Application Of Multiple Integral Finite Volume Method To Several Problems In Engineering

In the engineering field,partial differential equations(PDEs)have been widely used in the description of various practical problems.The practical problems can be mathematically depicted via the study of its determining solution,further providing an effect solution approach for these problems.Normally,there is no analytic expression for the solution of PDEs.Even if some PDEs have analytic solutions,the process for solving them is very complex.In recent years,the numerical solution study of PDEs has attracted lots of attentions from domestic and foreign scholars.In this paper,an improved finite volume method(multiple integral finite volume method)is used for the systematical and detailed investigation of two important engineering equations with initial and boundary value problems.Several kinds of flexible and adjustable numerical schemes with variable parameters are obtained respectively.Meanwhile,the appropriateness of the numerical schemes is also analyzed.Firstly,the discrete schemes for the nonlinear Burgers equation are constructed at the time n-level,through using the multiple integral finite volume method combined with three-point Lagrange interpolation polynomial and four-point Newton interpolation polynomial,respectively.We obtain three-point and four-point linear numerical schemes with variable parameters,and further prove the existence and uniqueness of solution for these numerical schemes.The range and optimal value of parameters?,?are analyzed and determined via numerical experiments.In addition,the error and energy analysis of current numerical solution are carried out.Some comparisons with other literatures are also presented in this paper,and results show that the error of four-point numerical scheme is smaller than the case of three-point numerical scheme.Secondly,the multiple integral finite volume method and the three-point Lagrange interpolation polynomial are used for the discreteness of Burgers equation on the n+1/2level of the time layer with the aim to obtain a non-linear numerical scheme with parameters.The accuracy of the scheme isO(?~2+h~2).The solvability and energy conservation of the numerical scheme are also proved.Numerical experiments show that the numerical scheme is effective.Finally,the discrete schemes with parameters for the Rosenau-RLW equation are construct-ed through using the multiple integral finite volume method and three-point and five-point Lagrange interpolation polynomial.The solvability,uniqueness of solution,conservation of energy,stability and convergence for the numerical scheme are strictly proved.In the numerical experiments,comparisons with other literatures are carried out,and results show that current scheme has better energy conservation and higher accuracy.It is believed that current work will provide a new and feasible numerical method for the future engineering research.