1 Introduction

Proper orthogonal decomposition (POD)^[1-15] is a most welcome and resultful numerical tool in the reduction of complicated systems and also takes an important role in the model order reduction for the classical Galerkin methods,finite element (FE) methods,and finite difference (FD) schemes^[16-34]. However,most existing model order reduction methods as mentioned were established by means of the POD bases formulated from the classical approximate solutions on the entire time interval $[0,T]$,before iterating the reduced-order approximate solutions on the identical interval $[0,T]$,which were the worthless repeated calculations.

The reduced-order POD finite volume element (FVE) formats for the two-dimensional (2D) parabolic equations^[35] and the 2D viscoelastic problem^[36] have been posed,but they also belonged to the repeated calculations on the same time interval $[0,T]$. In order to avoid the unrewarding repeated computations in the reduced-order POD formats,some reduced-order POD extrapolating FVE formats^[37-40] for the non-stationary Navier-Stokes problems,the non-stationary incompressible Boussinesq problems,Sobolev equations,and non-stationary Stokes problems have successively been established by Luo's research team since 2013. A POD-based reduced-order FE format^[32] without adopting extrapolation and iteration but including the worthless repeated computations for the 2D hyperbolic equations has been posed,but as far as we know,no article for the reduced-order POD extrapolating FVE format of the 2D hyperbolic equations has been reported. Therefore,in this study,we develop the reduced-order POD extrapolating FVE format for the 2D hyperbolic equations. The advantage of the format consists in that the POD basis is formulated with the set of snapshots composed of the few classical FVE solutions on the initial very short period of time $[0,T_0]$ ($T_0\ll T$) before seeking the other reduced-order solutions on the period of time $[T_0,T]$. This implies that the reduced-order POD format differs from most existing reduced-order POD formats as mentioned above and is important improvement and development for them.

Because the FVE method^[41-42],also named as a box method^[43] or a generalized difference approach^[44],is easier to operate than the FE method and the FD scheme and can deal with the complicated computational domains as well as can guarantee local mass conservation and has very satisfactory function in real-life applications. Therefore,the reduced-order POD extrapolating FVE format established here has more virtue than the reduced-order FE formulation^[32].

The plan of the paper is arranged as below. We first build the semi discrete variational format relative time and the fully discrete FVE format for the 2D hyperbolic equations and extract the set of snapshots from the very few FVE solutions on the first very short period of time in Section 2. Then,we formulate the POD basis with the set of snapshots and build the reduced-order POD extrapolating FVE format containing fewer degrees of freedom but holding sufficiently high accuracy in Section 3. Next,we furnish the error analysis of the reduced-order solutions and the algorithm procedure for solving the reduced-order format in Section 4. Then,we furnish a numerical example to confirm the correctness of theoretical conclusions in Section 5. Finally,we furnish main conclusions in Section 6.

2 Classical FVE method for hyperbolic equations and generation of snapshots

In the following,we first establish the semi discrete format relative time for the 2D hyperbolic equations and furnish the existence and the error analysis of its solutions,and then directly build the classical fully discrete FVE format from the semi discrete format relative time and estimate the errors of the FVE solutions. Thus,we can avoid the theoretical analysis for the semi discrete FVE format relative space variables so that the procedure becomes simpler than the available approaches^[44-45].

2.1 2D hyperbolic equations

Let $\Theta\subset{\mathbb{R}}^2$ be a bounded domain with the boundary $\partial\Theta$. The 2D hyperbolic equations can be stated as below.

Problem 1 Seek $u$ such that

(1)

where $u_{tt}=\frac{\partial^2 u}{\partial t^2}$,$\nu$ is a positive constant,$T$ is the time upper limit,$g(x,y,t)$ is a given source function,$\phi_0(x,y)$ and $\phi_1(x,y)$ are two given initial value functions,and $\phi(x,y,t)$ is a given boundary value function.

The hyperbolic equations hold the real-life application background, for example,describing the many nature phenomena such as the wave motion,shift in porous media,film vibrations,and acoustic vibrations in gas. For convenience,we suppose that $\phi_0(x,y)$, $\phi_1(x,y)$,and $\phi(x,y,t)$ are all zero functions in the subsequent theoretical study.

2.2 Semi discrete format relative time

The Sobolev spaces as well as their norms^[42] presented in the article are well known. Put $H=H_0^1(\Theta)$ and $A(u,w)=(\nabla u,\nabla w)$,where $(\cdot,\cdot)$ represents the inner product in the space $L^2(\Theta)$. Thus,the weak solution for Problem 1 may be stated as below.

