Uniform spectral convergence of the stochastic Galerkin method for the linear transport equations with random inputs in diffusive regime and a micro–macro decomposition-based asymptotic-preserving method
https://doi.org/10.1186/s40687-017-0105-1
© The Author(s) 2017
Received: 8 August 2016
Accepted: 17 March 2017
Published: 7 August 2017
Abstract
In this paper we study the stochastic Galerkin approximation for the linear transport equation with random inputs and diffusive scaling. We first establish uniform (in the Knudsen number) stability results in the random space for the transport equation with uncertain scattering coefficients and then prove the uniform spectral convergence (and consequently the sharp stochastic asymptotic-preserving property) of the stochastic Galerkin method. A micro–macro decomposition-based fully discrete scheme is adopted for the problem and proved to have a uniform stability. Numerical experiments are conducted to demonstrate the stability and asymptotic properties of the method.
Keywords
1 Background
In recent years, there have been extensive activities to study partial differential equations or engineering problems with uncertainties. Many numerical methods have been introduced. In this article, we are interested in the polynomial chaos (originally introduced in Wiener’s work [26])-based stochastic Galerkin method which has been shown to be competitive in many applications, see [4, 27, 28]. The stochastic Galerkin method has been used for linear transport equation with uncertain coefficients [25]. Here we are interested in the problem that contains both uncertainty and multiscale. The latter is characterized by the Knudsen number \(\varepsilon \), which, in the so-called optically thin region (\(\varepsilon \ll 1\)), due to high scattering rate of particles, leads the linear transport equation to a diffusion equation, known as the diffusion limit [1, 3, 18]. For the past decades, developing asymptotic-preserving (AP) schemes for (deterministic) linear transport equation with diffusive scaling has seen many activities, see for examples [5, 7–9, 17, 19, 20, 22]. Only recently AP scheme for linear transport equation with both uncertainty and diffusive scaling was introduced in [15] (in the framework of stochastic Galerkin method, coined as s-AP method). See more related recent works along this line in [6, 12, 13]. A scheme is s-AP if the stochastic Galerkin method for the linear transport equation, as \(\varepsilon \rightarrow 0\), becomes a stochastic Galerkin method for the limiting diffusion equation. It was realized in [15] that the deterministic AP framework can be easily adopted to study linear transport equations with uncertain coefficients. Moreover, as shown in [6, 12], kinetic equations, linear or nonlinear, could preserve the regularity in random space of the initial data at later time, which naturally leads to spectral accuracy of the stochastic Galerkin method.
When \(\varepsilon \ll 1\), however, the energy estimates and consequently the convergence rates given in [6, 12] depend on the reciprocal of \(\varepsilon \), which implies that one needs the degree of the polynomials used in the stochastic Galerkin method to grow as \(\varepsilon \) decreases. In fact, this is typical of a standard numerical method for problems that contain small or multiple scales. While AP schemes can be used with numerical parameters independent of \(\varepsilon \), to prove this rigorously is not so easy and has been done only in a few occasions [5, 14]. A standard approach to prove a uniform convergence is to use the diffusion limit, as was done first in [5] in the deterministic case and then in [11] for the uncertain transport equation. See also the review article [7]. However such approaches might not give the sharp convergence rate.
In this paper, we provide a sharp error estimate for the stochastic Galerkin method for problem (1). This requires a sharp (\(\varepsilon \)-independent) energy estimate on high-order derivatives in the random space for f, as well as \([f]-f\) where [f] is the velocity average of f defined in (5) which is shown to be bounded even if \(\varepsilon \rightarrow 0\). Then the uniform in \(\varepsilon \) spectral convergence naturally follows, without using the diffusion limit.
The s-AP scheme in [15] uses the AP framework of [8] that relies on the even- and odd-parity formulation of the transport equation. In this paper, we use the micro–macro decomposition-based approach (see [22]) to develop a fully discrete s-AP method. The advantage of this approach is that it allows us to prove a uniform (in \(\varepsilon \)) stability condition, as was done in the deterministic counterpart in [23]. In fact, we will show that one can easily adopt the proof of [23] for the s-AP scheme.
The paper is organized as follows. In Sect. 2 we summarize the diffusion limit of the linear transport equation. The generalized polynomial chaos-based stochastic Galerkin method for the problem is introduced in Sect. 3 and shown formally to be s-AP. The uniform in \(\varepsilon \) regularity of the stochastic Galerkin scheme is proven in Sect. 4, which leads to a uniform spectral convergence proof. The micro–macro decomposition-based fully discrete scheme is given in Sect. 5, and its uniform stability is established in Sect. 6. Numerical experiments are carried out in Sect. 7. The paper is concluded in Sect. 8.
