Shock capturing examples

Concluding, we provide some two-dimensional shock problems in order to demonstrate results, that require the presented shock capturing methods from Sec. 5 to stabilize the approximation. Hence, for both of the following tests we turn off any viscous or resistive effects and restrict to the advective parts of the governing equations. We begin with the original, inviscid version of the previously presented Orszag-Tang vortex and then show results for the magnetic rotor test , see e.g. [3]. In all simulations we compare the artificial viscosity results according to Sec. 5.1 with those obtained by the SIAC filter from Sec. 5.2.

0.0 0.1 0.2 0.3 0.4 0.5 Time 0.6 0.7 0.8 0.9 1.0 1.1 Normalized Total Entropy CFL=DFL=0.5

Figure 7.4.: Time evolution plot of the total entropy in the viscous Orszag-Tang vortex with N = 7 on 20 × 20 elements with diffusivity coefficients (7.16).

Inviscid Orszag-Tang vortex

The first shock problem describing the evolution of a turbulent plasma cloud is initialized as in (7.8) on the periodic domain Ω = [0, 1]2. Viscous effects as well as divergence error damping are turned off and we use the ES flux with LLF dissipation for all simulations, since it is better suited in the presence of discontinuities. Furthermore, we use CFL = 0.5, polynomials of degree N = 5 and 40 × 40 elements.

We note, that due to the absence of stabilizing viscous and resistive effects the simulation of this test crashes at t ≈ 0.2, because various shocks develop and interact all over the domain. Thus, we use the shock capturing mechanisms from Sec. 5 to regularize the approximation and compare the solution quality of both approaches.

We show the evolution of the density in the plots below (Fig. 7.5) smoothed by the derived two-dimensional SIAC filter with m = 3, k = 8 and a fixed ε = 1.6. We use the smoothing matrix Ξ from (5.40) and do the filtering adaptively as in (5.47) with the pressure as a shock indicator, σFIL_min = −9 and σ_maxFIL = −6. Further, we show the distribution of the cell-wise constant convex parameter κ from (5.47) at the same stages in Fig. 7.6, which confirms the correct tracking of shocks as they evolve.

In order to assess the performance of the two-dimensional Dirac-delta filter, we compare the simulation results to the ones obtained by the common artificial viscosity

Figure 7.5.: Time evolution of Orszag-Tang vortex density on 40 × 40 elements with

N = 5 filtered adaptively by Ξ with σ_minFIL = −9, σ_maxFIL = −6, m = 3, k = 8 and ε = 1.6.

Figure 7.6.: Time evolution of convex parameter κ on 40 × 40 elements with N = 5 for adaptive Dirac-delta filter Ξ with σ_minFIL= −9, σFIL_max= −6, m = 3, k = 8 and

ε = 1.6.

approach. For both methods we use the pressure as a shock indicator and introduce a smooth transition area as in (5.9) and (5.47), respectively. The according user defined parameters for the DOF energy indicator are σ_minDOF = −6, σ_maxDOF = −4 and ₀ = 0.1 in (5.9). We note, that for both shock capturing methods, the results vary sensitively with respect to the choice of parameters, which in the demonstrated simulations are set as optimal as possible in terms of an appropriate balance between smoothing oscillatory regions and avoiding too much dissipation.

Figure 7.7.: Orszag-Tang vortex density at T = 0.5 for CFL = 0.5, N = 5 and 40 × 40 elements smoothed by Dirac-delta filter (left) and artificial viscosity (right).

As we can see in the plots (Fig. 7.7), on first sight, the Dirac-delta filter performs slightly better than the artificial viscosity, since it is able to smooth out almost all spuri- ous oscillations, whereas in the right hand figure some mesh artifacts are still observable. In order to investigate the performance of both methods, we provide two slices (Figures 7.8 and 7.9), in which we cut through the density distribution at x = y and y = 0.3 to compare the profiles obtained by both shock capturing methods against a reference solution.

This highly resolved reference solution is computed by a second order MUSCL- Hancock finite volume method (see e.g. [137]) on 1024 × 1024 elements with the pub- licly available high performance application code FLASH (http://flash.uchicago.edu/ site/flashcode/).

Whereas the oscillations are smoothed out by both approaches, we see that the filtering technique produces more overshoots at shocks. On the other hand, the viscous approach is more dissipative and causes longer simulation times due to the expensive computation of the gradients in each Runge-Kutta step as well as the additional time step restriction. In the end, both approaches generate reasonable approximations for this shock test and it is up to the user, which method he or she prefers for the mentioned reasons.

Figure 7.8.: Orszag-Tang-Vortex density slice at x = y and T = 0.5 for CFL = 0.5, N = 5 and 40 × 40 elements.

Figure 7.9.: Orszag-Tang-Vortex density slice at y = 0.3 and T = 0.5 for CFL = 0.5, N = 5 and 40 × 40 elements.

Magnetic rotor

The second test case describes a rotating dense circle in a static fluid, that generates strong circular shock waves [3]. In general this benchmark problem is defined in the same periodic domain Ω = [0, 1]2, by the radius r = p(x − 0.5)2_{+ (y − 0.5)}2 _{and the slope}

s = r1−r

r1−r0. The initial primitive variables for the magnetic rotor are stated in Table 7.13,

where the unlisted quantities are initially zero in the entire domain and γ = 1.4.

% v1 v2 p B1 r < r0 10 u_r00  1 2 − y  u0 r0  x −1₂ 1 √5 4π r0≤ r ≤ r1 1 + 9s su_r00  1 2 − y  su0 r0  x −1₂ 1 √5 4π r > r1 1 0 0 1 √5_4π

Table 7.13.: Initial primitive states for the magnetic rotor test.

In our simulations we define r0 = 0.1, r1 = 0.115 and u0 = 2. We use CFL = 0.5, a polynomial degree of N = 4 and 100 × 100 elements. We show the density and pressure at T = 0.15 for both shock capturing methods in the plots below, Figures 7.10 and 7.11 respectively. Due to the strong circular shocks combined with the oscillatory split form evaluation of the EC volume flux this test case is extremely sensitive and unstable. Therefore, we apply the SIAC filtering matrix constructed by a Dirac-delta kernel with only one vanishing moment m = 1 and k = 5. Again, we smooth the approximation adaptively with the density as a shock indicator, σ_minFIL = −9, σFIL_max = −6 and a fixed

ε = 1.4. For the stabilization by artificial viscosity we use the same parameters as for

the previous test, i.e. σ_minDOF= −6, σDOF_max = −4 and 0= 0.1 in (5.9).

In Figures 7.10 and 7.11 we see, that both shock capturing techniques perform well in terms of stabilizing the approximation and regularizing oscillatory regions. As in the previous test, the artificial viscosity approach is more dissipative, but this time the filtered solution is polluted by small mesh artifacts, which is particularly visible at the generated Alfvén waves in the pressure profile. We note, that these artifacts do not occur when using the SIAC filter in combination with the standard DGSEM approximation, see e.g. [15].

Figure 7.10.: Magnetic rotor density at T = 0.15 for CFL = 0.5, N = 4 and 100 × 100 elements smoothed by Dirac-delta filter (left) and artificial viscosity (right).

Figure 7.11.: Magnetic rotor pressure at T = 0.15 for CFL = 0.5, N = 4 and 100 × 100 elements smoothed by Dirac-delta filter (left) and artificial viscosity (right).