Problem 2 Seek $u(t)\in H\,(0\le t\le T$) such that

(2)

It has been proved that Problem 2 has a unique solution that satisfies the following estimate^[47-49]:

(3)

where $|\cdot|_s$ represents the semi-norm in $H^s(\Theta)$, $\|\cdot\|_{H^m(H^s)}$ is the norm in $H^m(0,T;H^s(\Theta))$, $u^{(m)}(t)$ is the $m$th-derivative of $u$ relative to the time $t$,and $M_t=\sqrt{2\exp(t)/\min\{1,\nu\}}$.

It is well noting that $\|\cdot\|_1$ is equivalent to $\|\nabla\cdot\|_0$ in the space $H^1_0(\Theta)$,i.e.,we have the following inequality:

(4)

where $C_0$ is a constant independent of $v$.

Let $k$ be the time step increment that satisfies $k N=T$,let $v^n=v(x,y,t_n)$ represent the function value of $v(x,y,t)$ at time $t_n=nk ~ (0\le n\le N)$,and

Thus,the semi discrete format relative time $t$ for Problem 2 may be stated as below.

Problem 3 Seek $u^{n+1}\in H~ (n=1,2,\cdots,N-1)$ that satisfies

(5)

(6)

where $g^{n}= g(x,y,t_{n})$.

Let $\tilde{A}(u^{n+1},v)=2(u^{n+1},v)+k^2\nu A(u^{n+1},v)$, $\tilde{F}(v)=(4u^{n}-2u^{n-1},v) -k^2\nu A(u^{n-1},v)$ $+2k^2(g^{n},v)$. Then,Problem 3 can be restated as follows.

Problem 4 Seek $u^{n+1}\in H~ (n=1,2,\cdots,N-1)$ such that

(7)

(8)

For Problem 4,we have the following results.

Theorem 1 Problem $4,$ i$.$e$.,$ Problem $3$ has a unique set of solutions $\left\{u^{n}\right\}_{n=1}^N\subset H$$.$ Furthermore$,$ if $f\in H^3(0,$T$;$$ L^2(\Theta))$$,$ we have the following error estimates}:

(9)

Proof Since,for the fixed $k$, $\widetilde{A}(\cdot,\cdot)$ is a positive definite,symmetric,and bounded bilinear form on $H$,and $\widetilde{F}(\cdot)$ is also a bounded linear function on $U$,we deduce from the Lax-Milgram theorem^[50-52] that Problem 4,i.e.,Problem 3 has a unique set of solutions $\left\{u^{n}\right\}_{n=1}^N\subset H$.

By choosing $t=t_n$ in Problem 2 and by Taylor's formula,we achieve

(10)

where $ R_1=k^4(u^{(4)}(\xi_{1n})+u^{(4)}(\xi_{2n}))/24$,$ R_2=k^2(u^{(2)}(\xi_{3n})+u^{(2)}(\xi_{4n}))/4$,and ${\xi _{in}} \in ({t_{n - 1}},{t_n})(i = 1,2,3,4).$

Put $\mathcal{E}_n=u(t_n)-u^n\,(n=0,1,2,\cdots,N$). By subtracting (7) from (10),we have

(11)

By choosing $v=\mathcal{E}_{n+1}-\mathcal{E}_{n-1}$ in (11) and by the Hölder inequality,we obtain

(12)

Note that $\|\mathcal{E}_{n+1}-\mathcal{E}_{n-1}\|_0\le \|\mathcal{E}_{n+1}-\mathcal{E}_n\|_0+\|\mathcal{E}_n-\mathcal{E}_{n-1}\|_0$ and $\|\mathcal{E}_{n+1}-\mathcal{E}_{n-1}\|_0\le \|\mathcal{E}_{n+1}\|_0+\|\mathcal{E}_{n-1}\|_0\le C_0(\|\nabla \mathcal{E}_{n+1}\|_0+\|\nabla \mathcal{E}_{n-1}\|_0)$. Thus,from (12),we obtain

(13)

Note that $\mathcal{E}_1=\mathcal{E}_0=0$. By summing (13) from $1$ to $n$,we obtain

(14)

From (14),we immediately obtain

(15)

By summing (15) from $1$ to $n$,we achieve

(16)

Thus,by combining (14) and (16) with (3),we immediately obtain the error estimates (9) of Theorem 1.

