5 Numerical experiments
5.1 One-dimensional experiments
Example 5.1.1 (Smooth flow). Consider a one-dimensional flow in the domain Ω = [0,1]
with the following initial states
ρ0(x) = 1 + 0.9999999 sin(2πx), u0(x) = 0.9, p0(x) = 1.0 The boundary condition is set to be periodic. The exact solution is
ρ(x, t) = 1 + 0.9999999 sin(2π(x−0.9t)), u(x, t) = 0.9, p(x, t) = 1.0
In Table 5.1, we present the numerical results for the proposed method with and without the bound-preserving limiter. For the case k = 3, we take ∆t = CFL(∆x)4/3 so that the time error will not dominate. Since this is a high speed smooth flow with its lowest density near zero, the bound-preserving limiter does get turned on. As observed and discussed in [59] and [51], the SSP Runge-Kutta (RK) methods might degenerate the accuracy when the bound-preserving limiter is applied, while for the SSP multi-step (M-S) method, the full order of accuracy is recovered.
no limiter with limiter (RK) with limiter (M-S) k h L2 error order L2 error order L2 error order
1/20 3.50 e-2 – 4.85 e-2 – 4.55 e-2 –
1/40 8.72 e-3 2.00 1.06 e-2 2.11 1.05 e-2 2.12 1 1/80 2.17 e-3 2.01 2.55 e-3 2.05 2.55 e-3 2.03 1/160 5.43 e-4 2.00 6.13 e-4 2.03 6.11 e-4 2.06 1/320 1.36 e-4 2.00 1.50 e-4 2.04 1.49 e-4 2.03
1/20 2.16 e-3 – 3.47 e-3 – 2.40 e-3 –
1/40 2.77 e-4 2.97 5.05 e-4 2.78 2.79 e-4 3.10 2 1/80 3.48 e-5 2.99 9.42 e-5 2.42 3.48 e-5 3.00 1/160 4.35 e-6 3.00 1.87 e-5 2.33 4.35 e-6 3.00 1/320 5.44 e-7 3.00 3.68 e-6 2.35 5.44 e-7 3.00
1/20 8.99 e-5 – 1.82 e-4 – 1.52 e-4 –
1/40 5.60 e-6 4.00 2.09 e-5 3.12 5.82 e-6 8.03 3 1/80 3.46 e-7 4.02 2.30 e-6 3.18 3.50 e-7 4.05 1/160 2.16 e-8 4.00 2.70 e-7 3.09 2.19 e-8 4.00 1/320 1.35 e-9 4.00 1.86 e-8 3.86 1.46 e-9 3.91
Table 5.1: Example 5.1.1: One-dimensional accuracy test at T = 0.4 for the second-, third-and fourth-order DG methods with third-and without the BP limiters. The CFL number for the multistep method is one-third of that for the RK method. k is the degree of polynomial and h is the meshsize.
Moreover, in Figure 5.1, we plot the CPU time against theL2 error for polynomial degrees k = 1,2,3 with and without the BP limiter. The following two observations could be made:
(i) As one of the advantages of high order DG methods, for a given error, the computational time spent is diminishing with increasing order; (ii) The additional cost introduced by the BP limiter is very small.
Example 5.1.2 (Moderate blast wave). The relativistic blast wave problems are standard tests (see [22]) for a numerical relativistic hydrodynamical code. The computational domain is Ω = [0,1]. We first consider a moderate case, which has the initial states
(ρ0, u0, p0) =
(10.0,0.0,13.33), x <0.5
(1.0,0.0,0.0), x >0.5 (5.1) The initial discontinuity gives rise to a transonic rarefaction wave propagating left, a shock wave propagating right and a contact discontinuity in between. The numerical approximation in Figure 5.2 shows that our method can resolve these structures quite well. Small oscillations are observed around the contact discontinuity, since we have not applied any non-oscillatory
CPU time
L2 error
10-1 100 101 102
10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2
k=1
k=1, with limiter k=2
k=2, with limiter k=3
k=3, with limiter
Figure 5.1: Example 5.1.1: Log-log plot of the CPU time against theL2 error for polynomial degrees k= 1,2,3 with and without the BP limiter.
limiters such as the TVD/TVB or WENO limiters. For the same reason, in some of the following examples, we also observe spurious oscillations due to the lack of non-oscillatory limiters.
Example 5.1.3 (Strong relativistic blast wave). Next we examine a much more challenging blast wave example, of which the initial states are
(ρ0, u0, p0) =
(1.0,0.0,103), x < 0.5
(1.0,0.0,0.0), x >0.5 (5.2) This example is more relativistic than the previous one. The high relativity is due to the large enthalpy of the left state, which is h≃ 2.5×103 ≫1. This results in a thermodynamically relativistic configuration. The structure of the solution is the same as the moderate case, except for the formation of a very thin dense shell behind the shock in the density and a highly curved profile for the rarefaction fan in the velocity. The relativistic shock propagating at a Lorentz factor γ ≃ 6 [54]. In Figure 5.3, despite of small oscillations, we can see that our method resolves the curved part in the profile of the velocity very well and captures the thin shell in the density with little smearing.
x
ρ
0.2 0.4 0.6 0.8 1
0 2 4 6 8 10
(a) density ρ
x
p
0 0.2 0.4 0.6 0.8 1
0 2 4 6 8 10 12 14
(b) pressurep
x
u
0 0.2 0.4 0.6 0.8 1
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
(c) velocityu
Figure 5.2: Example 5.1.2: Blast wave with initial condition (5.1) atT = 0.5 on the mesh of 200 cells, approximated by the third-order RKDG method with the BP limiter. The squares represent the approximate cell averages and the solid line is the exact solution.
