
Example 1. Left panel:
The purpose of this research is to highlight the importance of periodically analyzing the data obtained from the technological sources used by customers, such as user comments on social networks and videos, using qualitative data analysis software. This research analyzes user sentiments, words, and opinions about virtual reality (VR) videos on YouTube in order to explore user reactions to such videos, as well as to establish whether this technology contributes to the sustainability of natural environments. User-generated data can provide important information for decision making about future policies of companies that produce video content. The results of our analysis of 12 videos revealed that users predominantly perceived these videos positively. This conclusion was supported by the findings of an opinion and text analysis, which identified positive reviews for videos and channels with many followers and large numbers of visits. The features such as the quality of the video and the accessibility of technology were appreciated by the viewers, whereas videos that are 100% VR and require special glasses to view them do not have as many visits. However, VR was seen to be a product which viewers were interested in and, according to Google, there are an increasing number of searches and sales of VR glasses in holiday seasons. Emotions of wonder and joy are more evident than emotions of anger or frustration, so positive feelings can be seen to be predominant.
Citation: Pedro R. Palos Sánchez, José A. Folgado-Fernández, Mario Alberto Rojas Sánchez. Virtual Reality Technology: Analysis based on text and opinion mining[J]. Mathematical Biosciences and Engineering, 2022, 19(8): 7856-7885. doi: 10.3934/mbe.2022367
[1] | John D. Towers . An explicit finite volume algorithm for vanishing viscosity solutions on a network. Networks and Heterogeneous Media, 2022, 17(1): 1-13. doi: 10.3934/nhm.2021021 |
[2] | Boris Andreianov, Kenneth H. Karlsen, Nils H. Risebro . On vanishing viscosity approximation of conservation laws with discontinuous flux. Networks and Heterogeneous Media, 2010, 5(3): 617-633. doi: 10.3934/nhm.2010.5.617 |
[3] | Giuseppe Maria Coclite, Carlotta Donadello . Vanishing viscosity on a star-shaped graph under general transmission conditions at the node. Networks and Heterogeneous Media, 2020, 15(2): 197-213. doi: 10.3934/nhm.2020009 |
[4] | Markus Musch, Ulrik Skre Fjordholm, Nils Henrik Risebro . Well-posedness theory for nonlinear scalar conservation laws on networks. Networks and Heterogeneous Media, 2022, 17(1): 101-128. doi: 10.3934/nhm.2021025 |
[5] |
Giuseppe Maria Coclite, Nicola De Nitti, Mauro Garavello, Francesca Marcellini .
Vanishing viscosity for a |
[6] | Karoline Disser, Matthias Liero . On gradient structures for Markov chains and the passage to Wasserstein gradient flows. Networks and Heterogeneous Media, 2015, 10(2): 233-253. doi: 10.3934/nhm.2015.10.233 |
[7] | Wen Shen . Traveling wave profiles for a Follow-the-Leader model for traffic flow with rough road condition. Networks and Heterogeneous Media, 2018, 13(3): 449-478. doi: 10.3934/nhm.2018020 |
[8] | Giuseppe Maria Coclite, Lorenzo di Ruvo, Jan Ernest, Siddhartha Mishra . Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes. Networks and Heterogeneous Media, 2013, 8(4): 969-984. doi: 10.3934/nhm.2013.8.969 |
[9] | Abraham Sylla . Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly nonlocal model. Networks and Heterogeneous Media, 2021, 16(2): 221-256. doi: 10.3934/nhm.2021005 |
[10] | Christophe Chalons, Paola Goatin, Nicolas Seguin . General constrained conservation laws. Application to pedestrian flow modeling. Networks and Heterogeneous Media, 2013, 8(2): 433-463. doi: 10.3934/nhm.2013.8.433 |
The purpose of this research is to highlight the importance of periodically analyzing the data obtained from the technological sources used by customers, such as user comments on social networks and videos, using qualitative data analysis software. This research analyzes user sentiments, words, and opinions about virtual reality (VR) videos on YouTube in order to explore user reactions to such videos, as well as to establish whether this technology contributes to the sustainability of natural environments. User-generated data can provide important information for decision making about future policies of companies that produce video content. The results of our analysis of 12 videos revealed that users predominantly perceived these videos positively. This conclusion was supported by the findings of an opinion and text analysis, which identified positive reviews for videos and channels with many followers and large numbers of visits. The features such as the quality of the video and the accessibility of technology were appreciated by the viewers, whereas videos that are 100% VR and require special glasses to view them do not have as many visits. However, VR was seen to be a product which viewers were interested in and, according to Google, there are an increasing number of searches and sales of VR glasses in holiday seasons. Emotions of wonder and joy are more evident than emotions of anger or frustration, so positive feelings can be seen to be predominant.
