element method;by using the central difference method in the time direction and combining with the initial conditions,a fully discrete finite element scheme that can be solved is obtained.Finally,by constructing two auxiliary problems(steady fourth-order problem-s and unsteady fourth-order problems similar to the original problem),using the structure of solutions of second order ordinary differential equations and the simi-larity theory of matrices,the optimal²error estimates for semi-discrete schemes are proved.For fully discrete schemes,the stability conditions of explicit schemes are given,and the²-seminorm estimates of explicit schemes are analyzed.The estimation is optimal for space variables and time variables(space variables are in H²-seminorm,time variables are in H^1,?norm).Numerical experiments verify the theoretical results.