2.3 Classical fully discrete FVE format

For building the classical fully discrete FVE format,we still need to adopt an FVE approximation to the spatial variables.

Let $\Im_h^*=\{V_z\}$ be the dual partition based on the quasi-uniform triangulation^[50-52],$\Im_h = \{K\}$ of $\Theta$,whose elements $V_z$ consist of the sub-domains $K_z$ sharing vertex $ z\in Z_h=\bigcup\limits_{K\in\Im_h}$ $Z_h(K)$, where $Z_h(K)$ represents a set of the vertices of $K$,$ z=(x_z,y_z)\in Z_h(K)$,and the quadrilaterals $K_z$ formed by jointing the barycentre $ z_K$ of $K\in \Im_h$ with segments to the centre points of the sides of $K$ (see Fig. 1).

The trial subspace $H_h$ and the test subspace $\widetilde{H}_h$ are,respectively,chosen as follows:

where $\mathcal{ P}_l(e)$ ($l=0,1$ and $e=K$ or $V_z$) are the $l$th polynomial subspaces on $e$.

It is evident that $H_h\subset H$ and the elements of the test subspace $\widetilde{H}_h$ are denoted by the following functions:

For $v\in H$,let $\Pi_hv$ be the interpolating projection of $v$ on the space $H_h$. From the interpolating theory^{[44, 50-52]},we achieve

(17)

where $C$ represents a positive constant that may be not equal at each place,independent of $h$ and $k$.

For $v\in H$,let $\Pi_h^*v$ be the interpolating projection of $v$ on $\widetilde{H}_h$,

(18)

From the interpolating theory^{[44, 50]},we achieve

(19)

In addition,$\Pi_h^*$ has the following results^[53-54].

Lemma 1 For any $w_h\in H_h$},

Put

(20)

where $\widetilde{A}(v_h,\phi_z)=\int_{\partial V_z}(\frac{\partial v_h}{\partial x}{\rm d}y -\frac{\partial v_h}{\partial y}{\rm d}x )$ and $ z=(x_z,y_z)$.

Furthermore,$\Pi^*_h$ still has the following known results^{[44, 53]}.

Lemma 2 $A(v_h,\Pi^*_h\widetilde{v}_h)$ is in a positive definite$,$ symmetric$,$ and bounded bilinear form},

Lemma 3 The interpolating projection $\Pi_h^*$ has the following results}:

Put $\mid\mid\mid \!v_h\!\mid\mid\mid _0=( v_h,\Pi_h^*v_h)^{1/2}$$.$ Thus$,$ for any $v_h\in H_h$$,$ $\mid\mid\mid\! v_h\!\mid\mid\mid _0$ is equivalent to $\|v_h\|_0$$,$ i.e.$,$ there are two positive constants $C_3$ and $C_4$ that satisfy

Thus,the classical FVE format of Problem 2 is stated as below.

Problem 5 Seek $u_h^{n+1}\in H_h~(1\le n\le N-1)$ that satisfies

(21)

(22)

For analyzing the errors of the solutions to Problem 5,we still need to define a generalized elliptic projection. For given solutions $u^n\in H\,(n=0,1,2,\cdots,N$) of Problem 3,let $R_h: U\to U_h$ satisfy that $R_hu^1=\Pi_hu^1$,$R_hu^0=\Pi_hu^0$,and

(23)

Then,by the standard FE method^[50-52],we have

(24)

(25)

For Problem 5,we have the next main results.

Theorem 2 Under the conditions of Theorem $1,$ Problem $5$ has a unique set of solutions $\{u_h^{n}\}_{n=1}^N\subset H_h$ that satisfies the following stabilization}:

(26)

where $\widetilde{C}_1=C_4nk/\min\{C_3,\nu/(2C_0C_4)\}\le C_4T/\min\{C_3,\nu/(2C_0C_4)\}$$.$ If $k^2=O(h)$$,$ the errors between the solutions $u_h^n$ of Problem $5$ and the solution $u(t)$ of Problem $2$ are

(27)

Proof By the same approach as Theorem 1,by Lemmas 2 and 3,we easily prove that Problem 5 has a unique set of solutions $\{u_h^{n}\}_{n=1}^N\subset H_h$. By choosing $v_h=u_h^{n+1}+u_h^{n-1}$ in Problem 5 and using the Hölder inequality and Lemmas 2 and 3,we obtain

(28)

By noting that $\mid\mid\mid u_h^{n+1}+u_h^{n-1}\mid\mid\mid _0\le C_4\|u_h^{n+1}+u_h^{n-1}\|_1\le C_4C_0\|\nabla (u_h^{n+1}+u_h^{n-1})\|_0$,from (28),we obtain

(29)

Note that $u_h^0=u_h^1=0$. By summing (29) from $1$ to $n$,we have

(30)

which yields (26).