This paper proposes an explicit finite volume scheme for first-order scalar conservation laws on a network having a single junction. The algorithm and analysis extend readily to networks with multiple junctions, due to the finite speed of propagation of the solutions of conservation laws. For the sake of concreteness, we view the setup as a model of traffic flow, with the vector of unknowns representing the vehicle density on each road. A number of such scalar models have been proposed, mostly differing in how the junction is modeled. An incomplete list of such models can be found in [4,5,6,7,8,11,12,13,14,15,16,18,19]. In this paper we focus on the so-called vanishing viscosity solution proposed and analyzed in [7] and [2]. The junction has
∂tuh+∂xfh(uh)=0,h=1,…,m+n. | (1.1) |
Incoming roads are indexed by
ui∈L∞(R−×R+;[0,R]),i∈{1,…,m},uj∈L∞(R+×R+;[0,R]),j∈{m+1,…,m+n}. | (1.2) |
Following [2] we make the following assumptions concerning the fluxes
The study of vanishing viscosity solutions on a network was initiated by Coclite and Garavello [7], who proved that vanishing viscosity solutions converge to weak solutions if
For the convenience of the reader, and to establish notation, we review some relevant portions of the theory of network vanishing viscosity solutions, as described in [2], where we refer the reader for a more complete development. Let
Gh(β,α)={min | (1.3) |
(Compared to [2], we list the arguments
\begin{equation} -L_h \le \partial G_h(\beta, \alpha)/\partial \beta \le 0, \quad 0\le \partial G_h(\beta, \alpha)/\partial \alpha \le L_h, \quad h \in \{1, \ldots, m+n\}. \end{equation} | (1.4) |
Definition 1.1 (Vanishing viscosity germ [2]). The vanishing viscosity germ
\begin{equation} \begin{split} &\sum\limits_{i = 1}^m G_i(p_{ \vec{u}}, u_i) = \sum\limits_{j = m+1}^{m+n} G_j(u_j, p_{ \vec{u}}), \\ &G_i(p_{ \vec{u}}, u_i) = f_i(u_i), \quad i = 1, \ldots, m, \\ &G_j(u_j, p_{ \vec{u}}) = f_j(u_j), \quad j = m+1, \ldots, m+n. \end{split} \end{equation} | (1.5) |
The definition of entropy solution requires one-sided traces along the half-line
\begin{equation} \begin{split} &\gamma_i u_i(\cdot) = u_i(\cdot, 0^-), \quad i \in \{1, \ldots, m \}, \\ &\gamma_j u_j(\cdot) = u_j(\cdot, 0^+), \quad j \in \{m+1, \ldots, m+n \}. \end{split} \end{equation} | (1.6) |
Definition 1.2 (
\begin{equation} q_h(v, w) = \mathrm{sign}(v-w)\left(f_h(v)-f_h(w) \right), \quad h = 1, \ldots, m+n. \end{equation} | (1.7) |
Given an initial condition
● For each
\begin{align} \int_{ \mathbb{R}_+} \int_{\Omega_h} \Bigl(\left|{u_h-c}\right| \partial_t \xi + q_h(u_h, c)\partial_x \xi \Bigr)\, dx\, dt + \int_{\Omega_h} \left|{u_{h, 0}-c}\right| \xi(x, 0) \, dx \geq 0. \end{align} | (1.8) |
● For a.e.
Associated with each
\begin{equation} k_h(x) = k_h, \quad x \in \Omega_h, \quad h \in \{1, \ldots, m+n\}. \end{equation} | (1.9) |
It is readily verified that viewed in this way,
Definition 1.2 reveals the relationship between the set
Definition 1.3 (
● The first item of Definition 1.2 holds.
