A Study Of Gridless Methods For Solving Partial Differential Equations

Gridless methods and their applications for the solution of partial differential equations are studied in this thesis. Only clouds of points instead of grids are distributed over the computational domain and the spatial derivatives of the considered partial equations are estimated using a least-square curve fit on local clouds of points. As an example, the implementation of the Laplace equation with the gridless method has been presented at first and the resulting large scale matrix equations are solved by GMRES algorithm. The numerical simulations of the flows over a cylinder are tested successfully with clouds of different scales, which shows the 'cloud' effects on the computational accuracy. The further extension of the method for solving Euler's equations is then presented with five-stage Runge-Kutta time-stepping scheme, studying the artificial dissipation model of unstructured mesh method. Convergence is accelerated by means of local time stepping, implicit residual smoothing. The numerical results have been obtained for the flows over N AC AGO 12 or RAE2822 airfoils. The thesis ends with researches of the object-oriented design patterns of gridless software, which bases on the cloud structure. In order to reuse the software design and make it more flexible, the base function classes have been divided, and a series of common design patterns have been put forward in the frame design of gridless software.