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
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
Aspects of discontinuous Galerkin methods for hyperbolic conservation laws
Finite Elements in Analysis and Design - Robert J. Melosh medal competition
Construction of orthogonal bases for polynomials in Bernstein form on triangular and simplex domains
Computer Aided Geometric Design
Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws
Applied Numerical Mathematics - Special issue: Workshop on innovative time integrators for PDEs
Imposing orthogonality to hierarchic higher-order finite elements
Mathematics and Computers in Simulation
Journal of Computational Physics
Journal of Computational Physics
Optimal Control of the Classical Two-Phase Stefan Problem in Level Set Formulation
SIAM Journal on Scientific Computing
Second-order accurate monotone finite volume scheme for Richards' equation
Journal of Computational Physics
Hierarchical slope limiting in explicit and implicit discontinuous Galerkin methods
Journal of Computational Physics
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A new approach to slope limiting for discontinuous Galerkin methods on arbitrary meshes is introduced. A local Taylor basis is employed to express the approximate solution in terms of cell averages and derivatives at cell centroids. In contrast to traditional slope limiting techniques, the upper and lower bounds for admissible variations are defined using the maxima/minima of centroid values over the set of elements meeting at a vertex. The correction factors are determined by a vertex-based counterpart of the Barth-Jespersen limiter. The coefficients in the Taylor series expansion are limited in a hierarchical manner, starting with the highest-order derivatives. The loss of accuracy at smooth extrema is avoided by taking the maximum of correction factors for derivatives of order p=1 and higher. No free parameters, oscillation detectors, or troubled cell markers are involved. Numerical examples are presented for 2D transport problems discretized using a DG method.