By subtracting Problem 5 from Problem 3 taking $v=w_h$ and by Lemma 2,we obtain the following error equations:

(31)

Put $u^{n}-u_h^{n}=(u^{n}-R_hu^{n})+(R_hu^{n}-u_h^{n})=\rho^n+\mathcal{E}^n$. Then,from (23) and (31),we obtain

(32)

By taking $w_h=\mathcal{E}^{n+1}-\mathcal{E}^{n-1}$ in (32) and by Lemma 1,we have

(33)

By Taylor's formula and (27),we obtain

(34)

With the inverse estimate inequality,we have $\|\nabla(\mathcal{E}^{n+1}-\mathcal{E}^{n-1})\|_0\le Ch^{-1}\|\mathcal{E}^{n+1}-\mathcal{E}^{n-1}\|_0 \le Ch^{-1}(\|\mathcal{E}^{n+1}-\mathcal{E}^{n}\|_0+\| \mathcal{E}^{n}-\mathcal{E}^{n-1}\|_0)$. Because $\|\nabla(\mathcal{E}^{n+1}-\mathcal{E}^{n-1})\|_0\le \|\nabla \mathcal{E}^{n+1}\|_0+\|\nabla \mathcal{E}^{n-1}\|_0$,by simplifying (34),we obtain

(35)

If $k^2=O(h)$,by summing (35) from $1$ to $n$ and using Lemma 3,we obtain

(36)

By applying the triangle inequality to (36),we obtain

(37)

By summing (37) from $1$ to $n-1$,we achieve

(38)

Combining (25) with (38) and Theorem 1 yields (27).

Remark 1 If the initial values $\phi_0(x,y)$ and $\phi_1(x,y)$ are non-zero,then we only need to choose $ u_h^i(x,y)=\Pi_h \phi_i(x,y)~ (i=0,1)$. We can achieve the same conclusions as Theorem 2. Therefore,as long as $g(x,y,t)$,$\nu $, $\phi_0(x,y)$,$\phi_1(x,y)$,$k$,$h$,and $H_h$ are appointed,we may achieve a set of solutions $\left\{u_h^n\right\}_{n=1}^N\subset H_h$ by Problem 5. We choose the set of snapshots $\{u_h^n\}_{n=1}^L$ which consists of the first $L$ solutions of the set of solutions $u_h^n~(1\le n\le N)$ to Problem 5.

Remark 2 When we solve the real-life engineering problems,we may extract the set of snapshots from engineering problems.

3 Generation of POD basis and reduced-order POD extrapolating FVE format

In the following,we adopt the POD method^{[24, 28, 35-36]} to formulate the POD basis and to build the reduced-order POD extrapolating FVE format for the 2D hyperbolic equations.

For the set of snapshots $\{u_{h}^n(x,y)\}_{n=1}^L$ in Section 2, put $U_i(x,y)= u_{h}^i(x,y)\,(i=1,2,\cdots,L )$ and

(39)

spanned by the set of snapshots $ \{U_i\}_{i=1}^ L $ at least one of which is non-zero. Let $ \{{\varphi}_j\}_{j=1}^l$ be an orthonormal basis of ${\cal V}$ with $l=\dim {\cal V}$. Then,each element in ${\cal V}$ can be denoted by

(40)

The POD approach is equivalent to seeking the orthonormal set $\{\varphi_j\}_{j=1}^l$ that satisfies

(41)

subject to

(42)

The set $ \{{\varphi}_j\}_{j=1}^d$ is referred to as the POD basis with the rank $d$.

We formulate the matrix ${ A}=(A_{ij})_{ L \times L }\in {\mathbb{R}}^{ L \times L }$ of the snapshots $\{U_i\}_{i=1}^ L $ by $ {A}_{ij}=(U_i,U_j)_{U}/L$. Because the matrix $ A$ is semi positive definite with the rank $l$,there exists a unique set of solution for (41) and (42) that satisfies the following result^{[24, 28, 35-36]}.

