5.4 Two-Step Shock-Capturing Strategy
5.4.1 Basic Properties
In this section, we discuss the basic properties of the two-step shock-capturing strategy by investigating the linear scalar transport equation and the nonlinear inviscid Burgers’ equation.
Scalar transport equation We consider the one-dimensional linear scalar transport equation for some scalar quantityψ(x1, t) :R→R, written in the form
∂ψ(x1, t)
∂t +u1∂ψ(x1, t)
∂x1
= 0, (5.29)
where the advection velocity is assumed to beu1 = 1. We choose the discontinuous initial conditions
ψ0(x1) =ψ(x1,0) =
{︄0.5, ifx1 ≤0.45,
1, ifx1 >0.45, (5.30)
on a pseudo-two-dimensional computation domainΩ = [0,1]×[0,1], being periodic inx2 -direction, with a characteristic grid size ofh= 0.1for a polynomial degree ofP = 2. In this setting, the initial discontinuity atx1= 0.45does not coincide with a cell boundary. We keep the complexity as low as possible for showing the basic properties so that we activate artificial viscosity by a simple on/off-switch
ε′j(x1, t) =
{︄0, ifSj ≤S0,
ε0, ifSj > S0, (5.31)
using a piece-wise constant version in contrast to Equation (5.23). We apply the explicit Euler scheme (4.7) for time-integration, where the maximum admissible time size∆tis calculated adaptively according to Equation (4.26).
This test case, being simple at first glance, is challenging in the context of shock-capturing strategies based on artificial viscosity, since the scalar transport equation (5.29) is not self-steepening in contrast to the Burgers’ equation (5.34) or the Euler equations (2.5). As a result, an initial discontinuity will never return to its former shape, once it is smoothed by artificial viscosity even if shock-capturing is deactivated. Consequently, a comparably high h/P-resolution is needed in order to obtain sub-cell resolution.
The first row of Figure 5.2 shows the temporal evolution of the initial jump profile (5.30). The solid and the dashed lines show the profiles for a simulation with deactivated and activated shock-capturing, respectively. The solution oscillates if shock-capturing is deactivated, whereas the profile is smooth and spread over several cells if shock-capturing is activated. The second row of Figure 5.2 shows the indicator values if shock-capturing is activated. We set the indicator value for activating artificial viscosity to
S0 = 6.25·10−4 1
P4, (5.32)
and the maximum amount of artificial viscosity to ε0= 10.0 h
P . (5.33)
0 0.2 0.4 0.6 0.8 1 0.4
0.6 0.8 1
1.2 ts= 0
x1
Scalarquantityψ
Shock-capturing off Shock-capturing on 0 0.2 0.4 0.6 0.8 1
0.4 0.6 0.8 1
1.2 ts= 16
x1
0 0.2 0.4 0.6 0.8 1 0.4
0.6 0.8 1
1.2 ts= 3500
x1
0 0.2 0.4 0.6 0.8 1 0
2 4 6 ·10−5
AV on ts= 0
x1
Modal-decayindicatorS
0 0.2 0.4 0.6 0.8 1 0
2 4 6 ·10−5
AV on ts= 16
x1
0 0.2 0.4 0.6 0.8 1 0
2 4 6 ·10−5
AV on ts= 3500
x1
Figure 5.2:Two-step shock-capturing strategy. The modal-decay indicator (5.6) in combination with a piece-wise constant artificial viscosity (AV) (5.31) for the one-dimensional scalar transport equation (5.29). As soon as the solution has been sufficiently smoothed, it stays constant for all time-stepsts. A downstream pollution can be recognized in cells where artificial viscosity has been active.
88
0 0.2 0.4 0.6 0.8 1 0.4
0.6 0.8 1
1.2 P= 2
x1
Scalarquantityψ
t= 0 t= 0.05 t= 0.25
(a) Evolution of the initial jump profile (5.30).
0 0.2 0.4 0.6 0.8 1 0.05
0.1 0.15 0.2 0.25
P = 2
x1
Timet
(b) Time-history of cells with active artificial viscosity.
Figure 5.3:Numerical solution and temporal tracking of cells with active artificial viscosity (marked with a cross) for the one-dimensional scalar transport equation (5.29).
In general, the values ofS0andε0have to be tuned for each test case. For that, a low-resolution run can be used in order to save computational costs. Once the values have been determined, this setting can be applied to high-resolution runs for multiple grid sizeshand polynomial degreesP.
