Some results on uniformly high-order accurate essentially nonoscillatory schemes
Applied Numerical Mathematics - Special issue in honor of Milt Rose's sixtieth birthday
On the scope of the method of modified equations
SIAM Journal on Scientific and Statistical Computing
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 effects of numerical viscosities. I: slowly moving shocks
Journal of Computational Physics
On postshock oscillations due to shock capturing schemes in unsteady flows
Journal of Computational Physics
Computations of slowly moving shocks
Journal of Computational Physics
Computational Considerations for the Simulation of Shock-Induced Sound
SIAM Journal on Scientific Computing
Unstable Godunov Discrete Profiles for Steady Shock Waves
SIAM Journal on Numerical Analysis
The Convergence Rate of Finite Difference Schemes in the Presence of Shocks
SIAM Journal on Numerical Analysis
Mathematics of Computation
A Remark on Numerical Errors Downstream of Slightly Viscous Shocks
SIAM Journal on Numerical Analysis
Elimination of First Order Errors in Shock Calculations
SIAM Journal on Numerical Analysis
Elimination of First Order Errors in Time Dependent Shock Calculations
SIAM Journal on Numerical Analysis
Analysis of First Order Errors in Shock Calculations in Two Space Dimensions
SIAM Journal on Numerical Analysis
Steady discrete shocks of high-order RBC schemes
Journal of Computational Physics
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The theoretical understanding of discrete shock transitions obtained by shock capturing schemes is very incomplete. Previous experimental studies indicate that discrete shock transitions obtained by shock capturing schemes can be modeled by continuous functions, so called continuum shock profiles. However, the previous papers have focused on linear methods. We have experimentally studied the trajectories of discrete shock profiles in phase space for a range of different high resolution shock capturing schemes, including Riemann solver based flux limiter methods, high resolution central schemes and ENO type methods. In some cases, no continuum profiles exists. However, in these cases the point values in the shock transitions remain bounded and appear to converge toward a stable limit cycle. The possibility of such behavior was anticipated in Bultelle, Grassin and Serre, 1998, but no specific examples, or other evidence, of this behavior have previously been given. In other cases, our results indicate that continuum shock profiles exist, but are very complicated. We also study phase space orbits with regard to post shock oscillations.