● For any
\begin{equation} \sum\limits_{h = 1}^{m+n} \left(\int_{ \mathbb{R}_+} \int_{\Omega_h} \left\{\left|{u_h - k_h }\right|\xi_t + q_h(u_h, k_h)\xi_x\right\} \, dx \, dt \right) \ge 0 \end{equation} | (1.10) |
for any nonnegative test function
Theorem 1.4 (Well posedness [2]). Given any initial datum
\begin{equation} \vec{u}_0 = (u_{1, 0}, \ldots, u_{m+n, 0}) \in L^{\infty}(\Gamma; [0, R]^{m+n}), \end{equation} | (1.11) |
there exists one and only one
In addition to the results above, reference [2] also includes a proof of existence of the associated Riemann problem. Based on the resulting Riemann solver, a Godunov finite volume algorithm is constructed in [2], which handles the interface in an implicit manner. This requires the solution of a single nonlinear equation at each time step. The resulting finite volume scheme generates approximations that are shown to converge to the unique
The main contribution of the present paper is an explicit version of the finite volume scheme of [2]. It differs only in the processing of the junction. We place an artificial grid point at the junction, which is assigned an artificial density. The artificial density is evolved from one time level to the next in an explicit manner. Thus a nonlinear equation solver is not required. (However, we found that in certain cases accuracy can be improved by processing the junction implicitly on the first time step.) Like the finite volume scheme of [2], the new algorithm has the order preservation property and is well-balanced. This makes it possible to employ the analytical framework of [2], resulting in a proof that the approximations converge to the unique entropy solution of the associated Cauchy problem.
In Section 2 we present our explicit finite volume scheme and prove convergence to a
For a fixed spatial mesh size
\begin{equation} \begin{split} &x_{\ell} = (\ell + 1/2) {\Delta} x, \quad \ell \in \{\ldots, -2, -1\}, \\ &x_{\ell} = (\ell - 1/2) {\Delta} x, \quad \ell \in \{1, 2, \ldots\}. \end{split} \end{equation} | (2.1) |
Each road
\begin{equation} \begin{split} &\Omega_i = \bigcup\limits_{\ell \le -1} I_\ell, \quad I_\ell : = (x_\ell - {\Delta} x/2, x_\ell + {\Delta} x/2] ~\text{ for}~ \ell \le -1 , \\ &\Omega_j = \bigcup\limits_{\ell \ge 1} I_\ell, \quad I_\ell : = [x_\ell - {\Delta} x/2, x_\ell + {\Delta} x/2) ~\text{for }~ \ell \ge 1 . \end{split} \end{equation} | (2.2) |
Our discretization of the spatial domain
\begin{equation} \begin{split} &U_{\ell}^{h, s} \approx u_h(x_\ell, t^s), \quad \ell \in \mathbb{Z} \setminus \{0\}, \\ &P^s \approx p_{\gamma \vec{u}(t^s)}. \end{split} \end{equation} | (2.3) |
We are somewhat artificially assigning a density, namely
Remark 1. We make the association
The initial data are initialized via
\begin{equation} \begin{split} &U^{h, 0}_{\ell} = {1\over {\Delta} x} \int_{I_{\ell}} u_{h, 0}(x) \, dx, \quad h \in \{1, \ldots, m+n \}, \\ &P^0 \in [0, R]. \end{split} \end{equation} | (2.4) |
Note that
\begin{equation} \left\{ \begin{split} &P^{s+1} = P^s - \lambda \left(\sum\limits_{j = m+1}^{m+n} G_j(U_{1}^{j, s}, P^s) - \sum\limits_{i = 1}^m G_i(P^s, U_{-1}^{i, s}) \right), \\ &U_\ell^{i, s+1} = U_\ell^{i, s} - \lambda \left(G_i(U_{\ell+1}^{i, s}, U_{\ell}^{i, s}) - G_i(U_{\ell}^{i, s}, U_{\ell-1}^{i, s}) \right), \quad i \in \{1, \ldots, m \}, \, \, \ell \le -1, \\ &U_\ell^{j, s+1} = U_\ell^{j, s} - \lambda \left(G_j(U_{\ell+1}^{j, s}, U_{\ell}^{j, s}) - G_j(U_{\ell}^{j, s}, U_{\ell-1}^{j, s}) \right), \quad j \in \{m+1, \ldots, m+n \}, \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \, \, \ell \ge 1.\\ \end{split} \right. \end{equation} | (2.5) |
