Semi-Lagrangian Methods
Semi-Lagrangian methods advect quantities by tracing characteristics backward from fixed grid points, avoiding the explicit CFL time-step limit.
Following characteristics without moving the grid
Advection can be viewed two ways: Eulerian, watching fluid pass fixed grid points, or Lagrangian, following moving parcels. Purely Lagrangian methods keep the physics simple but tangle their grids over time. Semi-Lagrangian methods take the best of both: they keep a fixed Eulerian grid but, each step, trace the characteristic backward from each grid point to find where the arriving parcel came from, then interpolate the old-time solution there to set the new value.
Because the method follows the physical flow of information rather than differencing across it, it is not bound by the CFL condition and can take time steps larger than the flow crosses a cell, a decisive advantage for advection-dominated problems.
The backward-trajectory step
Each step solves an ordinary differential equation for the characteristic backward over one time step from every grid point (the departure point), then interpolates the previous solution at that departure point. The interpolation order controls accuracy and dissipation: linear interpolation is diffusive, while cubic or higher (or shape-preserving) interpolation retains sharp structures at added cost.
Conservation and use
Basic semi-Lagrangian advection is not automatically conservative, since interpolation does not enforce a discrete balance; conservative semi-Lagrangian variants remap fluxes or cell integrals to restore conservation where it is required. The method is prized in atmospheric modeling and in Vlasov solvers for kinetic plasmas, where phase-space advection dominates and large steps are valuable.
- Backward characteristic tracing from fixed grid points
- No CFL step limit for the advective part
- Interpolation order sets accuracy and dissipation
- Conservative variants needed when balance must be exact
Semi-Lagrangian schemes are widely used in grid-based Vlasov and gyrokinetic solvers, where advection in high-dimensional phase space would make explicit CFL-limited stepping prohibitively expensive.