A Parallel Adaptive Finite-Element Semi-Lagrangian Advection Scheme for the Shallow Water Equations

A Computational Realization of a Semi-Lagrangian Method for Solving the Advection Equation

“…In spite of their advances for hyperbolic equations, these methods have the limitation consisting in a time step restriction, mainly for stability sake. On the other hand, during the last three decades the idea of applying the method of characteristics to advective quantities forward in time has rapidly developed and has gained popularity in many areas [9][10][11][12][13]. In contrast to traditional Eulerian schemes, semi-Lagrangian algorithms provide unconditional stability and allow using large time steps.…”

“…The simplest schemes use the approximation of a trajectory (or curvilinear characteristic) by a straight line and employ a low-order interpolation to compute a numerical solution. Nowadays, simplicity and efficiency of these schemes make them quite popular in different fields of numerical modeling like fluid dynamics applications [9,12,22], shallow water equations [10], fiber dynamics described by the Fokker-Planck equation [11], heatconduction equation [23], and so forth. Now modern semi-Lagrangian algorithms involve a higher-order approximation of a curvilinear characteristic and employ a higher-order interpolation; see, for example, [22].…”