Upwind Finite Volume Method On Rectangular Grids

Considering the convection-diffusion problems in two-dimensional spacewhere ? ? R2,diffusion coefficient:D = D(x,y)0<d0 ? D(x,y)? d1,convection coefficient:v = v(x,y)=(v1(x,y),v2(x,y)?.Then we make a rectang-ular partition on the computational domain and the fl is divided into some small re-ctangles.In this paper,we construct three upwind schemes for solving convection-diffusion equations.Including the pure upwinding finite volume method and the wei-ghted upwinding finite volume method.The dual nodes of the pure upwinding finite volume method are the midpoint of the rectangular grids.The weighted upwinding is based on a new dual partition who depending on the optimal weighted factor a,and the factor depends on the Pelcet number.We take bilinear finite element space as the trial function space,save as Uh,and the piecewise constant function space as the test function space,save as Vh.Than we have,for uh E Uh a(uh,wh)+ bi(uh,wh)=(f,wh),(?)wh ? Vh where i = 1,2,3.a(uh,wh)=?KP0*? ?h*?(?)KP0*Wh[-D?uh·n]ds,(f,wh)= ??fwhdxdy When i = 1,2,the format belongs to the pure upwinding FVM,and we havewhere uh+ and uh+ are the value of the pure upstream point and the value of the pure upstream.We will promote the conclusions of the pure upwinding FVM on triangular grids to the pure upwinding FVM on rectangular grids.In addition to stability,we can also get the following error estimatesTheorem 1(H~1 error estimate)|u =uh|1?Ch|u|2Theorem 2(max error estimate)?u-uh???Ch|u|2Inference 1(L~2 error estimate)?u-uh?0?Ch|u|2 Where u is the solution of the convection-diffusion problems,uh is the solution of FVM schemes.When i = 3,the format belongs to the weighted upwinding FVM,we havethan,about the weighted upwinding FVM on rectangular grids we have the following theoremsTheorem 3(H~1 error estimation)|u-uh|1?Ch|u|2Theorem 4(L~2 error estimation)?u-uh?0?Ch2(|u|2+h|u|3+h?u?w3,?)Theorem 5(max error estimation)?u-uh???Ch2|u|2Comparing the error estimation of the pure upwinding FVM and the the error estimation of weighted upwinding FVM we can see:for the pure upwinding FVM,the convergence order of L~2 and the convergence order of max error are all 1 order,but for the weighted upwinding,they have reached the best second order convergence.For the three different upwinding formats,we have did some numerical experiments.The experimental results of the error estimation are given respectively.The experimental results are consistent with the above theory.This also shows that the weighted upwinding FVM has a higher order of accuracy,and guarantees the convergences to achieve the best.