Application Of Multiple Integral Finite Volume Method To Some Engineering Problems

In recent years,many practical engineering problems can be described by partial differential equations.Thus the practical problems can be turned into mathematical problems.However,the exact solution of the partial differential equation cannot be found.Even if it can be solved,the boundary requirements are very strict,and the expressions of accurate solution are very complex.Therefore,in recent years,various numerical methods for solving partial differential equations are still hot issues for many scholars.In this paper,we use a numerical method,that is multiple integral finite volume method.With this method,we construct a new numerical scheme for the two equations Benjamin-Bona-Mahony-Burgers(BBMB)and Benjamin-Bona-Mahony-KdV(BBM-KdV)in recent years.We can get different numerical schemes by selecting the upper limit and lower limits of integration,and make theoretical analysis.Firstly,we study BBMB equation with Dirichlet boundary problem.At the time n,we use the multiple integral finite volume method and Lagrange three-point interpolation function to construct an linear numerical scheme with parameters.And we demonstrate the existence and uniqueness of numerical solution.Secondly,we use the same method to study the BBMB equation with the Dirichlet boundary problem at the time n+1/2,we choose the appropriate integration limit to obtain the nonlinear implicit numerical scheme with parameters.The numerical scheme has second order accuracy in time and space.We also demonstrate that the numerical scheme maintains the same energy attenuation as the original equation,the uniqueness,convergence and stability of the solution,and did the numerical experiments.Finally,we study the BBM-KdV equation with periodic boundary.We use the Lagrange three-point and five-point interpolation functions as approximation functions.And we use multiple integral finite volume method and choose the appropriate integration limit to get two-layer implicit and conservative numerical scheme.The accuracy of this scheme is O(?2+h2).Then we demonstrate the conservation of mass and energy of this discrete scheme.The existence,uniqueness,convergence and stability of the solution are analyzed theoretically.At the same time,we verify the correctness of the theorem by numerical experiments.Under the same condition,compared with other literatures,the error of this numerical scheme is much smaller.It can be seen that this scheme is effective.