In this section, we first reconstruct the optimal orthonormal POD basis from the elements of the snapshots via SVD technique and POD-method, and then establish the POD-based reduced-order FDEA with very few degrees of freedom and fully second-order accuracy for the traffic flow LWR model.

### 3.1 Form POD basis for snapshots

The set of snapshots {\{{\rho}_{i}^{l}\}}_{l=1}^{L} (1\le i\le I) can be used to constitute the following I\times L matrix:

\mathit{A}=\left(\begin{array}{cccc}{\rho}_{1}^{1}& {\rho}_{1}^{2}& \cdots & {\rho}_{1}^{L}\\ {\rho}_{2}^{1}& {\rho}_{2}^{2}& \cdots & {\rho}_{2}^{L}\\ \vdots & \vdots & \ddots & \vdots \\ {\rho}_{I}^{1}& {\rho}_{I}^{2}& \cdots & {\rho}_{I}^{L}\end{array}\right).

(8)

For the matrix \mathit{A}\in {R}^{I\times L}, we have the following SVD:

\mathit{A}=\mathit{U}\left(\begin{array}{cc}\mathbf{\Sigma}& 0\\ 0& 0\end{array}\right){\mathit{Y}}^{T},

(9)

where \mathit{U}=({\mathit{\varphi}}_{1},{\mathit{\varphi}}_{2},\dots ,{\mathit{\varphi}}_{I})\in {R}^{I\times I} and \mathit{Y}=({\mathit{\phi}}_{1},{\mathit{\phi}}_{2},\dots ,{\mathit{\phi}}_{L})\in {R}^{L\times L} consist of the orthonormal eigenvectors of \mathit{A}{\mathit{A}}^{T} and {\mathit{A}}^{T}\mathit{A}, respectively, \mathbf{\Sigma}=diag\{{\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{\ell}\}\in {R}^{\ell \times \ell} is the diagonal matrix, and {\sigma}_{i} (i=1,2,\dots ,\ell) are the positive singular values corresponding to *A* in a non-increasing order. They are connected to the eigenvalues of the matrices \mathit{A}{\mathit{A}}^{T} and {\mathit{A}}^{T}\mathit{A} in a manner such that {\lambda}_{i}={\sigma}_{i}^{2} (i=1,\dots ,\ell) and {\lambda}_{1}\ge {\lambda}_{2}\ge \cdots \ge {\lambda}_{\ell}.

Since the number of spatial nodes is far larger than that of time nodes extracted, *i.e.*, I\gg L, namely the order *I* for matrix \mathit{A}{\mathit{A}}^{T} is far larger than the order *L* for matrix {\mathit{A}}^{T}\mathit{A}, however, their non-zero eigenvalues are identical. Thus, we may first find the eigenvalues {\lambda}_{j} and the orthonormal eigenvectors {\mathit{\phi}}_{j} (j=1,2,\dots ,\ell) corresponding to the matrix {\mathit{A}}^{T}\mathit{A}, and then by the relationship

{\mathit{\varphi}}_{j}=\frac{1}{{\sigma}_{j}}\mathit{A}{\mathit{\phi}}_{j},\phantom{\rule{1em}{0ex}}j=1,2,\dots ,\ell ,

(10)

we may obtain the orthonormal eigenvectors {\mathit{\varphi}}_{j} (1\le j\le \ell \le L) corresponding to the non-zero eigenvalues for matrix \mathit{A}{\mathit{A}}^{T}.

Let

{\mathit{A}}_{M}=\mathit{U}\left(\begin{array}{cc}{\mathbf{\Sigma}}_{M}& 0\\ 0& 0\end{array}\right){\mathit{Y}}^{T},

where the diagonal matrix {\mathbf{\Sigma}}_{M}=diag\{{\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{M}\}\in {R}^{M\times M} consist of the first *M* main singular values. Define the norm of the matrix *A* as {\parallel \mathit{A}\parallel}_{2,2}={sup}_{\mathit{x}}\frac{{\parallel \mathit{A}\mathit{x}\parallel}_{2}}{{\parallel \mathit{x}\parallel}_{2}} (where {\parallel \cdot \parallel}_{2} is the norm of a vector). According to the relationship properties of the spectral radius and {\parallel \cdot \parallel}_{2,2} for the matrix, if M<\ell =rank\mathit{A} (\ell \le L), we have

\underset{rank(\mathit{B})\le M}{min}{\parallel \mathit{A}-\mathit{B}\parallel}_{2,2}={\parallel \mathit{A}-{\mathit{A}}_{M}\parallel}_{2,2}={\parallel \mathit{A}-\mathbf{\Phi}{\mathbf{\Phi}}^{T}\mathit{A}\parallel}_{2,2}=\sqrt{{\lambda}_{M+1}},

(11)

where \mathit{B}\in {R}^{I\times L}, \mathbf{\Phi}=({\mathit{\varphi}}_{1},{\mathit{\varphi}}_{2},\dots ,{\mathit{\varphi}}_{M}), and {\lambda}_{M+1} is (M+1)th eigenvalue of the matric \mathit{A}{\mathit{A}}^{T}. It is shown that {\mathit{A}}_{M} is an optimal representation of *A* and its error is \sqrt{{\lambda}_{M+1}}.

