High order semi-Lagrangian particle methods for transport equations: numerical analysis and implementation issues (Q2877714)

The subject of this paper is a transport equation of the following form NEWLINE\[NEWLINE{{\partial u}\over{\partial t}}+\text{div} (au)=0,\;\;x\in \mathbb R^d,\, t>0,\leqno(1) NEWLINE\]NEWLINE where \(a\) is given smooth velocity field. The authors propose to apply for an approximate solution of \((1)\) the remeshed particle method. The particle starts from a given regular grid \(x_i=i\Delta\), \(x\;i\in\mathbb Z^d\), moves according to the rule \(x_i^{n+1}=x_i+\bar{a}_i^n\Delta t\), where \(\bar{a}_i^n\) is evaluation of the field \(a\) at \(x_i\), and then returns on the grid after some operation of interpolation. The quality of this interpolation process plays a very important role in this algorithm: convergence and stability of the whole algorithm depends on it.NEWLINENEWLINEThe authors proppose to split the computational process according to space directions and then to build the algorithm with the help of one-dimensional transport processes. This allows to reach a much shorter computation time than in the case of a full multidimensional process. The paper contains some indications concerning the above mentioned interpolation process, as well as a discussion of stability and convergence of the method. Some information about methods of implementation on multiprocessor computers is given. A numerical illustration is added.