Non-oscillatory finite volume methods for conservation laws on unstructured grids.

This work focuses on the use of polyharmonic splines, a class of radial basis functions, in the reconstruction step of finite volume methods. We first establish the theory of radial basis functions as a powerful tool for scattered data approximation. We thereafter provide existing and new results on the approximation order and numerical stability of local interpolation by polyharmonic splines. These results provide the tools needed in the design of the Runge KuttaWeighted Essentially Non-Oscillatory (RK-WENO) method and the Arbitrary high order using high order DERivatives-WENO (ADER-WENO) method. In the RK-WENO method, a WENO reconstruction based on polyharmonic splines is coupled with Strong Stability Preserving (SSP) Runge-Kutta time stepping. The polyharmonic spline WENO reconstruction is also used in the spatial discretisation of the ADER-WENO method. Here, the time discretisation is based on a Taylor expansion in time where the time derivatives are replaced by space derivatives using the Cauchy-Kowalewski procedure. The high order °ux evaluation of the ADER-WENO method is achieved by solving generalized Riemann problems for the spatial derivatives across cell interfaces. Adaptive formulations of the RK-WENO and ADER-WENO methods are used to solve advection problems on unstructured triangulations. An a posteriori error indicator is used to design the adaptation rules for the dynamic modification of the triangular mesh during the simulation. In addition, the flexibility of the stencil selection strategy for polyharmonic spline reconstruction is utilised in developing a WENO reconstruction method with stencil adaptivity. Finally, order variation procedures are combined with mesh adaptation in order to handle regions of the computational domain where the solution is smooth in a different fashion from the vicinity of singularities and steep gradients with the goal of delivering accurate solutions with less computational effort and fewer degrees of freedom when compared to adaptive methods with fixed order of reconstruction.