Proposition 1 The POD basis of rank $d\leq l$ is defined by the positive eigenvalues arranged as $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_l>0$ and the associated eigenvectors $ v^i=(a_1^i,a_2^i,\cdots,a_L^i)\,(i=1,2,\cdots, l)$ of $ A$ as follows}:

(43)

that satisfies the next error estimate}

(44)

Put $H^d={\rm span}\{\varphi_1,\varphi_2,\cdots,\varphi_{d} \}$. Define a elliptic operator $P^d$: $H_h\to H^d$ such that for any $u_h\in H_h$,

Thus,from the continuation theorems of functional analysis^[51],there exists an extension operator $P^h$: $H\to H_h$ of $P^d$ such that $P^h|_{H_h}=P^d: H_h\to H^d$ satisfies,for any $v\in H$,

(45)

Thanks to (45),the operator $P^h$ meets the following property^[50]:

(46)

Further,$P^h$ also meets the following known inequality^{[24, 28]}:

(47)

In addition,we have the following known result^{[24, 28, 35-36, 46]}.

Lemma 4 If $u_h^{i}\in {\cal V}~(i=1,2,\cdots,L)$ are the solutions to Problem $5,$ then we have

(48)

Moreover$,$ for the solutions $u^{n}\in H\,(1\le n\le N)$ of Problem $3,$ we have the following estimates}:

(49)

Thus,by $H^d$,the reduced-order POD extrapolating FVE format of Problem 2 is stated as follows.

Problem 6 Seek $u_{d}^{n}\in H^d\,(1\le n\le N$) that satisfy

(50)

(51)

Let $u_{d}^n=\beta_1^n\varphi_1+\beta_2^n\varphi_2+\cdots+\beta_{d}^n\varphi_{d}~ (1\le n\le N)$. By $\Pi_h^*$,Problem 6 can be restated as follows.

Problem 7 Seek $(\beta_1^{n},\beta_2^{n},\cdots, \beta_{d}^{n})^{\rm T}\in R^d\,(1\le n\le N$) such that

(52)

(53)

where

Remark 3 Because $U_h$ is spanned by the piecewise linear function,the entire freedom-degrees (unknowns) of Problem 5 at each time node is $N_h$ (where $N_h$ represents the number of vertices of triangles in $\Im_h$,see Ref.^[44] or Ref.^[50]). However,the entire freedom-degrees of Problem 6 at each time node (when $n> L)$ is only $d$ $(d\ll l\leq L\ll N_h)$. The number $N_h$ in real-life engineering problems is larger than hundred of millions,but $d$ is very small and is only the number of the first few main eigenvalues (for instance,the numerical example in Section 5,$d=6$,but $N_h$ is larger than $4\times10^6$). This shows that Problem 6 is the reduced-order POD extrapolating FVE format that does not include repeated computation and totally differs from the now available reduced-order POD formats^[16-34].

4 Error estimates of solutions and algorithm procedure for Problem 6 4.1 Error estimates of reduced-order POD extrapolating FVE solutions

In the following,we devote to analyzing the errors for the reduced-order POD extrapolating FVE solutions. For this purpose,we need to review the following discrete Gronwall lemma.

Lemma 5^{[50, 56]} If $\{\widetilde{A}_n\}$ and $\{\widetilde{B}_n\}$ are two non-negative sequences$,$ $\{\widetilde{C}_n\}$ is a monotone positive sequence$,$ and they meet} $ \widetilde{A}_n+\widetilde{B}_n\le \widetilde{C}_n+\overline{\lambda}\sum\limits_{j=0}^{n-1}\widetilde{A}_j$ ( where} $\overline{\lambda}>0$ and} $n=1,2,\cdots$) and} $\widetilde{A}_0+\widetilde{B}_0\le \widetilde{C}_0$, then} $ \widetilde{A}_j+\widetilde{B}_j\le \widetilde{C}_j\exp(j\overline{\lambda})~(j=0,1,2,\cdots)$.

For Problem 6,we have the next conclusions.

