Limiters for high-order discontinuous Galerkin methods

We describe a limiter for the discontinuous Galerkin method that retains as high an order as possible, and does not automatically reduce to first order. The limiter is a generalization of the limiter introduced in [R. Biswas, K. Devine, J.E. Flaherty, Parallel adaptive finite element methods for conservation laws, Applied Numerical Mathematics 14 (1994) 255-284]. We present the one-dimensional case and extend it to two-dimensional problems on tensor-product meshes. Computational results for examples with both smooth and discontinuous solutions are shown. (C) 2007 Elsevier Inc. All rights reserved.