Define
\begin{equation} \lambda (m+n) L\le 1. \end{equation} | (2.6) |
If all of the
\begin{equation} \lambda \max(m, n) L\le 1. \end{equation} | (2.7) |
Remark 2. The algorithm (2.5) is an explicit version of the scheme of [2]. To recover the scheme of [2] from (2.5), one proceeds as follows:
● first substitute
● then advance each
The equation mentioned above (after substituting
\begin{equation} \sum\limits_{j = m+1}^{m+n} G_j(U_{1}^{j, s}, P^s) = \sum\limits_{i = 1}^m G_i(P^s, U_{-1}^{i, s}). \end{equation} | (2.8) |
The implicit portion of the scheme of [2] consists of solving (2.8) for the unknown
Remark 3. The convergence of the scheme (2.5) is unaffected by the choice of
\begin{equation} P^{0}_{\nu+1} = P^0_\nu - \lambda \left(\sum\limits_{j = m+1}^{m+n} G_j(U_{1}^{j, 0}, P^0_\nu) - \sum\limits_{i = 1}^m G_i(P^0_\nu, U_{-1}^{i, 0}) \right). \end{equation} | (2.9) |
Moreover, we found that this same fixed point iteration approach is a convenient way to solve (2.8) when implementing the implicit scheme of [2]. From this point of view the algorithm of this paper and the algorithm of [2] only differ in whether the first equation of (2.5) is iterated once, or iterated to (approximate) convergence. See Appendix A for a proof that the iterative scheme (2.9) converges to a solution of (2.8).
Remark 4. The CFL condition associated with the finite volume scheme of [2] is
\begin{equation} \lambda L \le 1/2. \end{equation} | (2.10) |
As soon as there are more than a few roads impinging on the junction, our CFL condition (2.6) imposes a more severe restriction on the allowable time step, which becomes increasingly restrictive as more roads are included. One could view this as the price to be paid for the simplified processing of the junction. On the other hand, most of the specific examples discussed in the literature are limited to
Let
\begin{equation} \begin{split} &u^{i, {\Delta}} = \sum\limits_{s\ge 0} \sum\limits_{\ell \le -1} \chi_{\ell}(x) \chi^s(t) U^{i, s}_\ell, \quad i \in \{1, \ldots, m \}, \\ &u^{j, {\Delta}} = \sum\limits_{s\ge 0} \sum\limits_{\ell \ge 1} \chi_{\ell}(x) \chi^s(t) U^{j, s}_\ell, \quad j \in \{m+1, \ldots, m+n \}.\\ \end{split} \end{equation} | (2.11) |
The discrete solution operator is denoted by
\begin{equation} S^{\Delta} \vec{u}_0 = \left(u^{i, {\Delta}}, \ldots, u^{m, {\Delta}}, u^{m+1, {\Delta}}, \ldots u^{m+n, {\Delta}} \right). \end{equation} | (2.12) |
The symbol
\begin{equation} \Gamma_{ \text{discr}} = \Bigl(\{1, \ldots, m \}\times\{\ell \in \mathbb{Z}, \ell \le -1 \} \Bigr) \bigcup \Bigl(\{m+1, \ldots, m+n \}\times\{\ell \in \mathbb{Z}, \ell \ge 1 \} \Bigr), \end{equation} | (2.13) |
and with the notation
\begin{equation} U^s = \left(U_{\ell}^{h, s} \right)_{(h, l)\in \Gamma_{ \text{discr}}}, \end{equation} | (2.14) |
Remark 5. In the case where
Lemma 2.1. Fix a time level
Proof. For
Fix
\begin{equation} \begin{split} &\partial U^{i, s+1}_{-1}/ \partial U^{i, s}_{-2} = \lambda \partial G_i(U^{i, s}_{-1}, U^{i, s}_{-2})/\partial U^{i, s}_{-2}, \\ &\partial U^{i, s+1}_{-1}/ \partial U^{i, s}_{0} = -\lambda \partial G_i(U^{i, s}_{0}, U^{i, s}_{-1})/\partial U^{i, s}_{0}, \\ &\partial U^{i, s+1}_{-1}/ \partial U^{i, s}_{-1} = 1- \lambda \partial G_i(U^{i, s}_{0}, U^{i, s}_{-1})/\partial U^{i, s}_{-1} + \lambda \partial G_i(U^{i, s}_{-1}, U^{i, s}_{0})/\partial U^{i, s}_{-1}. \end{split} \end{equation} | (2.15) |
That partial derivatives in the first two lines are nonnegative is a well-known property of the Godunov numerical flux. The partial derivative on the third line is nonnegative due to (1.4) and the CFL condition (2.6).
