A Study Of Parallel Difference Scheme For Parabolic Equation And An Iterative Method For Imperfect Contact Interface Problem

The content of this thesis consists three points:(1)Conservative parallel difference scheme devising and theoretical analysis for parabolic equations;(2)Positivity-preserving parallel difference scheme devising and theoretical analysis for parabolic equations;(3)An iterative method and its theoretical analysis for imperfect contact interface problems.In the first part,we generalize the conclusion for one dimension case by analysing the scheme proposed in the pioneers' work.First we propose a weighted numerical flux and the range of the weight,then we extend the scheme to 2-dimension and theoretical analysis are also given,at last we generalize the scheme to n-dimension(n>3).At the end of this part,some numerical tests are carried out to verify the theoretical results and the experiments demonstrate that this kind of schemes are of unconditional stability with second-order accuracy.What is more,they are intrinsically parallel and conservative.In the second part,we introduce the conception of " Similar to Implicit scheme based on Nodes(SIN)" first.By summarising the pioneers' work,we classify the frameworks of domain decomposition into two groups:UP-DOWN schemes and DOWN-UP schemes.We devise the schemes in one,two,three and higher-dimension by following these two devising ideas.All the schemes we proposed can deduce to one dimension case,especially we give the theoretical analysis for one dimension and give the efficient conditions for positivity-preserving scheme.What's more,we give the convergence analysis by using properties of compact spaces.At the end of this part,some numerical tests axe given to verify our theory.The numerical results indicate our schemes are of unconditional stability with second-order accuracy and the schemes are of intrinsic parallelism and positivity-preserving.In the third part,we discuss a kind of interface problems which has imperfect contact interface and propose an iterative method for it.Convergence in theory is given in general domains and the convergence speed is also be calculated in a special domain.The iterative procedure is convergent at a geometric rate and the iterative process is extremum-preserving.We demonstrate the robustness of the method with some numerical experiments in the end of this part.