A New Model Dimension Reduction Technique Based on Finite Volume Element and Proper Orthogonal Decomposition for Solving the 2D Hyperbolic Equation

Abstract

This article mainly researches the model dimension reduction in the finite volume element (FVE) method based on proper orthogonal decomposition (POD) for the two-dimensional (2D) hyperbolic equation. For this objective, an FVE method with unconditional stability and second-order temporal accuracy, and the existence, stability, and error estimates of the FVE solutions are first reviewed. Thereafter, most importantly, a new FVF model dimension reduction (FVEMDR) formulation is established by applying POD technology to lower the dimension of the vectors composed of unknown coefficients for the FVE solutions. The greatest contribution of this article is the theoretical analysis of the existence, unconditional stability, and error estimations for the FVEMDR solutions. Moreover, in computation, two sets of numerical simulations are provided to confirm the validity of the theoretical results and show the effectiveness of the FVEMRD formulation.

1. Introduction

Suppose that $\Omega \subset R^2$ indicates a bounded and connected region and its boundary is denoted by $\partial \Omega$. We research the following 2D hyperbolic equation [1,2].

Problem 1.

Seek $\varpi :[0,T_e]\to C^2\left(\overline{\Omega }\right)$ through the following system:

$$\left\{\begin{array}{cc}{\displaystyle {\varpi }_{tt}(x,y,t)+\Delta \varpi (x,y,t)=f(x,y,t),}\hfill & {\displaystyle (x,y,t)\in \Omega \times (0,T_e),}\hfill \\ {\displaystyle \varpi (\mathit{x},t)=0,}\hfill & {\displaystyle (x,y,t)\in \partial \Omega \times (0,T_e),}\hfill \\ {\displaystyle \varpi (x,y,0)={\varpi }_0(x,y),{\varpi }_t(x,y,0)={\varpi }_1(x,y),}\hfill & {\displaystyle (x,y)\in \Omega ,}\hfill \end{array}\right.$$

(1)

where $T_e$ is a given temporal limit, $C^2\left(\overline{\Omega }\right)$ denotes a function space composed of functions with second-order continuous derivatives onto $\overline{\Omega }$, ${\varpi }_{tt}(\mathit{x},t)={\displaystyle \frac{{\partial }^2\varpi (x,y,t)}{\partial t^2}}$, ${\varpi }_t(\mathit{x},t)={\displaystyle \frac{\partial \varpi (x,y,t)}{\partial t}}$, and $\Delta ={\displaystyle \frac{{\partial }^2}{\partial x^2}}+{\displaystyle \frac{{\partial }^2}{\partial y^2}}$, indicating the 2D Laplace operator, and the initial values ${\varpi }_0(x,y)$ and ${\varpi }_1(x,y)$, and the source term $f(x,y,t)$ are sufficiently smooth known functions.

The hyperbolic equation is a significant mathematical model with a practical application background. It can be normally applied to depict wave and acoustic phenomena in the natural world, for instance, fluid dynamics, displacement issues in porous medium, membrane vibrations, and the propagation of electromagnetic waves (see [1,2,3,4]). Consequently, it has received considerable attention (e.g., [1,2,3,4,5]). Nevertheless, when it contains the complex source term $f(x,y,t)$, initial value functions ${\varpi }_0(x,y)$ and ${\varpi }_1(x,y)$, or computational regions, it is highly difficult to compute its analytic solution. The most effective approaches are to calculate its numerical solutions [1,2,3,4,5].

Among all the numerical solutions used to compute the hyperbolic equation, the finite volume element (FVE) method (also referred to as the box technique [6] or generalized difference method [7,8]) is considered a highly valid numerical approach, since it is easily implemented and can offer flexibility in dealing with complex computational regions [9,10]. The most important aspect is that it may keep the conservation for local mass and energy. This is extremely desirable for many actual calculations. It has also been broadly used to compute numeric solutions for various partial differential equations (PDEs) [7,11].

Nevertheless, when the TVE method is utilized to compute the practical engineering computations, it generally possesses many (as many as millions) unknowns. To settle the puzzle of excessive unknowns in the traditional FVE computations for the hyperbolic equation, a dimension reduction technique [4] regarding the trial function space in the FVE method for the 2D hyperbolic equation was first set up through the proper orthogonal decomposition (POD).

However, as stated in [4,12], the dimension reduction methods of trial function subspace (i.e., finite element (FE) subspace) exist with some drawbacks, such as the complexity of theoretical analysis and the fact that the precision of its FVE solutions is influenced by the dimension reduction process for the trial function space. Consequently, this paper adopts an alternative strategy, which is to maintain the trial function space in the FVE method as unchanged, while only reducing the dimension of the vectors composed of the unknown solution coefficients for the FVE formulation, thus constructing the new FVE model dimension reduction (FVEMDR) formulation. The resulting FVEMDR formulation not only significantly reduces the unknowns but also maintains the trial function space as unchanged. This ensures that the precision of the FVEMDR solutions remains unaltered.