A similar calculation shows that the partial derivatives of
It remains to show that the partial derivatives of
\begin{equation} \begin{split} &\partial P^{s+1}/\partial U^{i, s}_{-1} = \lambda \partial G_i(P^s, U^{i, s}_{-1})/\partial U^{i, s}_{-1}, \\ &\partial P^{s+1}/\partial U^{j, s}_{1} = -\lambda \partial G_j(U^{j, s}_{1}, P^s)/\partial U^{j, s}_{1}, \\ &\partial P^{s+1}/\partial P^s = 1 - \lambda \left(\sum\limits_{j = m+1}^{m+n} \partial G_j(U^{j, s}_{1}, P^s)/\partial P^s - \sum\limits_{i = 1}^m \partial G_i(P^s, U^{i, s}_{-1})/\partial P^s \right)\\ &\qquad \qquad \quad \ge 1 - \lambda \left(\sum\limits_{j = m+1}^{m+n} \max(0, f'_j(P^s)) - \sum\limits_{i = 1}^m \min(0, f'_i(P^s)) \right). \end{split} \end{equation} | (2.16) |
The first two partial derivatives are clearly nonnegative. For the third partial derivative we have used the following readily verified fact about the Godunov flux:
\begin{equation} \min(0, f'_h(\beta)) \le {\partial G_h \over \partial \beta}(\beta, \alpha) \le 0 \le {\partial G_h \over \partial \alpha}(\beta, \alpha) \le \max(0, f'_h(\alpha)), \end{equation} | (2.17) |
and thus the third partial derivative is nonnegative due to (1.4) and the CFL condition (2.6).
Remark 6. If all of the fluxes
\begin{equation} \partial P^{s+1}/\partial P^s \ge 1 - \lambda \left(n \max(0, f'(P^s)) - m \min(0, f'(P^s)) \right), \end{equation} | (2.18) |
from which it is clear that
Lemma 2.2. Assuming that the initial data satisfies
Proof. The assertion is true for
\begin{equation} \begin{split} \tilde{U}^{h}_{\ell} = 0, \quad \tilde{P} = 0, \\ \hat{U}^{h}_{\ell} = R, \quad \hat{P} = R. \end{split} \end{equation} | (2.19) |
It is readily verified that
\begin{equation} \tilde{U}_{\ell}^h \le U^{0, h}_\ell \le \hat{U}_{\ell}^h, \quad \tilde{P} \le P^0 \le \hat{P}. \end{equation} | (2.20) |
After an application of a single step of the finite volume scheme, these ordering relationships are preserved, as a result of Lemma 2.1. Since
\begin{equation} \tilde{U}_{\ell}^h \le U^{1, h}_\ell \le \hat{U}_{\ell}^h, \quad \tilde{P} \le P^1 \le \hat{P}. \end{equation} | (2.21) |
This proves the assertion for
Given
\begin{equation} K^{h}_\ell = \begin{cases} k^i, \quad & \ell < 0~ \text{ and}~ h\in \{1, \ldots, m \} , \\ k^j, \quad & \ell > 0 ~~\text{and}~~ h\in \{m+1, \ldots, m+n \} .\\ \end{cases} \end{equation} | (2.22) |
and
Lemma 2.3. The finite volume scheme of Section 2 is well-balanced in the sense that each
Proof. For each fixed
Fix
\begin{equation} k^i - \lambda \left(G_i(p_{ \vec{k}}, k^{i}) - G_i(k^i, k^i) \right) = k^i - \lambda \left(f_i(k^{i}) - f_i(k^i) \right) = k^i. \end{equation} | (2.23) |
Here we have used the definition of
\begin{equation} p_{ \vec{k}} - \lambda \left(\sum\limits_{j = m+1}^n G_j(k^j, p_{ \vec{k}}) - \sum\limits_{i = 1}^m G_i(p_{ \vec{k}}, k^i) \right) = p_{ \vec{k}}, \end{equation} | (2.24) |
where we have applied the first equation of (1.5).
