Compact Volume Method For Two Kinds Of Interface Problems Of Forth-order Equations

Interface problems are widely used in many theoretical studies and engineer-ing computations,such as heat transfer in different media in the process of heat conduction,complex geological structure or multi-phase fluid in permeability me-chanics leading to the problem of miscible driving with intermittent permeability or diffusion coefficient,fluid movement in different permeability regions in reservoir simulation test.Therefore,the study of numerical solutions of interface problems has important theoretical and practical significance.Compact finite volume method not only has the advantages of pressiving local conservation and being simple to implement,but also has the advantage of higher accuracy.We mainly studies the compact volume method for two types of fourth-order equations interface problems.Because the fourth-order equations are converted into second-order systems by introducing intermediate variables when dealing with the fourth-order derivative terms,and then the compact volume method is adopted for each second-order equation,we first study the following one-dimensional second-order parabolic equation interface problem:(?) where β(x)≥βmin>0 is the piecewise constant,(?)In order to solve the above problems,the fourth-order compact volume scheme is used in space,and the second-order difference scheme is applied in time.The com-pact volume scheme of the problem is constructed,and the stability of the schemes is proved by using energy analysis method.The effectiveness of the method is verified by numerical examples.Secondly,the compact finite volume method for two kinds of fourth order e-quations is studied.(1)The following two-point boundary value interface problems for fourth-order equations are considered:(?) where β(x)≥βmin>0 is the piecewise constant,(?)In order to solve the above problem,the second-order derivative in space is treated as an intermediate variable,and the problem is transformed into a second-order system.For each equation in the system,the fourth-order compact volume method is used to discretize the second-order spatial derivative,and the compact volume scheme of the problem is obtained.The convergence analysis of the scheme is given,and the effectiveness of the method is verified by numerical examples.(2)The following initial boundary value interface problems for the fourth order parabolic equation is considered:(?) where β(x)≥βmin>0 is the segmentation constant,(?)For the abovc equation,the second-order spatial derivative is introduced as an intermediate variable,and the problem is transformed into the second-order system.In order to discretize each equation in the system,the first-rder time derivative is discretized by the difference with second-order accuracy about time,and the second-order spatial derivative is discretized by the fourth-order compact volume method.A compact volu1e scheme of the problem with truncation error of O(h4+γ2)is obtained.The effectiveness of the method is verified by numerical examples.