Stable spectral methods for conservation laws on triangles with unstructured grids Original Research Article

his paper presents an asymptotically stable scheme for the spectral approximation of linear conservation laws defined on a triangle. Lagrange interpolation on a general two-dimensional nodal set is employed and, by imposing the boundary conditions weakly through a penalty term, the scheme is proven stable in L2. This result is established for a general unstructured grid in the triangle. A special case, for which the nodes along the edges of the triangle are chosen as the Legendre Gauss—Lobatto quadrature points, is discussed in detail. The eigenvalue spectrum of the approximation to the advective operator is computed and is shown to result in an Ҫ(n−2) restriction on the time-step when considering explicit time-stepping.