Figure 1. Convergence of the Arbitrary Grid Legendre Galerkin Method on Various Grids

FIGURE 2. Divergence of the arbitrary grid Legendre Galerkin method for improperly imposed initial conditions. Time-stable boundary conditions for nite-diierence schemes solving hyperbolic systems: methodology and application to high-order compact schemes, JCP 111, 220, (1994). 25 To initialize the problem, we must construct an approximation to the initial condition f(x), based on the grid points x j (0 j N). We want to keep the exibility and rigid structure of the original grid distribution; however, the interpolation polynomial, based on the grid points x j , generally is not convergent. Therefore, we use the method outlined in (49) and (52). With this initialization, spectral convergence is recovered. 7 Conclusions A new technique for implementing spectral methods for hyperbolic equations has been developed that does not require grid points that are nodes of some Gauss quadrature formula. For this reason, this method is referred to as an arbitrary-grid spectral method. Both Galerkin and collocation formulations are presented for the conventional Legendre method, and a Galerkin formulation is presented for the conventional Laguerre method. The basis for the stability of the unstructured spectral schemes relies on a weighted energy norm in all cases. Stability is proven for the constant coeecient hyperbolic system. All unstructured spectral methods utilize a \weak" imposition of the boundary condition, similar to the technique used in the penalty formulations of the nite element method. With this imposition, the complete diierentiation matrix, including boundary conditions, is similar to (i.e., it has the same eigenvalues) the conventional diierentiation operator; therefore, this matrix behaves similarly. The new formulations are demonstrated on two scalar hyperbolic problems. The arbitrary-grid Legendre Galerkin method is used in both cases. Exponential accuracy is shown in both cases on arbitrary grids. Care must be exercised in the initialization procedure to ensure convergence of the new schemes. 24 Figure 1 shows the reenement study on ve diierent grids: