Parallel, adaptive finite element methods for conservation laws
Proceedings of the third ARO workshop on Adaptive methods for partial differential equations
Convergence to steady state solutions of the Euler equations on unstructured grids with limiters
Journal of Computational Physics
High-resolution conservative algorithms for advection in incompressible flow
SIAM Journal on Numerical Analysis
Total variation diminishing Runge-Kutta schemes
Mathematics of Computation
The Runge-Kutta discontinuous Galerkin method for conservation laws V multidimensional systems
Journal of Computational Physics
Compatible fluxes for van Leer advection
Journal of Computational Physics
A problem-independent limiter for high-order Runge—Kutta discontinuous Galerkin methods
Journal of Computational Physics
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws
Applied Numerical Mathematics - Special issue: Workshop on innovative time integrators for PDEs
On a robust discontinuous Galerkin technique for the solution of compressible flow
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Explicit and implicit FEM-FCT algorithms with flux linearization
Journal of Computational Physics
A vertex-based hierarchical slope limiter for p-adaptive discontinuous Galerkin methods
Journal of Computational and Applied Mathematics
Failsafe flux limiting and constrained data projections for equations of gas dynamics
Journal of Computational Physics
A multiscale discontinuous galerkin method
LSSC'05 Proceedings of the 5th international conference on Large-Scale Scientific Computing
Journal of Computational Physics
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In this paper, we present a collection of algorithmic tools for constraining high-order discontinuous Galerkin (DG) approximations to hyperbolic conservation laws. We begin with a review of hierarchical slope limiting techniques for explicit DG methods. A new interpretation of these techniques leads to an unconditionally stable implicit algorithm for steady-state computations. The implicit global problem for the mean values (coarse scales) has the computational structure of a finite volume method. The constrained derivatives (fine scales) are obtained by solving small local problems. The interscale transfer operators provide a two-way iterative coupling between the solutions to the global and local problems. Another highlight of this paper is a new approach to compatible gradient limiting for the Euler equations of gas dynamics. After limiting the conserved quantities, the gradients of the velocity and energy density are constrained in a consistent manner. Numerical studies confirm the accuracy and robustness of the proposed algorithms.