Jet schemes are semi-Lagrangian advection approaches that evolve parts of the jet of the solution (i.e., function values and higher derivatives) along characteristic curves. Suitable Hermite interpolations give rise to methods that are high order accurate, yet optimally local, i.e., the update for the data at any grid point uses information from a single grid cell only. Jet schemes can be systematically derived from an evolve-and-project methodology in function spaces, which in particular yields stability estimates. We present a comparison of the accuracy and computational cost of jet schemes with WENO and Discontinuous Galerkin schemes.
For interface evolution problems, jet schemes give rise to gradient-augmented level set methods (GALSM). These possess sub-grid resolution and yield accurate curvature approximations. We demonstrate how the optimal locality of jet schemes gives rise to a straightforward combination with adaptive mesh refinement (AMR), and provide an outlook on jet schemes for nonlinear Hamilton–Jacobi equations.