Mixed Finite Volume Methods For Solving One-dimensional Forth-order Evolution Equations

Finite volume methods discretize differential equations from their integral form of conservation law, which has a very good property of preserving the mass conservation. Because the methods have less cost and high accuracy, they are widely used in numerical partial differential equations.This paper consists of two parts, first, we consider the following forms of equations.Starting from the integral conservative form of the equation, we derive a mixed volume scheme. The truncation error estimates are discussed in detail.It's proved that the scheme has first order accuracy in discrete H~1semi-norm and discrete L~2 norm by a very clear method. Numerial exampleillustrates the effectiveness of the scheme.In second part, we consider the fourth order evolution equation with convective term.We use method of characteristics to approximate the convective term. And the other terms are discretized from the integral form of the equation. We discuss the truncation error and present an Uzawa iterative procedure. It'sproved that the scheme has first order accuracy in discrete H~1 semi-normand discrete L~2 norm by a very clear method .Finally, one example shows that the method is effective.