Theorem 3 Under conditions of Theorem $2,$ Problem $6$ has a sole set of solutions \[\{ u_d^n\} _{n = 1}^N \subset {H^d}\] that satisfies

(54)

(55)

The inequalities $(54)$ and $(55)$ imply that the solutions to Problem $6$ are stable$.$ If $k^2=O(h)$$,$ then the errors between the reduced-order POD extrapolating FVE solutions $u_{d}^n$ to Problem $6$ with the classical FVE solutions $u_h^n$ to Problem $5$ are

(56)

(57)

Proof Note that $H^d\subset U_h$. For a fixed $k$,let $A_{d}(u_{d}^{n+1},w_{d})=2(u_{d}^{n+1},\Pi_h^* w_{d})+ k^2\nu A(u_{d}^{n+1}$,$w_{d})$, $F_{d}(w_{d})=(4u_{d}^{n}-2u_{d}^{n-1},\Pi_h^* w_{d})+2k^2(g^{n},\Pi_h^* w_{d})-k^2\nu A(u_{d}^{n-1},w_{d})$. Thus,Problem 6 is restated as follows.

Problem 8 Seek $u_{d}^{n}\in H^d$ ($1\le n\le N$) such that

(58)

(59)

With Lemmas 2 and 3,we obtain

(60)

where $\widetilde{\alpha}=\min\{k^2\nu,2C_3\}$. This implies that $A_{d}(\cdot,\cdot)$ is a positive definite and symmetric bilinear form on $H^d$. It is obvious that $F_{d}(\cdot)$ and $A_{d}(\cdot,\cdot)$ are bounded. Therefore,from the Lax-Milgram theorem^{[50, 52]},we can deduce that Problem 8,i.e.,Problem 6 has one and only one set of solutions $\{u_{d}^n\}_{n=1}^N\subset H^d$.

By (46),we have

(61)

which yields (54). Let $Q_{d}^n=u_{d}^{n+1}-u_{d}^{n}$. Taking $w_{d}=u_{d}^{n+1}-u_{d}^{n-1}=(u_{d}^{n+1}-u_{d}^{n})+(u_{d}^{n}-u_{d}^{n-1})=Q_{d}^n+Q_{d}^{n-1}$ in Problem 6 yields that

(62)

By Lemmas 2 and 3 and the Hölder and Cauchy inequalities,from (62),we obtain

(63)

Note that $\mid\mid\mid Q_{d}^n-Q_{d}^{n-1}\mid\mid\mid _0\le \mid\mid\mid Q_{d}^n\mid\mid\mid _0+\mid\mid\mid Q_{d}^{n-1}\mid\mid\mid _0$ or $\mid\mid\mid Q_{d}^n-Q_{d}^{n-1}\mid\mid\mid _0\le \mid\mid\mid u_{d}^{n+1}\mid\mid\mid _0+$ $\mid\mid\mid u_{d}^{n-1}\mid\mid\mid _0$. From (63),we obtain

(64)

Note again that $H^2(L^2)\hookrightarrow\hookrightarrow L^\infty(L^2)$. By Taylor's formula,Theorem 2,and Lemma 4,we have

(65)

Thus,summing (64) from $L$ to $n$ ($n\le N-1$) and using (65) yield

(66)

Because $\mid\mid\mid Q_{d}^n\mid\mid\mid _0\,=\,\mid\mid \mid u_{d}^{n+1}-u_{d}^n\mid\mid\mid _0\,\ge\, \mid\mid\mid u_{d}^{n+1}\mid\mid\mid _0-\mid\mid\mid u_{d}^n\mid\mid\mid _0$,by summing (66) from $L$ to $n$ ($n\le N-1$), we obtain

(67)

which yields (55).

Note that $H^d\subset H_h$. By Problem 5,subtracting Problem 6,and choosing $w_h=w_{d}\in H^d$,we achieve

(68)

(69)

If $n=1,2,\cdots,L$,by Lemma 4 and (47),from (68),we have

(70)

Let $u_h^n-u_{d}^n=(u_h^n-P^du_h^n)+(P^du_h^n-u_{d}^n)=\varrho^n+\vartheta^n$. By Lemmas 2 and 3,we have

(71)

If $k^2=O(h)$,from the error equation (69),the properties of $P^d$,the Hölder inequality,Lemmas 2 and 3,and (47),we have

(72)

Combining (71) with (72),we have

(73)

Because,by Lemma 4 and Theorem 2,we have $\|\nabla\varrho^{i}\|_0\le \|\nabla(u_h^{i}-u^{i}) \|_0+\|\nabla(u^{i}-P^hu^{i})\|_0+\|\nabla P^h(u^{i}-u_h^{i})\|_0\le Ch$ and $\|\varrho^{i}\|_0\le Ch^2$. Therefore,summing (73) from $L$ to $n$ ($L\le n\le N-1$) and using Lemma 4 and Proposition 1,we obtain

(74)

When $k$ is sufficiently small and meets $Ck\le1/2$,from (74),we achieve

(75)

By applying Lemma 5 to (75),we achieve

(76)

i.e.,

(77)

(78)

Note that $\|\vartheta^{L}\|_0 =\|u_h^{L}-P^du_h^L\|_0 \le C\Big(k^2L\sum\limits_{j=d+1}^{l}\lambda_j\Big)^{\frac{1}{2}}$. If $k^2=O(h)$,summing (78) from $L$ to $n-1$,we have

(79)

Thus,we finish the proof of Theorem 3.