A substantial amount of numeric experiments (see those in [13,14,15,16,17,18,19]) show that the POD is one of the most valid tools for lowering the dimension of numeric formulations. It has also been used to set up the reduction dimension method of the trial function spaces for settling the Allen–Cahn equation [20], the unsteady incompressible Boussinesq problems [21], the parabolic equation [22], and the unsteady Stokes problems [23]. Nevertheless, the preceding reduction dimensionality methods almost belong to the reduction dimensional category for the finite element (FE) subspaces, namely the dimension reduction in the vectors composed of the FE basic functions. Hence, they are absolutely different from the FVEMDR formulation in this paper.

Although the dimension reduction models of the unknown FE solution coefficient vectors for the fourth-order parabolic equation [24], the nonlinear parabolic problem [25], the 2D fourth-order hyperbolic equation [26], and the extended Fisher–Kolmogorov equation [27] had been proposed separately, to the best of our knowledge, there have currently been no studies where the POD technology is utilized to cut down the dimensionality of the vectors made up of unknown solution coefficients in the FVE method for the hyperbolic equations and construct the FVEMDR formulation. Particularly, the FVEMDR formulation for the hyperbolic equation also possesses similar local conservation properties to the conventional FVE method and has many applications. Consequently, researching the FVEMDR formulation for the hyperbolic equation has great value.

The rest of this research is arranged as follows. We review the temporal semi-discretized (TSD) and FVE formulations with second-order temporal accuracy as well as the association with the theory results in Section 2. In Section 3, by rewriting the FVE formulation in functional form into the matrix model and applying the POD technology to cut down the dimension of the vectors made up of the unknown coefficients for the FVE solutions while maintaining the basic functions in trial function space as unvaried, we create a new FVEMDR formulation and prove the existence, unconditional stability, and error estimations for the FVEMDR solutions through matrix analysis. In Section 4, we resort to two groups of numeric simulations to prove the validity of the obtained theory outcomes as well as the effectiveness of the FVEMDR formulation. Lastly, in Section 5, we summarize the major conclusions of this paper and offer future research prospects.

2. Review the Temporal Semi-Discretized and Finite Volume Element Formulations

2.1. Variational Formulation and TSD Scheme

The Sobolev spaces and their norms adopted hereinafter are classical (see [28,29,30]). Let $\mathbb{W}=H_0^1(\Omega )$. Through the Green formula, we may draw the below weak form for Problem 1.

Problem 2.

For arbitrary $t\in (0,T_e)$, compute $\varpi \in \mathbb{W}$ from the following system:

$$\left\{\begin{array}{cc}{\displaystyle \left(\vartheta ,{\varpi }_{tt}\right)+a(\vartheta ,\varpi )=(\vartheta ,f),}\hfill & \forall \vartheta \in \mathbb{W},\hfill \\ {\displaystyle {\varpi }_t(x,y,0)={\varpi }_1(x,y),\varpi (x,y,0)={\varpi }_0(x,y),}\hfill & (x,y)\in \Omega ,\hfill \end{array}\right.$$

(2)

wherein $a(·,·)=(\nabla ·,\nabla ·)$ and $(\varpi ,\vartheta )={\int }_{\Omega }\varpi \vartheta \mathrm{d}\mathit{x}\mathrm{d}y$ indicates the $L^2$ inner product.

The existence and stability of the weak solutions for Problem 2 can be proven through the same technology as attesting Theorem 3.3.1 in [28].

We firstly review the TSD scheme for the 2D hyperbolic equations. Hence, we suppose that $N>0$ indicates a natural number, $\Delta t={\displaystyle \frac{T_e}{N}}$ is the temporal step, and ${\varpi }^n$ ($0⩽n⩽N$) are the approximation to $\varpi (x,y,t)$ at $t_n=n\Delta t$. Thereupon, the TSD scheme can be set up as below.

Problem 3.

Compute ${\left\{{\varpi }^n\right\}}_{n=1}^N\subset \mathbb{W}$ from the following system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{1}{\Delta t^2}({\varpi }^{n+1}-2{\varpi }^n+{\varpi }^{n-1},\vartheta )+\frac{1}{2}a({\varpi }^{n+1}+{\varpi }^{n-1},\vartheta )}\hfill \\ {\displaystyle =(f^n,\vartheta ),\forall \vartheta \in \mathbb{W},1⩽n⩽N-1,}\hfill \\ {\displaystyle {\varpi }^0={\varpi }_0,{\varpi }^1={\varpi }_1,\mathrm{in}\Omega .}\hfill \end{array}\right.\end{array}$$

(3)

The following conclusions of the existence, stability, and error estimates of the TSD solutions of the problem are given in [13] (Theorem 3.1.1).

Theorem 1.

