Conservation and Accuracy in Meshfree Generalized Finite Difference Methods.

Through this thesis, a comprehensive study of meshfree Generalized Finite Difference Methods (GFDMs) is done, which a focus on applications to fluid flow problems modeled by the incompressible Navier - Stokes equations. New methods are presented to solve over-determined problems, to introduce conservation, to improve accuracy in fluid flow solvers, and to improve accuracy in the Lagrangian movement process.

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The goal of this PhD is to improve various aspects of meshfree Generalized Finite Difference Methods (GFDMs). In this thesis, different meshfree GFDMs are compared, and their potential to solve over-determined problems is presented. A new method is presented that introduces conservation of fluxes in a meshfree setting, which reduces the problem of lack of conservation that has plagued meshfree methods. Special attention is paid on the application of meshfree GFDMs to simulate fluid flow modeled by the incompressible Navier - Stokes equations. A new meshfree GFDM scheme for the same is presented which improves local accuracy, and shows better approximations to the mass conservation condition. Further, different aspects of meshfree Lagrangian frameworks are studied, and new methods to improve accuracy in the Lagrangian movement process are also presented.