By combining Theorem 3 with Theorem 2,we immediately achieve the following conclusion.

Theorem 4 Under the conditions of Theorem $3,$ the errors between the solution $u$ to Problem $2$ with the solutions $u_{d}^n$ to Problem $6$ are as follows}:

Remark 4 For Theorems 3 and 4,we have the following further applications:

(i) The errors in Theorem 3 act as the judgement determining the number $d$ of the POD basis and the element number $L$ of snapshots, i.e.,if $k^2=O(h)$ and $k\le 1$,then $d$ and $L$ should be chosen to satisfy $\Big(L\sum\limits_{m=d+1}^l\lambda_m\Big)^{\frac{1}{2}}=O(k)$. Therefore,the number $L$ of snapshots cannot be too large. The best value should satisfy $\sqrt{L}<5$ (usually $L=20$) so that it does not affect the total errors and can reduce the computational load when solving the eigenvalue problem.

(ii) The error terms $(n-L)kh\,(n=L+1,L+2,\cdots,N$) in Theorems 3 and 4 are induced by extrapolation and act as a judgement to renew the POD basis,i.e.,if $(n-L)^2h^2\le L\sum\limits_{m=d+1}^l\lambda_m ~(n=L+1,L+2,\cdots,N)$, $u_{d}^n~(1\le n\le N)$ is just the solution to Problem 6 that meets the accuracy requirement. Otherwise,i.e.,if $(n-L)^2h^2> L\sum\limits_{m=d+1}^l\lambda_m ~(n=L+1,L+2,\cdots, N)$,it is necessary to renew the POD basis.

(iii) The condition $k^2=O(h)$ in Theorems 3 and 4 is not strict, which is only an assumption for the theoretical analysis and may be replaced with other conditions.

4.2 Algorithm procedure for solving reduced-order POD extrapolating FVE format

The algorithm procedure for solving the reduced-order POD extrapolating FVE format includes the following seven steps.

Step 1 For the given $\nu$,$g(x,y,t)$,$\phi_0(x,y)$, $\phi_1(x,y)$,$\phi(x,y,t)$,$T$,and the needed accuracy $\delta=\max\{k^2,h^2\}$ $=k^2$ (since the condition $k^2=O(h)$), by solving the next classical FVE format at the first few $L$ steps (usually choose $L=20$),

we achieve the snapshots $ U_n(x,y)=u_h^n\,(1\le n\le L)$,which can be obtained from the engineering problems or the previous given results.

Step 2 Form the matrix $ A=({A}_{ij})_{L\times L}$ meeting ${A}_{ij}=(\nabla U_i,\nabla U_j)/L=(\nabla u_h^i,\nabla u_h^{j})/L$.

Step 3 Seek the eigenvalues of $ A$ arranged as \[{\lambda _1} \geqslant {\lambda _2} \geqslant \cdots \geqslant {\lambda _l} > 0(l = \dim \{ {U_1},{U_2}, \cdots ,{U_L}\} )\] and the corresponding eigenvectors $ v^j=(a_1^j,a_2^j,\cdots,a_L^j)^{\rm T}\,(j=1,2,\cdots,l)$.

Step 4 Confirm the number $d$ of the POD bases and $L$ that meet $L\sum\limits_{m=d+1}^l\lambda_m\le k^2$.

Step 5 Formulate the POD basis $\varphi_j=\sum\limits_{i=1}^{L}a_i^ju_h^i/{\sqrt{L\lambda_j}} ~(1\le j\le d)$.

Step 6 By solving the next reduced-order POD extrapolating FVE format that only contains $d$ unknowns at each time node,

we achieve $(\beta_1^n,\beta_2^n,\cdots,\beta_{d}^n)^{\rm T}\in {\mathbb{R}}^d$ ($1\le n\le N$). Further,we achieve the reduced-order POD extrapolating FVE solutions $u_{d}^n=\sum\limits_{i=1}^d\beta_i^n\varphi_i(x,y)\,(1\le n\le N)$.