Under the case of $f\in L^2(0,T_e;L^2(\Omega ))$, a sole sequence of solutions $\left\{{\varpi }^n\right\}\subset \mathbb{W}$ may be computed through Problem 3 to satisfy the following boundedness $(i.e., stationrity)$:

$$\parallel \nabla {\varpi }^n{\parallel }_0⩽c.$$

where $c>0$ represents a real number independent of $\Delta t$, but dependent on the initial conditions ${\varpi }_0$ and ${\varpi }_1$, and the source function f. And under the case of $f\in$$H^3(0,T_e;L^2(\Omega ))$, the following error estimations hold:

$$\begin{array}{c}{\displaystyle \Delta t\parallel \nabla (\varpi \left(t_n\right)-{\varpi }^n){\parallel }_0+{\parallel \varpi \left(t_n\right)-{\varpi }^n\parallel }_0⩽c\Delta t^2,}\hfill \end{array}$$

(4)

wherein $\varpi \left(t_n\right)=\varpi (x,y,t_n)$$(1⩽n⩽N)$.

Remark 1.

Theorem 1 implies that the TSD solutions are stable and can reach an optimal order error estimate.

2.2. The Finite Volume Element Formulation

To set up the FVE model, it is necessary to further utilize the FVE technique to discretize the spatial variables to Problem 3. To this end, we suppose that ${\mathfrak{F}}_h=\left\{E\right\}$ is a consistent triangulation onto $\Omega$, where $h=max\{{[{(x_1-x_2)}^2+{(y_1-y_2)}^2]}^{1/2}:(x_1,y_1),(x_2,y_2)\in E,\forall E\in {\mathfrak{J}}_h\}$ (see [7]), and resort to the technique in [7] to construct a control volume set ${\mathfrak{F}}_h^{\ast }$ based on ${\mathfrak{F}}_h$ as follows.

(1)

By linking the barycenter ${\mathit{z}}_E$ of $E\in {\mathfrak{F}}_h$ with the mid-points of the edges of E with line segments, we divide E as three quadrilateral elements $E_z$ (see Figure 1a), where $\mathit{z}=(x_z,y_z)\in {\mathbb{Z}}_h\left(E\right)$, ${\mathbb{Z}}_h\left(E\right)$ is the vertex set of E, ${\mathbb{Z}}_h={\bigcup }_{E\in {\mathfrak{F}}_h}{\mathbb{Z}}_h\left(E\right)$, and the set composed of the interior vertices in ${\mathbb{Z}}_h$ is denoted by ${\mathbb{Z}}_h^{\circ }$.

(2)

Through per vertex $\mathit{z}\in {\mathbb{Z}}_h$, we can construct a control volume $V_z$ composed of the union of the sub-domains $K_z$ to share the vertex $\mathit{z}$ (see Figure 1b) and the set of control volumes covering the region $\Omega$ is known as the dual subdivision ${\mathfrak{F}}_h^{\ast }$ based on the triangulation ${\mathfrak{F}}_h$.

The trial function subspace ${\mathbb{W}}_h$ and the test function subspace ${\tilde{\mathbb{W}}}_h$ in the FVE method are separately worded below:

$$\begin{array}{ccc}& & {\displaystyle {\mathbb{W}}_h=\{{\vartheta }_h\in C\left(\overline{\Omega }\right)\cap \mathbb{W}:{\vartheta }_h{\mid }_E\in {\mathbb{P}}_1\left(E\right),\forall E\in {\mathfrak{F}}_h\},}\hfill \\ & & {\displaystyle {\tilde{\mathbb{W}}}_h=\left\{{\vartheta }_h\in L^2(\Omega ):{\vartheta }_h{{|}_{V_z\in {\mathfrak{F}}_h^{\ast }}\in {\mathbb{P}}_0\left(V_z\right);{\vartheta }_h|}_{V_z}=0,\mathrm{if}\partial \Omega \cap V_z\ne \varnothing \right\},}\hfill \end{array}$$

where ${\mathbb{P}}_s\left(e\right)(s=0,1)$ represent the multinomial spaces defined on e ($e=E$ or $V_z$) with degree $⩽s$.

It is evident that ${\mathbb{W}}_h\subset \mathbb{W}$. If ${\Pi }_{h}v$ denotes an interpolating function of $\forall v\in \mathbb{W}$ in the trial function space ${\mathbb{W}}_h$, then with the interpolating theorem [28] (Theorem 1.3.13), we get

$$|v-{\Pi }_h{v|}_r⩽c{h}^{2-s}{\left|v\right|}_2,s=0,1,\mathrm{when}v\in H^2(\Omega ),$$

(5)

where $c>0$ also denotes a real number independent of the temporal step $\Delta t$ and spatial subdivision parameter h.

