High Order Accurate Numerical Methods For Cahn-Hilliard Equations

Cahn-Hilliard equation is an important forth order nonlinear di?usion equa-tion, Since we cannot hope for analytical solutions, we must resort to numericalmethods to solve the Cahn-Hilliard equation. Even numerical solutions are noteasy to obtain. This is partly because the strong nonlinear behavior and thedependence of the solution on the initial data u(x,0) is very sensitive.Our mainwork is that discuss the numerical solutions.First in preface, we introduce the backgrounds and significances of Cahn-Hilliard equation, the present studies of home and abroad reserchers are summa-rized, introduces main idea of our work, spread out rudimentary knowledge thatbe used.The main body of the article composed of three chapters,which study initial-boundary value problem of Cahn-Hilliard equation and a kind of Cahn-Hilliardequation with methods have high degree of accuracy.The chapter two considers the one dimension initial-boundary value prob-lem of Cahn-Hilliard equation. It using B spline Galerkin method discretes thespacial variate. Consequently, it constructs the finite element formats for thisfourth order equation, then proves the two famous properties that the total massconserved and the total free energy decreases with time. We proves that ap-proximate solution is bounded at maximum norm by Sobolev lemma and obtainsO(h~4 +Ï„~2) errors.The chapter three,we constructs the two kinds compact finite di?erencescheme for Cahn-Hilliard equation. The two new schemes also inherit the conser-vation of mass properties. Then, by method of inductive assumption, we provethe stability and convergence of the numerical solution, the order of error isO(h~4 +Ï„~2) too. The chapter four, we use B-spline method to solve the EOM equation whichalso describe the phase separation, prove the stability and convergence of thenumerical solution with energy method.