A hybridized discontinuous Petrov–Galerkin scheme for scalar conservation laws

We present a hybridized discontinuous Petrov–Galerkin (HDPG) method for the numerical solution of steady and time-dependent scalar conservation laws. The method combines a hybridization technique with a local Petrov–Galerkin approach in which the test functions are computed to maximize the inf-sup condition. Since the Petrov–Galerkin approach does not guarantee a conservative solution, we propose to enforce this explicitly by introducing a constraint into the local Petrov–Galerkin problem. When the resulting nonlinear system is solved using the Newton–Raphson procedure, the solution inside each element can be locally condensed to yield a global linear system involving only the degrees of freedom of the numerical trace. This results in a significant reduction in memory storage and computation time for the solution of the matrix system, albeit at the cost of solving the local Petrov–Galerkin problems. However, these local problems are independent of each other and thus perfectly scalable. We present several numerical examples to assess the performance of the proposed method. The results show that the HDPG method outperforms the hybridizable discontinuous Galerkin method for problems involving discontinuities. Moreover, for the test case proposed by Peterson, the HDPG method provides optimal convergence of order kC1. Copyright © 2012 John Wiley & Sons, Ltd.