Furthermore, the test function space ${\tilde{\mathbb{W}}}_h$ can be generated through the following basic functions:

$${\varphi }_z(x,y)=\left\{\begin{array}{c}1,(x,y)\in V_z,\hfill \\ {\displaystyle 0,(x,y)\in \overline{\Omega }\backslash V_z,}\hfill \end{array}\mathit{z}=(x_z,y_z)\in Z_h^{\circ }.\right.$$

whereupon, $\forall v_h\in {\tilde{\mathbb{W}}}_h$ may be indicated as

$$v_h=\sum _{\mathit{z}\in Z_h^{\circ }}{v}_h\left(\mathit{z}\right){\varphi }_z.$$

If we assume that ${\Pi }_h^{\ast }v$ indicates an $L^2$ interpolation function of $\forall v\in \mathbb{W}$ in the test subspace ${\tilde{\mathbb{W}}}_h$, namely that

$${\Pi }_h^{\ast }v=\sum _{\mathit{z}\in Z_h^{\circ }}v\left(\mathit{z}\right){\varphi }_z,$$

(6)

then, from the interpolation theorem [28] (Theorem 1.3.13), we get

$$\parallel v-{\Pi }_h^{\ast }{v\parallel }_0⩽Ch{\left|v\right|}_1.$$

(7)

Furthermore, the interpolation function ${\Pi }_h^{\ast }v$ has the following properties.

Lemma 1.

If ${\vartheta }_h\in {\mathbb{W}}_h$, then

$${\displaystyle {\int }_K({\vartheta }_h-{\Pi }_h^{\ast }{\vartheta }_h)\mathrm{d}\mathit{x}\mathrm{d}y=0,K\in {\mathfrak{F}}_h;}$$

$$\parallel {\vartheta }_h-{\Pi }_h^{\ast }{\vartheta }_h{\parallel }_{L^q(\Omega )}⩽c{h}_K{\parallel {\vartheta }_h\parallel }_{W^{1,q}(\Omega )},1⩽q⩽\infty .$$

From [13] (Lemmas 3.1.3 and 3.1.4), we also get two other properties of the interpolation operator ${\Pi }_h^{\ast }$.

Lemma 2.

The bilinear functional $a({\vartheta }_h,{\Pi }_h^{\ast }{\vartheta }_h)$ is bounded, symmetric, and positive definite on ${\mathbb{W}}_h$, namely

$$a({\vartheta }_h,{\Pi }_h^{\ast }{v}_h)=a(v_h,{\Pi }_h^{\ast }{\vartheta }_h)=a({\vartheta }_h,v_h),\forall {\vartheta }_h,v_h\in {\mathbb{W}}_h,$$

contenting

$$a({\vartheta }_h,v_h)⩽\parallel \nabla {\vartheta }_h{\parallel }_0\parallel \nabla v_h{\parallel }_0,a({\vartheta }_h,{\vartheta }_h)={\parallel \nabla {\vartheta }_h\parallel }_0^2,\forall {\vartheta }_h,v_h\in {\mathbb{W}}_h.$$

Lemma 3.

The interpolation function ${\Pi }_h^{\ast }v$ possesses the following properties.

(i) There holds symmetry below:

$$(u_h,{\Pi }_h^{\ast }{\overline{u}}_h)=({\Pi }_h^{\ast }{u}_h,{\overline{u}}_h),\forall u_h,{\overline{u}}_h\in {\mathbb{W}}_h$$

(ii) $\forall u\in H^{\ell }(\Omega )$$(\ell =0,1)$ and ${\vartheta }_h\in U_h,$ and there hold the following error estimations:

$$|(u,{\vartheta }_h)-(u,{\Pi }_h^{\ast }{\vartheta }_h)|⩽c{h}^{\ell +s}{\parallel u\parallel }_{\ell }{\parallel {\vartheta }_h\parallel }_s,s=0,1.$$

(iii) If we set ${\mid \mid \mid {\vartheta }_h\mid \mid \mid }_0={\mid ({\vartheta }_h,{\Pi }_h^{\ast }{\vartheta }_h)\mid }^{1/2}$, then $\mid \mid \mid ·\mid \mid {\mid }_0$ is equivalent to ${\parallel ·\parallel }_0$ in ${\mathbb{W}}_h$, namely there are two real numbers $c_0>0$ and $c_1>0$ meeting

$$c_0\parallel {\vartheta }_h{\parallel }_0⩽\mid \mid \mid {\vartheta }_h\mid \mid {\mid }_0⩽c_1{\parallel {\vartheta }_h\parallel }_0,\forall {\vartheta }_h\in {\mathbb{W}}_h.$$

Thus, the FVE model may be created below.

Problem 4.

