Physics – Computational Physics
Scientific paper
2012-04-03
Physics
Computational Physics
24 pages, 21 figures
Scientific paper
We introduce the $C$-method, a simple scheme for adding localized, space-time smooth, artificial viscosity to nonlinear systems of conservation laws which propagate shock waves, rarefactions, and contact discontinuities. In particular, we focus our attention on the compressible Euler equations which form a 3x3 system 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 both the location and strength of the added viscosity. Near shock discontinuities, $C(x,t)$ is large and localized, and transitions smoothly in space-time to zero away from the shock. This simple approach has two fundamental features: (1) our regularization is at the continuum level--i.e., the level of he partial differential equations (PDE)-- so that any higher-order numerical discretization scheme can be employed, and (2) we avoid the use of Riemann-type or characteristic solvers near strong shocks. Additionally, the $C$-method has a straightforward generalization to the multi-D setting, and does not require dimensional splitting. We demonstrate the effectiveness of the $C$-method with two significantly different numerical implementations and apply these to a collection of classical problems. First, we use a classical continuous finite-element implementation using second-order discretization in both space and time, FEM-C, and second, we employ a simplified WENO scheme within our $C$-method framework, WENO-C. Both 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 characteristic solvers near strong shocks.
Reisner Jon
Serencsa Jonathan
Shkoller Steve
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