Denote the *L* column vectors of matrix *A* by {\mathit{a}}^{l}={({\rho}_{1}^{l},{\rho}_{2}^{l},\dots ,{\rho}_{m}^{l})}^{T} (l=1,2,\dots ,L) and {\mathit{\epsilon}}_{l} (l=1,2,\dots ,L) by unit column vectors except that the *l* th component is 1, while the other components are 0. Then we have

{\parallel {\mathit{a}}^{l}-{\mathit{a}}_{M}^{l}\parallel}_{2}={\parallel (\mathit{A}-\mathbf{\Phi}{\mathbf{\Phi}}^{T}\mathit{A}){\mathit{\epsilon}}_{l}\parallel}_{2}\le {\parallel \mathit{A}-\mathbf{\Phi}{\mathbf{\Phi}}^{T}\mathit{A}\parallel}_{2,2}{\parallel {\mathit{\epsilon}}_{l}\parallel}_{2}=\sqrt{{\lambda}_{M+1}},

(12)

where {\mathit{a}}_{M}^{l}={\sum}_{j=1}^{M}({\mathit{\varphi}}_{j},{\mathit{a}}^{l}){\mathit{\varphi}}_{j}, ({\mathit{\varphi}}_{j},{\mathit{a}}^{l}) is the canonical inner product for vectors {\mathit{\varphi}}_{j} and {\mathit{a}}^{l}. Inequality (12) shows that {\mathit{a}}_{M}^{l} are the optimal approximation to {\mathit{a}}^{l} and their errors are all \sqrt{{\lambda}_{M+1}}. Then \mathbf{\Phi}=({\mathit{\varphi}}_{1},{\mathit{\varphi}}_{2},\dots ,{\mathit{\varphi}}_{M}) (M\ll L) is an orthonormal basis corresponding to *A*, which is known as an orthonormal POD basis.

### 3.2 Establish the POD-based reduced-order FDEA for the traffic flow LWR model

In order to establish the POD-based reduced-order FDEA for the traffic flow LWR model, it is necessary to introduce the following symbols:

\begin{array}{r}{\mathit{\rho}}_{I}^{n}={({\rho}_{1}^{n},{\rho}_{2}^{n},\dots ,{\rho}_{I}^{n})}^{T},\phantom{\rule{2em}{0ex}}{\mathit{\alpha}}_{I}^{n}={({\alpha}_{1}^{n},{\alpha}_{2}^{n},\dots ,{\alpha}_{M}^{n})}^{T},\\ {\rho}_{I}^{\ast n}={({\rho}_{1}^{\ast n},{\rho}_{2}^{\ast n},\dots ,{\rho}_{I}^{\ast n})}^{T}=\mathbf{\Phi}{\mathit{\alpha}}_{I}^{n},\end{array}

(13)

and to rewrite (4) as the following iterative scheme:

\{\begin{array}{l}{\rho}_{i}^{n+1}={\rho}_{i}^{n}+\mathrm{\u25b3}{t}^{2}{u}_{m}^{2}{(1-\frac{2{\rho}_{i}^{n}}{{\rho}_{m}})}^{2}\frac{{\rho}_{i+1}^{n}-2{\rho}_{i}^{n}+{\rho}_{i-1}^{n}}{2\mathrm{\u25b3}{x}^{2}}\\ \phantom{{\rho}_{i}^{n+1}=}-{u}_{m}\mathrm{\u25b3}t(1-\frac{2{\rho}_{i}^{n}}{{\rho}_{m}})\frac{{\rho}_{i+1}^{n}-{\rho}_{i-1}^{n}}{2\mathrm{\u25b3}x},\phantom{\rule{1em}{0ex}}1\le i\le I-1,1\le n\le N-1,\\ {\rho}_{i}^{0}={\rho}^{0}({x}_{i}),\phantom{\rule{1em}{0ex}}i=1,2,\dots ,I,\\ {\rho}_{0}^{n}={\rho}_{0}(0,{t}_{n}),\phantom{\rule{2em}{0ex}}{\rho}_{J}^{n}={\rho}_{0}(J,{t}_{n}),\phantom{\rule{1em}{0ex}}n=1,2,\dots ,N.\end{array}

(14)

Thus, the first equation of (14) is rewritten as the following iterative scheme of vector form:

{\mathit{\rho}}_{I}^{n+1}={\mathit{\rho}}_{I}^{n}+\mathit{Q}\left({\mathit{\rho}}_{I}^{n}\right){\mathit{\rho}}_{I}^{n},\phantom{\rule{1em}{0ex}}1\le n\le N-1,

(15)

where \mathit{Q}({\mathit{\rho}}_{I}^{n}) is a matrix determined by the second and third terms on the right hand of the first equation in (14).