Calculate ${\left\{{\varpi }_h^n\right\}}_{n=1}^N\subset {\mathbb{W}}_h$ from the following system of equations:

$$\begin{array}{c}{\displaystyle \frac{1}{\Delta t^2}({\varpi }_h^{n+1}-2{\varpi }_h^n+{\varpi }_h^{n-1},{\Pi }_h^{\ast }{\vartheta }_h)+\frac{1}{2}a({\varpi }_h^{n+1}+{\varpi }_h^{n-1},{\Pi }_h^{\ast }{\vartheta }_h)}\hfill \\ {\displaystyle =(f^n,{\Pi }_h^{\ast }{\vartheta }_h),\forall {\vartheta }_h\in {\mathbb{W}}_h,1⩽n⩽N-1,}\hfill \end{array}$$

(8)

$$\begin{array}{ccc}& & {\displaystyle {\varpi }_h^0={\Pi }_h{\varpi }_0(x,y),{\varpi }_h^1={\Pi }_h{\varpi }_1(x,y),(x,y)\in \Omega .}\hfill \end{array}$$

(9)

The following conclusions for the existence, stability, and error estimations for the FVE solutions about Problem 4 have been given in [13] (Theorem 3.1.5).

Theorem 2.

Under the entirely same hypotheses as Theorem 1, a sole solution set ${\left\{{\varpi }_h^n\right\}}_{n=1}^N\subset {\mathbb{W}}_h$ may be calculated through Problem 4 to satisfy the following unconditional stability:

$$\begin{array}{c}{\displaystyle \parallel \nabla {\varpi }_h^n{\parallel }_0⩽c,n=1,2,\dots ,N.}\hfill \end{array}$$

(10)

Furthermore, under the case of $\Delta t=O\left(h\right)$, the following error bounds hold:

$$\parallel \varpi \left(t_n\right)-{\varpi }_h^n{\parallel }_0+h{\parallel \nabla (\varpi \left(t_n\right)-{\varpi }_h^n)\parallel }_0⩽c\left(\Delta t^2+h^2\right),n=1,2,\dots ,N.$$

(11)

Remark 2.

Theorem 2 explains that the FVE solutions have unconditional stability and can theoretically attain an optimal order error estimation.

2.3. The Matrix Representation for the FVE Formulation

The most pivotal step of building the FVEMDR formulation is to reword the FVE formulation in matrix form. To achieve this goal, we hypothesize that M is the dimensionality of the trial function subspace ${\mathbb{W}}_h$. Noticing that the bilinear functional $A(·,·)\equiv [\Delta t^{2}a(·,·)+(·,{\Pi }^{\ast }·)+({\Pi }^{\ast }·,·)]$ is bounded and positive definite, we claim that it is also an inner product in ${\mathbb{W}}_h$. Accordingly, through the standard orthogonal basic existence [30] (Proposition 1.6.21) and the Schmidt orthogonalization principle [30] (Section 1.6.3), we assert that the trial function space ${\mathbb{W}}_h$ defines a series of orthonormal basic functions ${\left\{{\xi }_i\right\}}_{i=1}^M$ under the inner product $A(·,·)$. Whereupon, the trial function space ${\mathbb{W}}_h$ may be restated below:

$$\begin{array}{ccc}& & {\displaystyle {\mathbb{W}}_h=\left\{{\vartheta }_h\in \mathbb{W}\cap C\left(\overline{\Omega }\right):{\vartheta }_h{|}_E\in {\mathbb{P}}_1\left(E\right),\forall E\in {\mathfrak{F}}_h\right\}\equiv \mathrm{span}\left\{{\xi }_1,{\xi }_2,\dots ,{\xi }_M\right\}.}\hfill \end{array}$$

Through the basic functions ${\left\{{\xi }_i\right\}}_{i=1}^M$, the FVE solutions ${\varpi }_h^k$ may be worded as

$${\varpi }_h^n=\sum _{i=1}^M{\xi }_i{W}_i^n={({\xi }_1,{\xi }_2,\dots ,{\xi }_M)}^T·{(W_1^n,W_2^n,\dots ,W_M^n)}^T=\mathit{\xi }·{\mathit{W}}^n.$$

(12)

where the vector $\mathit{\xi }={({\xi }_1,{\xi }_2,\dots ,{\xi }_M)}^T$ is composed of ${\xi }_1,{\xi }_2,$$\dots ,$${\xi }_M$ and the vectors ${\mathit{W}}^n=$$(W_1^n,W_2^n,\dots ,$$W_M^n{)}^T$ are composed of the unknown coefficients in the FVE solutions ${\varpi }_h^n$.

Noticing that $({\xi }_i,{\Pi }^{\ast }{\xi }_j)={\displaystyle \frac{1}{2}}[({\xi }_i,{\Pi }^{\ast }{\xi }_j)+({\Pi }^{\ast }{\xi }_i,{\xi }_j)]$ can be drawn by Lemma 1, the FVE formulation of functional form can be equivalently reexpressed as the matrix form below.

Problem 5.