To further test the bound preserving property of our method, we consider a more extreme example. The initial condition now is
(ρ0, u0, p0) =
(1.0,0.0,104), x < 0.5
(1.0,0.0,0.0), x >0.5 (5.3) with the specific heat ratio Γ = 4/3. In this case the enthalpy of the left state ish≃4×104. In the profile of the density, the width of the thin shell is approximately 2.5×10−3 and the
x
ρ
0.2 0.4 0.6 0.8 1
0 2 4 6 8 10
(a) density ρ
x
p
0.2 0.4 0.6 0.8
0 200 400 600 800 1000
(b) pressurep
x
u
0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8 1
(c) velocityu
Figure 5.3: Example 5.1.3: Strong blast wave with initial condition (5.2) atT = 0.4 approx-imated by the third-order RKDG method with the BP limiter on the mesh with 200 cells.
The square represents the approximate cell averages and the solid line is the exact solution.
rarefaction part in the velocity becomes even more curved. In Figure 5.4, we can see the good performance of our proposed method.
Example 5.1.4 (Shock-heating problem). The shock-heating problem (see [7, 54, 21, 22]) is a standard benchmark problem to examine the ability of the method to deal with strong shocks. A warm wall is located at x = 0 and cold gas flows in at x = 1.0. When the gas
x
ρ
0.2 0.4 0.6 0.8 1
0 5 10 15
(a) densityρ, 400 cells
x
ρ
0.2 0.4 0.6 0.8 1
0 5 10 15
(b) densityρ, 800 cells
x
u
0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
(c) velocityu, 400 cells
x
u
0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
(d) velocityu, 800 cells
Figure 5.4: Example 5.1.3: Strong blast wave with initial condition (5.3) atT = 0.3 approxi-mated by the third-order RKDG method with the BP limiter. The mesh is decomposed into 400 cells (left column) and 800 cells (right column). The square represents the approximate cell averages and the solid line is the exact solution.
gets reflected on the wall, a reverse strong shock forms and propagates to the right. In our experiment, the inflow velocity is set to be vin = 0.999999, with the corresponding Lorentz factor γ = 707.1. The initial density is ρ0 = 1.0 and the initial pressure p0 = 0.0. The specific heat ratio in this example is Γ = 4/3. The analytical solution can be found in [21].
In Figure 5.5, we observe that, due to the BP limiter, near the physical bounds, there is
no oscillation and the bounds are well preserved. The shocks are sharply captured and the oscillations behind the shock may be eliminated with the non-oscillatory techniques such as the TVB or the WENO limiter. There is an undershoot in the left end point of the density, which is the so-called “wall-heating” effect. All these show the effectiveness of the proposed BP limiter in the ultra-relativistic regime with very strong shocks.
x
ρ/1000
0.2 0.4 0.6 0.8
0 0.5 1 1.5 2 2.5 3
(a) density ρ
x
p/100000
0.2 0.4 0.6 0.8
0 1 2 3 4 5 6 7
(b) pressurep
x
u
0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8 1
(c) velocityu
Figure 5.5: Example 5.1.4: One-dimensional shock-heating problem on the mesh of 200 cells, at T = 1.5 with vin = 0.999999, approximated by the third-order RKDG method with the BP limiter. The squares represent the numerical results and the solid line is the exact solution.
Example 5.1.5 (One-dimensional Riemann problem with non-zero transverse velocity).
The last one-dimensional test is the Riemann problem with non-zero transverse velocity, see [54] and [25]. The domain is still Ω = [0,1] and the initial states are
(ρ0, u0, v0, p0) =
(1.0,0.0,0.9,103), x <0.5
(1.0,0.0,0.9,10−2), x > 0.5 (5.4) Compared with Example 5.1.3, where there is no transverse velocity, the density has a smaller jump and a wider dense shell, however the distance between the tail of the rarefaction and the contact discontinuity is much smaller, which requires very high resolution to resolve the structure. The purpose of this example is to show the robustness of the BP limiter even when the transverse speed is nonzero and close to the speed of light. Admittedly, to correctly restore the right wave speed, it is unavoidable to refine the mesh (see the comparison between 400 cells and 6400 cells in Figures 5.6 and 5.7 and also results in [54] with the adaptive mesh refinement technique), but this goes beyond the purpose of this paper and we will not pursue it here. Our results on meshes of 400 and 6400 cells are presented in Figure 5.6 and Figure 5.7. These results match those in [54] and [25] well.
We further test this problem with a transverse velocity v = 0.999. In this case, the dense shell in the density is much thinner and the transverse velocity increases from v = 0.999 to v = 0.99967 at the rarefaction part, which corresponds to a Lorentz factor γ ≃39. We see in Figure 5.8, our method is still robust in this severe case.