Chapter 3. A Conservative Solver for Boltzmann Equation Based
3.7 Numerical Results
Test 1 is a 2-d Maxwell type of elastic collisions, benchmarked by Bobylev-Krook-Wu (BKW) exact solutions. The initial density distribution is
f (v, 0) = v
2
πσ2exp(−v 2
/σ2) . (3.31)
This problem has an exact solution [57] f (v, t) = 1 2πs2 2s − 1 + 1 − s 2s v2 σ2 exp − v 2 2sσ2 , (3.32)
where s = 1 − 12exp(−σ2t/8). In the test, we choose the scaling parameter σ = π/6 such that the truncation domain is well chosen by Ωv = [−π, π].
We let it run for 600 time steps with ∆t = 0.1. This example is used to test the accuracy by calculating the relative L2 errors compared to its exact
solution and relative entropy verifying that the numerical solution will converge to the true equilibrium. See Figure 3.4 for the evolution of the marginal density distributions; Figure 3.5 and Figure 3.6 shows the relative L2 errors
and relative entropy, respectively. The marginal density distribution is defined as fx(vx ∈ Ik) = 1 (∆v)2 Z Ik Z In/2 f (v, t)dvxdvy.
The relative L2 error is defined as R Ωv|fh(v, t) − f (v, t)| 2dv1/2 R Ωv|f (v, t)| 2dv1/2 ,
and the relative entropy given by Hrel(t) = Z Ωv f (v, t) log f (v, t) − fM(v) log fM(v)dv = Z Ωv f (v, t) logf (v, t) fM(v) dv , (3.33) where fM(v) is the true equilibrium density distribution, is expected to con-
verge to zero which implies the solution converges to the true equilibrium in the sense of L1.
Figure 3.4: Test 1: Comparison of solutions at time t = 0, 1, 5, 10, 15s. n=44 per direction; solid line: exact solution, stars: p.w. constant approximation
Remark. Through Test 1, we would like to mention the positivity issue of numerical solutions. The true density distributions are expected to be pos- itive for any given positive time, if initially so. Our numerical tests show that,
Figure 3.5: Test 1: Relative L2 er- rors, compared with true solution, for different number of mesh ele- ments
Figure 3.6: Test 1: Relative En- tropy for different number of mesh elements
positivity can be achieved if we apply piecewise constant basis functions. The conservation laws (here, only mass due to the zero-th order of basis polynomi- als) are expected to hold but only for a short time, and will be seriously broken in the long run (see more details from the results of the next test problem). With invoke of our conservation routine, the conservations are guaranteed but the positivity is inevitably broken. This seems a common issue for almost all numerical solvers known so far. But fortunately, the negativity only occurs at the tails of the distribution functions, and as long as the “negative energy” (second order moment of the negative part of the density function) stays un- der controlled by a small ratio to the “positive energy”, the accuracy of the numerical approximations is guaranteed.
used to show the conservation routines. The initial states we take are convex combinations of two shifted Maxwellian distributions.
Figure 3.7: Test 2: Initial Probability Distribution: two shifted Maxwellians
Truncate the velocity domain Ω = [−4.5, 4.5]2 and set number of nodes in each velocity direction n = 32, 40. The initial density function is a convex combination of two Maxwellians
f0(v) = λM1(v) + (1 − λ)M2(v) , (3.34)
with Mi(v) = (2πTi)−d/2e −|v−Vi|2
2Ti , T1 = T2 = 0.16, V1 = [−1, 0], V2 = [1, 0] and
λ = 0.5.
We test for n = 32 and n = 40, for 1000 time steps to compare the results and see the long time behavior as well. The probability density distri- bution functions are reconstructed with splines.
From Figure 3.10 and Figure 3.11, we can see, the scheme with piecewise constant test functions, as expected conserves moments for short time; in the
Figure 3.8: Test 2: Evolution of pdf without conservation routines.
Figure 3.9: Test 2: Evolution of pdf with conservation routines
Figure 3.10: Test 2: Evolution of mass
Figure 3.11: Test 2: Evolution of kinetic energy
long run, due to the truncation, the tails of the density functions are lifted up and thus moments are expected to lose. At the same time, finer grids indeed give more accuracy. Since the basis polynomials are only zero order, it’s expected that mass is much better conserved than higher order moments. Through the comparison of Figure 3.8 and Figure 3.9 we see, after long time, with no conservation routine, the density distribution collapses due to the
truncation of the domain. While with the invoke of conservation routines, the density function stays stable when reaching equilibrium. So, the conservation routine works and is necessary for stability. However, the cost we pay is the loss of positivity.
Test 3 is initialized by a sudden jump on temperatures, i.e. a jump discontinuity in its initial and far from equilibrium, as shown in Figure 3.12. The initial state is given by
f0(v) = 1 2πT1 exp(−|v| 2 2T1 ) , v1 ≤ 0 1 2πT2 exp(−|v| 2 2T2 ) , v1 > 0
with T1 = 0.3 and T2 = 0.6. The collision is of type 2d hard spheres.
Figure 3.12: Test 3: Initial density function
With truncated domain Ωv = [−5, 5], n = 44 in each direction, the
DG solution well captures the discontinuity and converges to equilibrium. See Figure 3.13 and Figure 3.14.
Figure 3.13: Test 3: DG solutions Figure 3.14: Test 3: The entropy decay of DG solutions
Test 4 is testing on the 3D homogeneous Boltzmann equation with Maxwell molecular potential, with initial
f0(v) = 1 2(2πσ2)3/2 exp −|v − 2σe| 2 2σ2 + exp −|v + 2σe| 2 2σ2 ,
where parameters σ = π/10 and e = (1, 0, 0). Ωv = [−3.4, 3.4]3, n = 30.
Figure 3.15 shows the evolution of the marginal density distributions, which is defined as fx(vx∈ Ik) = 1 (∆v)3 Z Ik Z In/2 Z In/2 f (v, t)dvxdvydvz.
Figure 3.16 shows the decay of entropy to its equilibrium state.
Figure 3.17 shows the relaxations of directional temperature, which as expected converge to the averaged temperature.
Figure 3.15: Test 4: Evolution of marginal distributions at t = 0, 1, 2.5, 5s; dots are the piece- wise constant value on each el- ement; solid lines are spline re- constructions
Figure 3.16: Test 4: Entropy decay
Chapter 4
Computations of Spectral Gaps for Linearized
Boltzmann Operators
The quantitative information on the spectral gaps for the linearized Boltzmann operator is of primary importance on justifying the Boltzmann model and study of relaxation to equilibrium. In this chapter, for the first time in this field, we provide numerical evidence on the existence of spectral gaps and corresponding approximate values.