Compute ${\left\{{\mathit{W}}^n\right\}}_{n=1}^N\subset {\mathbb{R}}^M$ and ${\left\{{\varpi }_h^n\right\}}_{n=1}^N$$\subset {\mathbb{W}}_h$ from the following system:

$$\left\{\begin{array}{c}{\displaystyle {\mathit{W}}^1=\left(({\Pi }_h{\varpi }_1,\mathit{\xi })\right),{\mathit{W}}^0=\left(({\Pi }_h{\varpi }_0,\mathit{\xi })\right),}\hfill \\ {\displaystyle {\mathit{W}}^{n+1}-{\mathit{W}}^n={\mathit{W}}^n-{\mathit{W}}^{n-1}-\Delta t^2\mathit{D}{\mathit{W}}^n+\Delta t^2{\mathit{F}}^n,1⩽n⩽N-1,}\hfill \\ {\displaystyle {\varpi }_h^n={\mathit{W}}^n·\mathit{\xi },1⩽n⩽N,}\hfill \end{array}\right.$$

(13)

where the matrix $\mathit{D}={\left((\nabla {\xi }_i,\nabla {\xi }_j)\right)}_{M\times M}$ is positive definite and ${\displaystyle {\mathit{F}}^n=\left((f^n,{\Pi }^{\ast }\mathit{\xi })\right)}$.

We can draw the following conclusions to Problem 5.

Theorem 3.

Under the entirely same hypotheses as Theorem 2, an FVE solution vector set ${\left\{{\mathit{W}}^n\right\}}_{n=1}^N$$\subset {\mathbb{R}}^M$ and an FVE solution set ${\left\{{\varpi }_h^n\right\}}_{n=1}^N$$\subset {\mathbb{W}}_h$ may be solely calculated through Problem 5 to satisfy the following unconditional boundedness, namely unconditional stability:

$$\begin{array}{ccc}& & {\displaystyle \parallel {\mathit{W}}^n\parallel ⩽c,1⩽n⩽N,}\hfill \end{array}$$

where $\parallel {\mathit{W}}^n\parallel$$(1⩽n⩽N)$ indicate the Euclid norms for the vectors ${\mathit{W}}^n$.

Proof. 

By multiplying (12) via the FE basic vectors $\mathit{\xi }$, we draw

$$\begin{array}{ccc}& & {\displaystyle {\mathit{W}}^n={\varpi }_h^n\mathit{\xi }/{\parallel \mathit{\xi }\parallel }^2,1⩽n⩽N.}\hfill \end{array}$$

Thereupon, via Theorem 2 and (12), we claim that Problem 5 possesses a sole FVE solution vector set ${\left\{{\mathit{W}}^n\right\}}_{n=1}^N$$\subset {\mathbb{R}}^M$ as well as a sole FVE solution set ${\left\{{\varpi }_h^n\right\}}_{n=1}^N$$\subset {\mathbb{W}}_h$.

By the inverse estimation [28] (Corollary 1.3.2) and (10) in Theorem 2, we can attest

$$\begin{array}{c}\hfill {\displaystyle \parallel {\mathit{W}}^n\parallel =|{\varpi }_h^n|\parallel \mathit{\xi }\parallel /\parallel \mathit{\xi }\parallel ⩽c\parallel {\varpi }_h^n{\parallel }_{\infty }⩽c{\parallel \nabla {\varpi }_h^n|}_0⩽c,1⩽n⩽N.}\end{array}$$

This means that the FVE solution vector set ${\left\{{\mathit{W}}^n\right\}}_{n=1}^N$ of Problem 5 is unconditionally bounded, namely it shows unconditional stability. This ends the attestation of Theorem 3. □

Remark 3.

If the initial values ${\varpi }_0$ as well as ${\varpi }_1$, the source function f, and the spatial and temporal partition parameters h and $\Delta t$ are given, an FVE solution set ${\left\{{\varpi }_h^n\right\}}_{n=1}^N$$\subset {\mathbb{W}}_h$ could be computed through Problem 5. Nevertheless, when Problem 5 is applied to compute the hyperbolic equation in actual application, it often has a lot of (as many as millions) unknowns. Consequently, it is highly needed to adopt the POD technology to cut down the dimensionality for the FVE formulation (namely Problem 5) and build a new FVEMDR formulation.

3. The Finite Volume Element Model Dimension Reduction Formulation

3.1. Construction for the POD Basic Vectors

The POD basic vectors may be constructed through the below flow-chart.

(1)

Compute an FVE solution vector set ${\left\{{\mathit{W}}^n\right\}}_{n=1}^L$ through Problem 5 at the first L temporal nodes to arrange the matrix $\mathit{B}=({\mathit{W}}^1,{\mathit{W}}^2,$$\dots ,{\mathit{W}}^L,{\tilde{\mathit{W}}}^{L+1})$, where ${\tilde{\mathit{W}}}^{L+1}=({\mathit{W}}^L-$${\mathit{W}}^{L-1})/\Delta t$. In calculation of actual engineering, the matrix $\mathit{B}$ may be made up of the measurement values at the nodes in partition ${\mathfrak{F}}_h$ without the need to compute the FVE solution vectors.