With the notation
\begin{equation} Q_{\ell+1/2}^{h}[U^s, \hat{U}^s] = G_h(U^{h, s}_{\ell+1} \vee \hat{U}^{h, s}_{\ell+1}, U^{h, s}_{\ell} \vee \hat{U}^{h, s}_{\ell}) - G_h(U^{h, s}_{\ell+1} \wedge \hat{U}^{h, s}_{\ell+1}, U^{h, s}_{\ell} \wedge \hat{U}^{h, s}_{\ell}). \end{equation} | (2.25) |
Lemma 2.4. Let
\begin{equation} \begin{split} &\sum\limits_{i = 1}^m {\Delta} x {\Delta} t\sum\limits_{s = 1}^{+\infty} \sum\limits_{\ell < 0} \left|{U^{i, s}_{\ell}-\hat{U}^{i, s}_{\ell}}\right| \left(\xi_{\ell}^{s}-\xi_{\ell}^{s-1} \right)/{\Delta} t\\ &+\sum\limits_{i = 1}^m {\Delta} x {\Delta} t \sum\limits_{s = 0}^{+\infty} \sum\limits_{\ell \le 0} \, Q^i_{\ell-1/2}[U^s, \hat{U}^s] \left(\xi^{s}_{\ell} - \xi^{s}_{\ell-1} \right)/{\Delta} x\\ &+\sum\limits_{j = m+1}^{m+n} {\Delta} x {\Delta} t \sum\limits_{s = 1}^{+\infty} \sum\limits_{\ell > 0} \left|{U^{i, s}_{\ell}-\hat{U}^{i, s}_{\ell}}\right| \left(\xi_{\ell}^{s}-\xi_{\ell}^{s-1} \right)/{\Delta} t\\ &+\sum\limits_{j = m+1}^{m+n} {\Delta} x {\Delta} t \sum\limits_{s = 0}^{+\infty} \sum\limits_{\ell \ge 0} \, Q^j_{\ell+1/2}[U^s, \hat{U}^s] \left(\xi^{s}_{\ell+1} - \xi^{s}_{\ell} \right)/{\Delta} x\\ & +{\Delta} x {\Delta} t \sum\limits_{s = 1}^{+\infty} \, \left|{P^s-\hat{P}^s}\right| \left(\xi_{0}^{s}-\xi_{0}^{s-1} \right)/{\Delta} t \ge 0.\\ \end{split} \end{equation} | (2.26) |
Proof. From the monotonicity property (Lemma 2.1), a standard calculation [9] yields
\begin{equation} \begin{split} &\left|{U_{\ell}^{i, s+1}-\hat{U}_{\ell}^{i, s+1}}\right| \le \left|{U_{\ell}^{i, s}-\hat{U}_{\ell}^{i, s}}\right|\\ &\qquad \qquad \qquad \qquad - \lambda \left(Q^i_{\ell+1/2}[U^s, \hat{U}^s] - Q^i_{\ell-1/2}[U^s, \hat{U}^s] \right), \quad \ell \le -1, \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \, \, \, \, i \in \{1, \ldots, m \}, \\ &\left|{U_{\ell}^{j, s+1}-\hat{U}_{\ell}^{j, s+1}}\right| \le \left|{U_{\ell}^{j, s}-\hat{U}_{\ell}^{j, s}}\right|\\ &\qquad \qquad \qquad \qquad - \lambda \left(Q^j_{\ell+1/2}[U^s, \hat{U}^s] - Q^j_{\ell-1/2}[U^s, \hat{U}^s] \right), \quad \ell \ge 1, \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad j \in \{m+1, \ldots, m+n \}, \\ &\left|{P^{s+1}-\hat{P}^{s+1}}\right| \le \left|{P^{s}-\hat{P}^{s}}\right| - \lambda \left(\sum\limits_{j = m+1}^{m+n} Q_{1/2}^j[U^s, \hat{U}^s] - \sum\limits_{i = 1}^m Q_{-1/2}^i[U^s, \hat{U}^s]\right). \end{split} \end{equation} | (2.27) |
We first multiply each of the first and second set of inequalities indexed by
Lemma 2.5. Suppose that
Proof. The proof that the first condition of Definition 1.2 holds is a slight adaptation of a standard fact about monotone schemes for scalar conservation laws [9].
