Accurate and efficient numerical methods for nonlocal problems

In this thesis, we study several nonlocal models to obtain their numerical solutions accurately and efficiently. In contrast to the classical (local) partial differential equation models, these nonlocal models are integro-differential equations that do not contain spatial derivatives. As a result, these nonlocal models allow their solutions to have discontinuities. Hence, they can be widely used for fracture problems and anisotropic problems. This thesis mainly includes two parts. The first part focuses on presenting accurate and efficient numerical methods. In this part, we first introduce three meshless methods including two global schemes, namely the radial basis functions collocation method (RBFCM) and the radial ba- sis functions-based pseudo-spectral method (RBF-PSM) and a localized scheme, namely the localized radial basis functions-based pseudo-spectral method (LRBF-PSM), which also gives the development process of the RBF methods from global to local. The comparison of these methods shows that LRBF-PSM not only avoids the Runge phenomenon but also has similar accuracy to the global scheme. Since the LRBF-PSM uses only a small subset of points, the calculation consumes less CPU time. Afterwards, we improve this scheme by adding enrichment functions so that it can be effectively applied to discontinuity problems. This thesis abbreviates this enriched method as LERBF-PSM (Localized enriched radial basis functions-based pseudo-spectral method). In the second part, we focus on applying the derived methods from the first part to nonlocal topics of current research, including nonlocal diffusion models, linear peridynamic models, parabolic/hyperbolic nonlocal phase field models, and nonlocal nonlinear Schrödinger equations arising in quantum mechanics. The first point worth noting is that in order to verify the meshless nature of LRBF-PSM, we apply this method to solve a two-dimensional steady-state continuous peridynamic model in regular, irregular (L-shaped and Y-shaped) domains with uniform and non-uniform discretizations and even extend this method to three dimensions. It is also worth noting that before solving nonlinear nonlocal Schrödinger equations, according to the property of the convolution, these partial integro-differential equations are transformed into equivalent or approximate partial differential equations (PDEs) in the whole space and then the LRBF-PSM is used for the spatial discretization in a finite domain with suitable boundary conditions. Therefore, the solutions can be quickly approximated