(2)

Compute a standard orthogonal eigenvector set ${\widehat{\mathit{\phi }}}_i$$(1⩽i⩽r=:$rank$\left(\mathit{B}\right))$ related to the eigenvalue series ${\lambda }_1⩾{\lambda }_2⩾\dots ⩾{\lambda }_r>0$ of the matrix ${\mathit{B}}^T\mathit{B}$ through the singular value decomposition [31].

(3)

Compute d ($d\ll r$) most main normalized vectors $\left\{{\mathit{\phi }}_1,{\mathit{\phi }}_2,\dots ,\right.$$\left.{\mathit{\phi }}_d\right\}$ of the matrix $\mathit{B}{\mathit{B}}^T$ by the formulas ${\mathit{\phi }}_i=\mathit{B}{\widehat{\mathit{\phi }}}_i/\sqrt{{\lambda }_i}$$(1⩽i⩽d)$, which are named as the POD basic vectors, to make up the matrix $\mathbf{\Phi }=({\mathit{\phi }}_1,{\mathit{\phi }}_2,$$\dots ,$${\mathit{\phi }}_d)$.

By Section 5.1.2 in [28], we assert that the matrix Φ possesses the following property.

$$\parallel \mathit{B}-\mathbf{\Phi }{\mathbf{\Phi }}^T{\mathit{B}\parallel }_{2,2}=\sqrt{{\lambda }_{d+1}},$$

(14)

where ${\displaystyle {\parallel \mathit{B}\parallel }_{2,2}=\underset{\mathit{\vartheta }\in {\mathbb{R}}^L}{sup}\frac{\parallel \mathit{B}\mathit{\vartheta }\parallel }{\parallel \mathit{\vartheta }\parallel }}$ and $\parallel \mathit{\vartheta }\parallel$ also denotes the Euclidean norm to vector $\mathit{\vartheta }$.

Through (14), we may attest

$$\begin{array}{c}{\displaystyle \parallel {\mathit{W}}^n-\mathbf{\Phi }{\mathbf{\Phi }}^T{\mathit{W}}^n\parallel = \parallel (\mathit{B}-\mathbf{\Phi }{\mathbf{\Phi }}^T\mathit{B}){\mathit{e}}^n\parallel ⩽ \parallel {\mathit{e}}^n\parallel \parallel \mathit{B}-\mathbf{\Phi }{\mathbf{\Phi }}^T{\mathit{B}\parallel }_{2,2}}\hfill \\ ⩽\sqrt{{\lambda }_{d+1}},1⩽n⩽L,\hfill \end{array}$$

(15)

$$\begin{array}{c}\parallel {\tilde{\mathit{W}}}^{L+1}-\mathbf{\Phi }{\mathbf{\Phi }}^T{\tilde{\mathit{W}}}^{L+1}\parallel = \parallel (\mathit{B}-\mathbf{\Phi }{\mathbf{\Phi }}^T\mathit{B}){\mathit{e}}^{L+1}\parallel ⩽ \parallel {\mathit{e}}^{L+1}\parallel \parallel \mathit{B}-\mathbf{\Phi }{\mathbf{\Phi }}^T{\mathit{B}\parallel }_{2,2}\hfill \\ ⩽\sqrt{{\lambda }_{d+1}},\hfill \end{array}$$

(16)

wherein ${\mathit{e}}^n$$(n=1,2,\dots ,L+1)$ denote the $(L+1)$-dimensional unit vectors for the n-th component 1.

3.2. Creation of the Finite Volume Element Model Dimension Reduction Formulation

If we suppose that ${\mathit{W}}_d^n={(W_{1d}^n,W_{2d}^n,\dots ,W_{Md}^n)}^T$ indicate the M-dimension vectors and ${\mathit{\beta }}^n={({\beta }_1^n,{\beta }_2^n,\dots ,{\beta }_d^n)}^T$ stand for the d-dimension coefficient vectors, then the FVEMDR solutions may be denoted by ${\varpi }_d^n=\mathit{\xi }·\mathbf{\Phi }{\mathit{\beta }}^n$$(1⩽n⩽N)$. Whereupon, the first L FVEMDR solution coefficient vectors can be straightforwardly procured through

$${\mathit{W}}_d^n=\mathbf{\Phi }{\mathbf{\Phi }}^T{\mathit{W}}^n=:\mathbf{\Phi }{\mathit{\beta }}^n,1⩽n⩽L,$$

where ${\mathit{W}}^n$$(n=1,2,\dots ,L)$ are just the first L FVE solution coefficient vectors.

By replacing the solution coefficient vectors ${\mathit{W}}^n$$(n=L+1,L+2,\dots ,N)$ to Problem 5 with ${\mathit{W}}_d^n=\mathbf{\Phi }{\mathit{\beta }}^n$ and using the orthogonality for the POD basic vectors in $\mathbf{\Phi }$, a new FVEMDR formulation is set up below.

Problem 6.

