A Hermite-Type Adaptive Semi-Lagrangian Scheme

Adaptive semi-Lagrangian schemes for solving the Vlasov equation in the phase space have recently been developed. They include wavelet techniques (Gutnic et al., 2004; Gutnic et al., 2005), the moving mesh method (Sonnendrücker et al., 2004), and hierarchical finite element decomposition (Campos Pinto and Mehrenberger, 2004; Campos Pinto and Mehrenberger, 2005). One main advantage of the latter method is that the underlying dyadic partition of cells allows for an efficient parallelization. It has been implemented with a biquadratic Lagrange interpolation. But the use of higher-order methods is not straightforward in that context. The same problem in fact occurs in the case of semi-Lagrangian schemes on unstructured grids. One solution there was to use a Hermitetype interpolation (see (Besse and Sonnendrücker, 2003) and also (Nakamura and Yabe, 1999)). We propose here to do the same in the adaptive context. Thanks to a well chosen Hermite interpolation recently found (Hong and Schumaker, 2004), we thus obtain a more accurate scheme. The paper is organized as follows: Section 2 presents two interpolating operators that we designed for our numerical scheme. First, we recall the Lagrange operator and then we present the new Hermite one. Section 3 briefly recalls our uniform and adaptive semi-Lagrangian schemes. Section 4 focuses on the crucial point of the computational cost of the two operators. We propose efficient algorithms to compute the interpolated value as a sequence of assignments. Section 5 completes the definition of our adaptive scheme. For each operator, criteria for compressing cells of the adaptive mesh are provided. Finally, Section 6 shows our experimental results before concluding.