Conservative methods for the numerical solution of the Vlasov equation are developed in the context of the one-dimensional splitting. In the case of constant advection, these methods and the traditional semi-Lagrangian ones are proven to be equivalent, but the conservative methods offer the possibility to add adequate filters in order to ensure the positivity. In the non constant advection case, they present an alternative to the traditional semi-Lagrangian schemes which can suffer from bad mass conservation, in this time splitting setting.

This paper addresses non-linear gyrokinetic simulations of ion temperature gradient (ITG) turbulence in tokamak plasmas. The electrostatic Gysela code is one of the few international 5D gyrokinetic codes able to perform global, full-f and flux-driven simulations. Its has also the numerical originality of being based on a semi-Lagrangian (SL) method. This reference paper for the Gysela code presents a complete description of its multi-ion species version including: (i) numerical scheme, (ii) high level of parallelism up to 500k cores and (iii) conservation law properties.

SUMMARYThis paper presents an interpolation operator on unstructured triangular meshes that verifies the properties of mass conservation, P 1 -exactness (order 2), and maximum principle. This operator is important for the resolution of the conservation laws in computational fluid dynamics by means of mesh adaptation methods as the conservation properties are not verified throughout the computation. Indeed, the mass preservation can be crucial for the simulation accuracy. The conservation properties are achieved by local mesh intersection and quadrature formulae. Derivatives reconstruction are used to obtain an order 2 method. Algorithmically, our goal is to design a method that is robust and efficient. The robustness is mandatory to apply the operator to highly anisotropic meshes. The efficiency will permit the extension of the method to dimension 3. Several numerical examples are presented to illustrate the efficiency of the approach.