2 The diffusion limit
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\([ {{{\mathcal {L}}}} f] =0\), for every \(f\in L^2([-1,1])\);
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The null space of f is \({{{\mathcal {N}}}} ({{{\mathcal {L}}}}) = \text{ Span } \{\,\phi \mid \phi =[ \phi ]\,\}\);
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The range of f is \({{{\mathcal {R}}}} ({{{\mathcal {L}}}}) = {{{\mathcal {N}}}} ({{{\mathcal {L}}}})^{\bot }=\{\,f\mid [f]=0\, \} ;\)
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\({{{\mathcal {L}}}}\) is nonpositive self-adjoint in \(L^2((-1,1); \phi ^{-1}\,\mathrm {d}v)\), i.e. , there is a positive constant \(s_m\) such that$$\begin{aligned} \langle f, {{{\mathcal {L}}}} f\rangle _{\phi } \le - 2 s_m \Vert f\Vert ^2_{\phi },\quad \forall \, f\in {{{\mathcal {N}}}} ({{{\mathcal {L}}}})^{\bot }; \end{aligned}$$(7)
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\({{{\mathcal {L}}}}\) admits a pseudo-inverse, denoted by \({{{\mathcal {L}}}}^{-1}\), from \({{{\mathcal {R}}}} ({{{\mathcal {L}}}})\) to \({{{\mathcal {R}}}} ({{{\mathcal {L}}}})\).
3 The gPC-stochastic Galerkin approximation
4 The regularity in the random space and a uniform spectral convergence analysis of gPC-SG method
4.1 Notations
4.2 Regularity in the random space
We will study the regularity of f with respect to the random variable z. To this aim, we first prove the following lemma. For simplicity, we state and prove the following lemma only for one-dimensional case. Proof for high-dimensional case is identical except the change of coefficient.
Lemma 4.1
Proof
Now we are ready to prove the following regularity result.
Theorem 4.1
Proof
Theorem 4.1 shows the derivatives of the solution with respect to z can be bounded by the derivatives of initial data. In particular, the \(\Vert D^k f\Vert _{\Gamma }\) bound is independent of \(\varepsilon \)! This is crucial for our later proof that our scheme is s-AP. However, this estimate alone is not sufficient to guarantee that the whole gPC-SG method has a spectral convergence uniform in \(\varepsilon \) (since there is \(O(1/\varepsilon ^2)\) coefficient in front of the projection error, such we need \(O(\varepsilon ^2)\) estimation of \([ f] - f\) to cancel this coefficient). To this aim, we first provide the following lemma.
Lemma 4.2
Proof
Now we are ready to prove the following theorem.
Theorem 4.2
Proof
Remark 4.1
We remark that all the above lemma and theorems are proved for \(z\in {{\mathbb {R}}}\) and \(\sigma \) depending only on z. However, our conclusions and techniques are not limited to these cases. For \(z\in {{\mathbb {R}}}^d\), it is straightforward to prove and for \(\sigma (x,z)\) also a function of x, we only need to modify the proof of Lemma 4.2 by using the same technique as in the proof of Theorem 4.1.
4.3 A spectral convergence uniformly in \(\varvec{\varepsilon }\)
For the truncation error \(r_M\), we have the following lemma
Lemma 4.3
Proof
Lemma 4.4
Proof
Finally, we are now ready to state the main convergence theorem:
Theorem 4.3
Proof
Remark 4.2
Theorem 4.3 gives a uniformly in \(\varepsilon \) spectral convergence rate; thus, one can choose M independent of \(\varepsilon \), a very strong s-AP property. If the scattering is anisotropic, namely \(\sigma \) depends on \(\nu \), then one usually obtains a convergence rate that requires \(M\gg \varepsilon \) (see for example [12]). In such cases the proof of s-AP property is much harder, and one usually needs to use the diffusion limit, see [5] in the case of deterministic case and [11] in the random case.
5 The Full discretization
As pointed out in [15], by using the gPC-SG formulation, one obtains a vector version of the original deterministic transport equation. This enables one to use the deterministic AP scheme. In this paper, we adopt the AP scheme developed in [22] for gPC-SG system (16).
One of the most important and challenge problems for linear transport equation is the treatment of boundary conditions; here, we refer to the early work by Jin and Levermore [10] and more recent work by Lemou and Méhats [21] for their study of AP property and numerical treatment of physical boundary conditions.
We take a uniform grid \(x_i = ih, i = 0, 1, \ldots N\), where \(h=1/N\) is the grid size, and time steps \(t^n=n \Delta t\). \(\rho ^n_{i}\) is the approximation of \(\rho \) at the grid point \((x_i, t^n)\), while \(g^{n+1}_{i+\frac{1}{2}}\) is defined at a staggered grid \(x_{i+1/2} = (i+1/2)h, i = 0, \ldots N-1\).
We also notice that \([ \hat{g}^n_{i+\frac{1}{2}}]=0\) for every n which will be used later.
6 The uniform stability
One important property for an AP scheme is to have a stability condition independent of \(\varepsilon \), so one can take \(\Delta t \gg O(\varepsilon )\) when \(\varepsilon \) becomes small. In this section we prove such a result. The proof basically follows that of [23] for the deterministic problem.
