Application Of Moving Mesh Method And Layer-adapted Grids For Several Singularly Perturbed Problems

Singularly perturbed problems are arising from fields such as ffuid mechanics,elasticity, acoustics, optics, chemical reaction, optimal control, etc. The charac-teristics of such problems are that the differential equations contain perturbedparameters, which can reffect the physical character of the equations or can beintroduced artificially. The solution of such problems changes quickly in somelocal region of the solving region. Moving mesh method and layer-adapted gridscan solve these problems effectively. In recent years, many scholars at home andabroad have achieved many important results for such problems.In this paper, we study the application of moving mesh method and layer-adapted grids for several singularly perturbed problems based on several refer-ences. In the foreword, We introduce the background of singularly perturbedproblems, and research status of moving mesh method and layer-adapted gridsfor these problems, then introduce the paper's work brieffy.The main body is composed of three parts. In the first part, we constructdifference scheme on moving mesh of a class of singularly perturbed problem, andprove that the numerical solution of difference scheme is more than first-order con-vergence by Richardson extrapolation. We improve two algorithms, and comparethe accuracy and running time of the two algorithms by numerical experiments.In traditional algorithms, the initial mesh of moving meshes is uniform mesh. Wechange it to B-S mesh which is suitable for the model problem, and compare therunning time and accuracy under the two conditions by numerical experiments.We get the result that B-S mesh is better than uniform mesh as initial mesh inrunning time and accuracy.In the second part, we construct Crank-Nicolson difference scheme on mov-ing mesh for a class of non-constant singularly perturbed convection-diffusionproblem with source term, and introduce the iterative method and algorithm ofmoving mesh. We compare error on moving mesh and uniform mesh by numericalexperiments, and prove that it can decrease numerical oscillation effectively thatusing Crank-Nicolson difference scheme on moving mesh.In the last part, We construct difference scheme of an elliptic singularlyperturbed problem. Using decomposition of solution, we analysis each part onsubregions and prove that the numerical solution of difference scheme is uniformly first-order convergent to the exact solution by comparison principle on B-S mesh.