Step 7 If $h^2(n-L)^2\!\le\! L\sum\limits_{m=d+1}^l\lambda_m ~(L+1\le n\le N)$,then $u_{d}^{n}~(1\le n\le N)$ are the solutions to Problem 6 that satisfy the accuracy requirement. Otherwise,i.e.,if\[{h^2}{(n - L)^2} > L\sum\limits_{m = d + 1}^l {{\lambda _m}} (L + 1 \leqslant n \leqslant N)\],let $W_i=u_{d}^{i}~(i=n-L,n-L-1,\cdots,n-1)$ and return to Step 2.

5 Numerical experiment

In the following,we use the numerical experiment to confirm the feasibility and efficiency of the reduced-order POD extrapolating FVE format and the correctness of the theoretical conclusions.

We choose the computational domain as $\overline{\Theta}=[0, 2]\times[0,2]$ and discretize $\overline{\Theta}$ into right triangle elements with the diameter $h=\sqrt{2}\times10^{-3}$, constituting the triangularization $\Im_h$ with the number of node $N_h=4\times 10^{6}$. We choose the barycenter dual decomposition as the dual decomposition $\Im^*_h$. For the condition $k^2=O(h)$ in Theorem 3 satisfied,we choose the time step $k=4\times10^{-2}$.

We specify $\nu=1$,the source function $g(x,y,t)=0$,and the initial values as follows:

and the boundary value

According to the algorithm procedure for solving the POD-based reduced-order POD extrapolating FVE format in Subsection 3.4,by solving the classical FVE format,we first achieve twenty classical FVE solutions $u_h^n$ ($n=1,2,\cdots,20$) as the snapshots. Then, we achieve the error $\Big(20k^{2}\sum\limits_{j=7}^{20}\lambda_j\Big)^{1/2}$ $\leq 3\times10^{-3}$ when $d=6$ and $k=0.04$. This means that we only need to choose the first six POD bases to span subspace $H^d$. Finally,we achieve two reduced-order POD extrapolating FVE solutions at $t=250k=10$ and $t=500k=20$,but update the POD basis once when $t=15$,which are drawn in the right charts in Figs. 2 and 3,respectively. We also achieve the numerical FVE solutions $u_h^n$ by solving the classical FVE format when $t=10$ and $t=20$, which are drawn in the left charts in Figs. 2 and 3,respectively. By comparing the two charts in Figs. 2 and 3,we find that the right charts are distinctly slower dissipation than the left ones. It shows that the reduced-order POD extrapolating FVE solutions are more stable and better than the classical FVE solutions due to the classical FVE format with four millon unknowns at each time node, but the reduced-order POD extrapolating FVE format with very few (only six) unknowns at each time node to lessen the computational load and alleviate the accumulation of the truncated errors. This implies that the reduced-order POD extrapolating FVE format is far more feasible and efficient than the classical FVE format for solving the 2D hyperbolic equations.

Because the POD bases are arranged according to the eigenvalues in a non-increasing order,when we take first six eigenvalues,the reduced-order POD extrapolating FVE solutions include main information. Thus,it is reasonable that the error curves in Fig. 4 smoothly decrease. Therefore,Fig. 4 exhibits the errors between the classical FVE solutions $u_h^n$ and the reduced-order POD extrapolating FVE solutions $u_{d}^n$ with 20 different numbers of POD bases associated the norm in $L^2(\Theta)$ at $t=10$ and 20, respectively. It is shown that the numerical conclusions are consistent with the theoretical results because the numerical and theoretical errors are not larger than $3\times10^{-3}$. This means that the reduced-order POD extrapolating FVE format is very effective for solving the 2D hyperbolic equations.

6 Conclusions

In this study,we build the reduced-order POD extrapolating FVE format by means of the POD method for the 2D hyperbolic equations, furnish the error estimates between the classical FVE solutions and the reduced-order POD extrapolating FVE solutions and the algorithm procedure for solving the reduced-order POD extrapolating FVE format,and present an example to confirm the correctness of the theoretical results and illustrate the feasibility and efficiency of the reduced-order POD extrapolating FVE format. It is sufficiently shown that the reduced-order POD extrapolating FVE format can greatly lessen the degrees of freedom so that it can alleviate the accumulation of the truncated errors and the computational load. Moreover,it is shown that the reduced-order POD extrapolating FVE format has very extensive applications.