期刊
JOURNAL OF COMPUTATIONAL PHYSICS
卷 235, 期 -, 页码 912-933出版社
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jcp.2012.08.027
关键词
Artificial viscosity; Numerical shock-capturing; Conservation laws; Euler equations; Contact discontinuities
资金
- National Science Foundation [DMS-1001850]
- United States Department of Energy through Idaho National Laboratory LDRD Project [NE-156]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1001850] Funding Source: National Science Foundation
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. Published by Elsevier Inc.
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