Active Flux Methods on Manifolds

In recent years, Active Flux method, first introduced by T. Eymann and P. Roe, has been adapted to solve many problems for hyperbolic systems of PDEs on orthogonal and polygonal meshes. Typically, in Active Flux two types of mesh variables are used: cell averages and point values at nodes and edges of the mesh (in 2D planar case). The evolution of cell averages is approximated with a finite volume scheme for conservative form of hyperbolic equations, and the evolution of point values is handled with a finite difference scheme for the characteristic form of equations. In such way, both conservative and characteristic nature of the equations is captured in the numerical method.
In this talk, we introduce the generalization of Active Flux method on triangular meshes to hyperbolic problems on a sphere. For the finite volume part, we rewrite the fluxes using the tangent vectors instead of normals to get a geometry-compatible scheme. For the point values update part, we use local projections of spherical triangles and introduce a quasi-polynomial reconstruction of mesh functions on planar projected triangles to find the gradients. Some tests for linear hyperbolic problems on a sphere are demonstrated. It is worth noting that the use of local projections allows to generalize this method to problems on an arbitrary manifold.