If {\mathit{\rho}}_{I}^{n} (n=1,2,\dots ,L) in (13) are replaced for {\rho}_{I}^{\ast n}={({\rho}_{1}^{\ast n},{\rho}_{2}^{\ast n},\dots ,{\rho}_{I}^{\ast n})}^{T}=\mathbf{\Phi}{\mathbf{\Phi}}^{T}{\mathit{\rho}}_{I}^{n}, *i.e.*, {\mathit{\alpha}}_{I}^{n}={\mathbf{\Phi}}^{T}{\mathit{\rho}}_{I}^{n} (n=1,2,\dots ,L) and {\mathit{\rho}}_{I}^{n} (n=L+1,L+2,\dots ,N) in (15) are replaced for {\mathit{\rho}}_{I}^{\ast n}={({\rho}_{1}^{\ast n},{\rho}_{2}^{\ast n},\dots ,{\rho}_{I}^{\ast n})}^{T}=\mathbf{\Phi}{\mathit{\alpha}}_{I}^{n} (n=L+1,L+2,\dots ,N), we obtain the following POD-based reduced-order FDEA which only contains *M* degrees of freedom on every time level (n>L):

\{\begin{array}{l}{\mathit{\alpha}}_{I}^{n}={\mathbf{\Phi}}^{T}{\mathit{\rho}}_{I}^{n},\phantom{\rule{1em}{0ex}}n=1,2,\dots ,L,\\ \mathbf{\Phi}{\mathit{\alpha}}_{I}^{n+1}=\mathbf{\Phi}{\mathit{\alpha}}_{I}^{n}+\mathit{Q}(\mathbf{\Phi}{\mathit{\alpha}}_{I}^{n})\mathbf{\Phi}{\mathit{\alpha}}_{I}^{n},\phantom{\rule{1em}{0ex}}n=L,L+1,\dots ,N-1,\end{array}

(16)

where {\mathit{\rho}}_{I}^{n} (n=1,2,\dots ,L) is a given vector formed by the first *L* solutions in (14). Since all columns in **Φ** are orthonormal vectors, the second equation in (16) multiplied by {\mathbf{\Phi}}^{T} is reduced into the following POD-based reduced-order FDEA:

\{\begin{array}{l}{\mathit{\alpha}}_{I}^{n}={\mathbf{\Phi}}^{T}{\mathit{\rho}}_{I}^{n},\phantom{\rule{1em}{0ex}}n=1,2,\dots ,L,\\ {\mathit{\alpha}}_{I}^{n+1}={\mathit{\alpha}}_{I}^{n}+{\mathbf{\Phi}}^{T}\mathit{Q}(\mathbf{\Phi}{\mathit{\alpha}}_{I}^{n})\mathbf{\Phi}{\mathit{\alpha}}_{I}^{n},\phantom{\rule{1em}{0ex}}n=L,L+1,\dots ,N-1.\end{array}

(17)

After {\mathit{\alpha}}_{I}^{n} (n=1,2,\dots ,N) are obtained from the system of (17), the POD reduced-order FDEA solutions for the traffic flow LWR model are presented as follows:

{\mathit{\rho}}_{I}^{\ast n}={({\rho}_{1}^{\ast n},{\rho}_{2}^{\ast n},\dots ,{\rho}_{I}^{\ast n})}^{T}=\mathbf{\Phi}{\alpha}_{I}^{n},\phantom{\rule{1em}{0ex}}n=1,2,\dots ,N.

(18)

Further, we can obtain the POD-based reduced-order FDEA solutions of the flow *q* and the equilibrium speed *u* as follows, respectively:

{q}_{i}^{\ast n}={u}_{m}(1-\frac{{\rho}_{i}^{\ast n}}{{\rho}_{m}}){\rho}_{i}^{\ast n},\phantom{\rule{2em}{0ex}}{u}_{i}^{\ast n}={u}_{m}(1-\frac{{\rho}_{i}^{\ast n}}{{\rho}_{m}}),\phantom{\rule{1em}{0ex}}i=1,2,\dots ,I,n=1,2,\dots ,N.

(19)

**Remark 3.1** The system of equations (16) or (17) has no repeating computation and is different from the existing POD-based reduced-order numerical computational methods (see, *e.g.*, [14–20, 22–31]) based on POD technique.