It takes several time-steps until the oscillations reach a certain magnitude and are detected by the indicator. Once the indicator value exceeds the user-defined activation limit, artificial viscosity is added locally. This happens multiple times until the oscillations are damped to a certain level so that the indicator values will always be below the activation limitS0. High frequencies are still present in the cells through which the shock has been propagating, leading to a downstream pollution of the computational domain.
Figure 5.3a shows the temporal evolution of the initial discontinuity (5.30) at several times t={0,0.05,0.25}. Untilt≈0.07, the indicator marks cells containing a shock and activates artificial viscosity as shown in Figure 5.3b. Later on, no additional smoothing is applied when neglecting the two outliers. Once the shock has been smoothed to a certain level, it will never return to its former shape in the case of scalar transport.
Figure 5.4 shows refinement studies for grid sizes ofh={0.1,0.05,0.025}and for polynomial degrees ofP ={2,4,10}. The shock becomes steeper for a largerh/P-resolution. ForP = 10, we obtain sub-cell resolution, meaning that the shock can be resolved within one single cell (Persson and Peraire, 2006; Barter and Darmofal, 2010; Lv et al., 2016).
Inviscid Burgers’ equation In the same manner like for the scalar transport equation (5.29), we analyze the one-dimensional nonlinear inviscid Burgers’ equation
∂ψ(x1, t)
∂t +ψ(x1, t)∂ψ(x1, t)
∂x1 = 0, (5.34a)
ψ0=ψ(x1,0) = 1
2 +sin(2πx1) . (5.34b)
We choose a periodic computational domainΩ = [0,1]with a characteristic grid size ofh= 0.1.
The actual computations are performed in a pseudo-two-dimensional setting. Like in the scalar
0 0.2 0.4 0.6 0.8 1 0.4
0.6 0.8 1
1.2 P = 2
x1
Scalarquantityψ
h= 0.1 h= 0.05 h= 0.025
(a)h-refinement study.
0 0.2 0.4 0.6 0.8 1 0.4
0.6 0.8 1
1.2 h= 0.1
x1
Scalarquantityψ
P = 2 P = 4 P = 10
(b)P-refinement study.
Figure 5.4:Refinement studies for the one-dimensional scalar transport equation (5.29) using the two-step shock-capturing strategy. The shock thickness scales withδs ∼h/P. The shock is contained within a single cell for a large enoughh/P-resolution, which is known as sub-cell resolution. All curves are plotted att= 0.25.
0 0.2 0.4 0.6 0.8 1
−1 0 1
2 P= 10
x1
Scalarquantityψ
t= 0 t= 0.25 t= 0.5
(a) Evolution of the initial sine wave (5.34b).
0 0.2 0.4 0.6 0.8 1 0
0.2 0.4
0.6 P= 10
x1
Timet
(b) Time-history of cells with active artificial viscosity.
Figure 5.5:Numerical solution and temporal tracking of cells with active artificial viscosity (marked with a cross) for the one-dimensional inviscid Burgers’ equation (5.34).
transport case, the shock-capturing strategy consists of the modal-decay indicator (5.6) and a simple artificial viscosity on/off-switch (5.31). The on/off-limit is set toS0= 6.25·10−4/P4 and the maximum artificial viscosity is set toε0= 1.0h/P.
Figure 5.5a shows the temporal evolution of the sine wave (5.34b) for a polynomial degree ofP = 10for the points in timet={0,0.25,0.5}. Since the Burgers’ equation is self-steepening, the moving front of the sine wave transforms from its initial shape into a shock front. Then, artificial viscosity is activated by the shock indicator and the shock front is flattened. Next, the shock will steepen again and the procedure is repeated. The height of the shock decreases with every diffusion step. This behavior could be verified for low and high polynomial degrees.
ForP = 10, the shock front can be resolved within a single cell, obtaining sub-cell resolution, see again Figure 5.5a. It can clearly be seen that the indicator is still active in cells through which the shock front has been propagating, see Figure 5.5b. The results match well with the results of Persson and Peraire (2006, Figure 1).
90
2 4 6 8 10−3
10−2
h= 0.1
Polynomial degreeP
Time-stepsize∆t
conv. (ref.) shock-cap. off conv./diff. (ref.) shock-cap. on
2 4 6 8
10−4 10−3
h= 0.05
Polynomial degreeP
Time-stepsize∆t
Figure 5.6:Maximum stable time-step sizes using the two-step shock-capturing strategy for the one-dimensional scalar transport equation (5.29). The numerically determined results are compared to a basic convection-diffusion time-step restriction, which serves as a reference, see Equation (5.35).