Nonlinear Finite Volume Method Preserving Maximum Principle

For solving the diffusion equation with discontinuous coefficient,we mainly develop three kind of nonlinear finite volume method in this paper,which all can preserve maximum principle.Firstly,we construct a new scheme,cell-center-discontinuous-edge coupling nonlinear finite volume scheme preserving maximum principle,whose primary unknowns include both cell-center unknowns and edge unknowns on discontinuous line.The system overcomes the defect of the existing schemes preserving maximum principle,which have restrictions on diffusion coefficient or the location of discontinuous line.We prove that the scheme satisfies the discrete maximum principle.Based on this,the existence of a solution for the scheme is also obtained.Numerical results are presented to show that our scheme obtains almost second order accuracy for solution on random meshes,preserves discrete maximum principle,and is superior to the existing scheme preserving the maximum principle on dealing with the problems with discontinuous coefficient on distorted meshes.Secondly,we design a cell-center-edge coupling nonlinear finite volume scheme preserving maximum principle.Not only does the system take cell-center unknowns as basic unknowns,but also the whole cell-edge unknowns.By using the continuity of normal flux,we observe the relation formulation between cell-center unknowns with cell-edge unknowns.Numerical results prove that this scheme results in second order accuracy for solution and satisfies discrete maximum principle.Thirdly,we transform the nonlinear finite volume scheme preserving positivity into a new scheme to satisfy maximum principle.Comparing the matrix element between finite volume schemes preserving maximum and schemes preserving positivity,we separate a part of the two-point flow of the positive format and transform coefficients into nonnegative ones.Then the flow is transformed into a structure preserving maximum principle.Numerical results testify that this nonlinear finite volume scheme makes the solution convergent rate be up to second and preserves maximum principle.