Point-wise hierarchical reconstruction for discontinuous Galerkin and finite volume methods for solving conservation laws

We develop a new hierarchical reconstruction (HR) method [17, 28] for limiting solutions of the discontinuous Galerkin and finite volume methods up to fourth order of accuracy without local characteristic decomposition for solving hyperbolic nonlinear conservation laws on triangular meshes. The new HR utilizes a set of point values when evaluating polynomials and remainders on neighboring cells, extending the technique introduced in Hu, Li and Tang [9]. The point-wise HR simplifies the implementation of the previous HR method which requires integration over neighboring cells and makes HR easier to extend to arbitrary meshes. We prove that the new point-wise HR method keeps the order of accuracy of the approximation polynomials. Numerical computations for scalar and system of nonlinear hyperbolic equations are performed on two-dimensional triangular meshes. We demonstrate that the new hierarchical reconstruction generates essentially non-oscillatory solutions for schemes up to fourth order on triangular meshes. ‡(E-mail: [email protected]) Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556. Research supported in part by NSF grant DMS-0800612. §(E-mail: [email protected]) School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332. Research supported in part by NSF grant DMS-0810913. ∗∗(E-mail: [email protected]) Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556. ∥(E-mail: [email protected]) Computational Mathematics Group, Pacific Northwest National Laboratory, 902 Battelle Boulevard, Richland, WA 99352. Research was supported by the Advanced Scientific Computing Research Program of the U.S. Department of Energy Office of Science. ¶(E-mail: [email protected]) Division of Applied Mathematics, Brown University, Providence RI 02912. Research supported in part by ARO grant W911NF-08-1-0520 and NSF grant DMS-0809086.