The Research And Application Of Positivity Scheme For Convection-diffusion Equation On Deformed Meshes

The positive scheme of convection-diffusion equation is mainly researched in this paper.Firstly,for the discretization of diffusive flux under the two-dimensional case,the existed positive scheme[1]is taken as the discrete scheme in this paper.For the discretization of convective flux,a new corrected upwind scheme with second order accuracy is proposed.The method based on the Taylor series expansion at the center of upstream cell.Then the approximation of value at the cell-edge midpoint is finished by using some auxiliary unknowns on some related edges of the cell to reconstruct the gradient.However,the approximate value can not guarantee the nonnegativity in the numerical facet completely,thereby using iterative step to correct.The numerical experiments show that the method achieves second-order accuracy with the case that diffusive coefficient is continuous,discontinuous and anisotropic when we solve the diffusion-dominated or convection-dominated problems on deformed meshes.However,there exist two problems in the above scheme.First,the gradient reconstruction is calculated by using related auxiliary unknowns on some edges of the cell,which these auxiliary unknowns include edge midpoint unknowns and vertex unknowns.However,the calculation of vertex unknown maybe negative,therefore,an improved scheme is proposed to avoid this occurrence completely.Second,although the above corrected upwind scheme and improved scheme sat-isfy the requirement of positivity-preserving,the appearance of corrected step is still considered as a special limitation in essence.Hence,another positive scheme without any limitation is proposed,which is suitable for arbitrary star-shaped meshes.The method based on the second-order reconstruction for the centered value of upstream cell.The positivity of nonlinear coefficient can be preserved completely.Then,the positive scheme of the steady convection-diffusion equation is extended to the unsteady convection-diffusion-reaction equation,and the proof for existence of solution of discrete scheme is given.Numerical experiments show that the results have second-order accuracy and reduce solving time when the larger time step is taken with the reason that the fully implicit scheme is used to discretize scheme.Finally,numerical simulation for highly radionuclide waste repository is car-ried out.We use the fully implicit positive scheme to simulate this coupled problem.Compared with the traditional explicit nine point scheme,implicit nine point scheme and explicit positive scheme,we find that our fully implicit positive scheme has its obvious advantages.