Computer Methods in Applied Mechanics and Engineering
Uniformly high order accurate essentially non-oscillatory schemes, 111
Journal of Computational Physics
Uniformly high-order accurate nonoscillatory schemes
SIAM Journal on Numerical Analysis
Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Non-oscillatory central differencing for hyperbolic conservation laws
Journal of Computational Physics
The behavior of flux difference splitting schemes near slowly moving shock waves
Journal of Computational Physics
The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection
Computer Methods in Applied Mechanics and Engineering
Weighted essentially non-oscillatory schemes
Journal of Computational Physics
Accurate upwind methods for the Euler equations
SIAM Journal on Numerical Analysis
Efficient implementation of weighted ENO schemes
Journal of Computational Physics
Nonoscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws
SIAM Journal on Scientific Computing
Weighted essentially non-oscillatory schemes on triangular meshes
Journal of Computational Physics
Comparison of Several Difference Schemes on 1D and 2D Test Problems for the Euler Equations
SIAM Journal on Scientific Computing
Journal of Computational Physics
A subcell remapping method on staggered polygonal grids for arbitrary-Lagrangian-Eulerian methods
Journal of Computational Physics
Accurate monotonicity- and extrema-preserving methods through adaptive nonlinear hybridizations
Journal of Computational Physics
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications
Journal of Computational Physics
Shock capturing with PDE-based artificial viscosity for DGFEM: Part I. Formulation
Journal of Computational Physics
Efficient implementation of essentially non-oscillatory shock-capturing schemes, II
Journal of Computational Physics
Hi-index | 31.45 |
We introduce a new methodology for adding localized, space-time smooth, artificial viscosity to nonlinear systems of conservation laws which propagate shock waves, rarefactions, and contact discontinuities, which we call the C-method. We shall focus our attention on the compressible Euler equations in one space dimension. The novel feature of our approach involves the coupling of a linear scalar reaction-diffusion equation to our system of conservation laws, whose solution C(x,t) is the coefficient to an additional (and artificial) term added to the flux, which determines the location, localization, and strength of the artificial viscosity. Near shock discontinuities, C(x,t) is large and localized, and transitions smoothly in space-time to zero away from discontinuities. Our approach is a provably convergent, spacetime-regularized variant of the original idea of Richtmeyer and Von Neumann, and is provided at the level of the PDE, thus allowing a host of numerical discretization schemes to be employed. We demonstrate the effectiveness of the C-method with three different numerical implementations and apply these to a collection of classical problems: the Sod shock-tube, the Osher-Shu shock-tube, the Woodward-Colella blast wave and the Leblanc shock-tube. First, we use a classical continuous finite-element implementation using second-order discretization in both space and time, FEM-C. Second, we use a simplified WENO scheme within our C-method framework, WENO-C. Third, we use WENO with the Lax-Friedrichs flux together with the C-equation, and call this WENO-LF-C. All three schemes yield higher-order discretization strategies, which provide sharp shock resolution with minimal overshoot and noise, and compare well with higher-order WENO schemes that employ approximate Riemann solvers, outperforming them for the difficult Leblanc shock tube experiment.