For clarity in this section we assume \(\sigma ^a =S=0\). The main theoretical result about the stability is the following theorem:
Theorem 6.1
Remark 6.1
Since the right-hand side of (87) has a lower bound when \(\varepsilon \rightarrow 0\) (and the lower bound being that of the stability condition of discrete diffusion equation (86)), the scheme is asymptotically stable and \(\Delta t\) remains finite even if \(\varepsilon \rightarrow 0\).
6.1 Notations and useful lemma
6.2 Energy estimates
Now we provide the details of the energy estimate. The proof is similar to that for deterministic problem in [23].
7 Numerical examples
In this section, we present several numerical examples to illustrate the effectiveness of our method.
7.1 Example 1
Example 1. Errors of the mean (solid line) and standard deviation (dash line) of \(\rho \) with respect to the gPC order at \(\varepsilon = 10^{-8}\): \(\Delta x = 0.04\) (squares), \(\Delta x = 0.02\) (circles), \(\Delta x = 0.01\) (stars)
In Fig. 1, we plot the errors in mean and standard deviation of the gPC numerical solutions at \(t = 0.01\) with different gPC orders. Three sets of results are included: solutions with \(\Delta x = 0.04\) (squares), \(\Delta x = 0.02\) (circles), \(\Delta x = 0.01\) (stars). We always use \(\Delta t = 0.0002/3\). One observes that the errors become smaller with finer mesh. One can see that the solutions decay rapidly in N and then saturate where spatial discretization error dominates. It is then obvious that the errors due to gPC expansion can be neglected at order \(M = 4\) even for \(\varepsilon = 10^{-8}\). The solution profiles of the mean and standard deviation are shown on the left and right of Fig. 2, respectively.
Example 1. The mean (left) and standard deviation (right) of \(\rho \) at \(\varepsilon =10^{-8}\), obtained by the gPC Galerkin at order \(M=4\) (circles), the stochastic collocation method (crosses) and limiting analytical solution (103)
Example 1. The mean (left) and standard deviation (right) obtained by gPC-Galerkin (circle) and collocation method (cross) at time \(t=0.01\)
Example 1. Differences in the mean (solid line) and standard deviation (dash line) of \(\rho \) with respect to \(\varepsilon ^2\), between limiting analytical solution (103) and the 4th-order gPC solution with \(\Delta x = 0.04\) (squares), \(\Delta x = 0.02\) (circles) and \(\Delta x = 0.01\) (stars)
In Fig. 4, we examine the difference between the solution \(t = 0.01\) obtained by the 4th-order gPC method with \(\Delta x = 0.01, \Delta t = \Delta x^2/12\) and limiting analytical solution (103). As expected, we observe the differences become smaller as \(\varepsilon \) is smaller in a quadratic fashion, before the numerical errors become dominant.
7.2 Example 2: mixing regime
\(\varepsilon (x)\)
7.3 Example 3: random initial data
Example 3. The mean (left) and standard deviation (right) obtained by gPC-Galerkin (circle) and collocation method (cross) at time \(t=0.1, \varepsilon =10^{-8}\)
Example 3. The mean (left) and standard deviation (right) obtained by gPC-Galerkin (circle) and collocation method (cross) at time \(t=0.1, \varepsilon =1\)
One can see a good agreement between the gPC-SG solutions and the solutions by the collocation method.
7.4 Example 4: random boundary data
Example 4. The mean (left) and standard deviation (right) obtained by gPC-Galerkin (circle) and collocation method (cross) at time \(t=0.1, \varepsilon =10^{-8}\)
Example 4. The mean (left) and standard deviation (right) obtained by gPC-Galerkin (circle) and collocation method (cross) at time \(t=0.1, \varepsilon =10\)
7.5 Example 5: 2D random space
The mean (left) and standard deviation (right) of \(\rho \) at \(\varepsilon = 10^{-8}\), obtained by 5th-order gPC Galerkin (circles) and the stochastic collocation method (crosses). The random input has dimension \(d=2\)
Errors of the mean (solid line) and standard deviation (dash line) of \(\rho \) with respect to gPC order, with the \(d=2\)-dimensional random input
8 Conclusions
In this paper we establish the uniform spectral accuracy in terms of the Knudsen number, which consequently allows us to justify the stochastic asymptotic-preserving property of the stochastic Galerkin method for the linear transport equation with random scattering coefficients. For the micro–macro decomposition-based fully discrete scheme we also prove a uniform stability result. These are the first uniform accuracy and stability results for the underlying problem.
It is expected that our uniform stability proof is useful for more general kinetic or transport equations, which is the subject of our future study.
Acknowledgements
Research was supported by NSF Grants DMS-1522184, DMS-1514826 and the NSF RNMS: KI-Net (DMS-1107291 and DMS-1107444). Shi Jin was also supported by NSFC Grant No. 91330203 and by the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin-Madison with funding from the Wisconsin Alumni Research Foundation.
Notes
Declarations
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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