A semi-Lagrangian scheme for convection equations sing the finite element method

This paper considers a semi-Lagrangian scheme (SLS) for solving convection equations. The transport equations, as written in the Lagrangian form at each time step, are approximated on the basis of a weak form using a finite element method representation with various coordinate functions: delta functions, piecewise-linear functions, and various interpolation methods: those based on piecewise-linear functions and third order B-splines. For the two-dimensional case, the calculation of two-dimensional B-spline reduces to solving a set of the algebraic equations of the dimension less than initial system allows one to reduce the calculation time. For the schemes considered, test calculations have been conducted, the results for the one- and two-dimensional cases are presented.