The proof is completed by verifying the second condition of Definition 1.2. Let
With Lemmas 2.1 through 2.5 in hand it is possible to repeat the proof of Theorem 3.3 of [2], which yields Theorem 2.6 below.
Theorem 2.6. For a given initial datum
We found that if
Initialization of
● Initialize
● Initialize
Example 1. This example demonstrates the appearance of a spurious bump when
Example 1. Left panel:
One can also get rid of the spurious bump by choosing
I thank two anonymous referees for their comments and suggestions.
In this appendix we prove that the fixed point iterations (2.9) converge to a solution of the equation (2.8). Let
\begin{equation} u_i = U_{-1}^{i, s} for i = 1, \ldots, m , \quad u_j = U_{1}^{j, s} ~\text{for }~ j = m+1, \ldots, m+n . \end{equation} | (A.1) |
Then (2.8) takes the form
\begin{equation} \Phi_{{\vec u}}^{ \text{out}}(p) = \Phi_{\vec{u}}^{ \text{in}}(p), \end{equation} | (A.2) |
where
\begin{equation} \Phi_{\vec{u}}^{ \text{in}}(p) = \sum\limits_{i = 1}^m G_i(p, u_i), \quad \Phi_{{\vec u}}^{ \text{out}}(p) = \sum\limits_{j = m+1}^{m+n} G_j(u_j, p). \end{equation} | (A.3) |
This notation agrees with that of [2], where it was shown that
With the simplified notation introduced above, the iterative scheme (2.9) becomes
\begin{equation} p_{\nu+1} = p_{\nu} - \lambda \left( \Phi_{{\vec u}}^{ \text{out}}(p_{\nu}) - \Phi_{\vec{u}}^{ \text{in}}(p_{\nu}) \right), \quad p_0 \in [0, R]. \end{equation} | (A.4) |
Proposition 1. The sequence
Proof. Note that
\begin{equation} \begin{split} &0 \le G_i(0, u_i), \quad G_i(R, u_i) = 0, \\ &G_j(u_j, 0) = 0, \quad 0 \le G_j(u_j, R). \end{split} \end{equation} | (A.5) |
Thus,
\begin{equation} \begin{split} &0 \le \Phi_{\vec{u}}^{ \text{in}}(0), \quad \Phi_{\vec{u}}^{ \text{in}}(R) = 0, \\ & \Phi_{{\vec u}}^{ \text{out}}(0) = 0, \quad 0 \le \Phi_{{\vec u}}^{ \text{out}}(R). \end{split} \end{equation} | (A.6) |
Define
\begin{equation} \Psi_{ \vec{u}}(p) = \Phi_{{\vec u}}^{ \text{out}}(p) - \Phi_{\vec{u}}^{ \text{in}}(p), \end{equation} | (A.7) |
and observe that
\begin{equation} \Psi_{ \vec{u}}(0) \le 0 \le \Psi_{ \vec{u}}(R). \end{equation} | (A.8) |
Define
\begin{equation} \Pi(p) = p - \lambda \Psi_{ \vec{u}}(p). \end{equation} | (A.9) |
\begin{equation} \Pi'(p) = 1 - \lambda \Psi'_{ \vec{u}}(p) \ge 1 - \lambda (m+n)L. \end{equation} | (A.10) |
Thanks to (A.10) and the CFL condition (2.6), it is clear that
\begin{equation} 0 \le \Pi(0) \le \Pi(p) \le \Pi(R) \le R ~\text{for}~ p \in [0, R] . \end{equation} | (A.11) |
Thus
\begin{equation} p_{\nu +1} = \Pi(p_{\nu}), \quad p_0 \in [0, R]. \end{equation} | (A.12) |
We can now prove convergence of the sequence
Now consider the case where
\begin{equation} \begin{split} p_{\nu+1}-p_{\nu} & = \Pi(p_{\nu}) - \Pi(p_{\nu-1}) \\ & = {\Pi(p_{\nu}) - \Pi(p_{\nu-1})\over p_{\nu} - p_{\nu-1} } (p_{\nu} - p_{\nu-1}). \end{split} \end{equation} | (A.13) |
Since
\begin{equation} \mathrm{sign}(p_{\nu+1}-p_{\nu}) = \mathrm{sign}(p_{\nu}-p_{\nu-1}) = \ldots = \mathrm{sign}(p_{1}-p_{0}). \end{equation} | (A.14) |
In other words, the sequence
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