The Positivity-preserving Interpolation Algorithm Of The Vertex Unknowns For Diffusion Problems

In this paper,we mainly study the interpolation algorithm of the vertex auxiliary un-knowns in centered finite volume schemes for the diffusion equation.Firstly,some interpo-lation algorithms of vertex unknowns are introduced,including EMPFA,LPEW(LPEWI&LPEWⅡ),etc.And the positivity of the interpolation algorithms is discussed briefly.Then in order to construct a positivity-preserving interpolation algorithm for the vertex unknowns by suggesting a new algorithm for the parameter τk in EMPFA,a new interpolation algorith-m is obtained.When the diffusion coefficient is a scalar,the new interpolation algorithm can be strictly proved to be positivity-preserving on the triangular mesh.Numerical examples verify the effectiveness of the interpolation algorithm and the correctness of the theoretical results.Then,based on LPEWⅡ,and by using the new algorithm of τk in EMPFA-τk another new interpolation algorithm LPEWⅡ-τk is obtained.Under certain assumptions,it can be proved that LPEWⅡ-τk is positivity-preserving on polygonal mesh,and the numerical results are consistent with the theoretical analysis.Finally,based on LPEWⅡ-τk,a nonlinear extremum-preserving scheme is constructed.When the diffusion coefficient is scalar,it can be strictly proved that the scheme satisfies the discrete extremum-preserving principle,and the theoretical analysis is validated by numerical experiments.