Compute ${\left\{{\mathit{\beta }}^n\right\}}_{n=1}^N\subset {\mathbb{R}}^d$ and ${\left\{{\varpi }_d^n\right\}}_{n=1}^N\subset {\mathbb{W}}_h$ from the following system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {\mathit{\beta }}^n={\mathbf{\Phi }}^T{\mathit{W}}^n,1⩽n⩽L;}\hfill \\ {\displaystyle {\mathit{\beta }}^{n+1}-{\mathit{\beta }}^n={\mathit{\beta }}^n-{\mathit{\beta }}^{n-1}-\Delta t^2{\mathbf{\Phi }}^T\mathit{D}\mathbf{\Phi }{\mathit{\beta }}^n+\Delta t^2{\mathbf{\Phi }}^T{\mathit{F}}^n,L⩽n⩽N-1.}\hfill \\ {\displaystyle {\varpi }_d^n=\mathit{\xi }·\left(\mathbf{\Phi }{\mathit{\beta }}^n\right),1⩽n⩽N,}\hfill \end{array}\right.\end{array}$$

(17)

where ${\mathit{W}}^n$$(n=1,2,\dots ,L)$ indicate the FVE solution coefficient vectors computed from Problem 5 at the initial L temporal nodes, and $\mathit{D}$ and ${\displaystyle {\mathit{F}}^n}$ are given by Problem 5.

Remark 4.

After comparing the FVE formulation with the FVEMDR formulation, it is clearly observable that each round of iteration for the FVE formulation involves M unknowns, whereas each round of iteration for the FVEMDR formulation only has d unknowns $(d\ll L\ll M)$. Accordingly, the unknowns in the FVEMDR formulation are far less than those in the FVE formulation. However, the FVEMDR formulation has the same FE basic vector $\mathit{\xi }$ as the FVE formulation. Thereupon, the FVEMDR formulation has the same precision as the FVE formulation under the case that the $\sqrt{{\lambda }_{d+1}}$ is sufficiently small. In another words, although the unknowns to the FVEMDR formulation are highly reduced, under the case that the $\sqrt{{\lambda }_{d+1}}$ is sufficiently small, the FVEMDR solutions maintain the same precision as the FVE formulation. Consequently, compared with the FVE formulation, the FVEMDR formulation has an absolute advantage.

3.3. The Theoretic Analysis of the Finite Volume Element Model Dimension Reduction Solutions

For Problem 6, we draw the following conclusions.

Theorem 4.

Under the entirely same hypotheses as Theorem 2, an FVEMDR solution set ${\left\{{\varpi }_d^n\right\}}_{n=1}^N$$\subset {\mathbb{W}}_h$ is solely computed with Problem 6 to satisfy the following unconditional boundedness, namely unconditional stability:

$$\parallel \nabla {\varpi }_d^n{\parallel }_0 ⩽ c,n=1,2,\dots ,N,$$

(18)

and the following error estimates:

$$\begin{array}{c}\hfill {\displaystyle \parallel {\varpi }_d^n-\varpi \left(t_k\right){\parallel }_0+ h{\parallel \nabla ({\varpi }_d^n-\varpi \left(t_k\right))\parallel }_0⩽c\left(\Delta t^2+h^2+\sqrt{{\lambda }_{d+1}}\right),n=1,2,\dots ,N.}\end{array}$$

(19)

Proof. 

The attestation for Theorem 3 is accomplished through the following three steps.

Step 1.

Attestation for the existence of the FVEMDR solutions.

(a)

Under the case of $1⩽n⩽L$, the first and third sub-equations in (17) visibly show a sole FVEMDR solution set ${\left\{{\varpi }_d^n\right\}}_{n=1}^L\subset {\mathbb{W}}_h$.

(b)

Under the case of $L+1⩽n⩽N$, through ${\mathit{W}}_d^n=\mathbf{\Phi }{\mathit{\beta }}^n$, the second and third equations in (17) may be reworded into the following system:

$$\left\{\begin{array}{c}{\mathit{W}}_d^{n+1}-{\mathit{W}}_d^n={\mathit{W}}_d^n-{\mathit{W}}_d^{n-1}-\Delta t^2\mathit{D}{\mathit{W}}_d^n+\Delta t^2{\mathit{F}}^n, L⩽n⩽N-1,\hfill \\ {\displaystyle {\varpi }_d^n=\mathit{\xi }·{\mathit{W}}_d^n,L⩽n⩽N.}\hfill \end{array}\right.$$

(20)

whereupon, the system of Equation (20) also visibly shows a sole FVEMDR solution set ${\left\{{\varpi }_d^n\right\}}_{n=L+1}^N\subset {\mathbb{W}}_h$.

By combining (a) with (b), we assert that Problem 6 possesses a sole FVEMDR solution set ${\left\{{\varpi }_d^n\right\}}_{n=1}^N\subset {\mathbb{W}}_h$.

Step 2.

The attestation for the unconditional stability of the FVEMDR solutions.

(i)

Under the case of $1⩽n⩽L$, through the boundedness of the POD basic vectors in $\mathbf{\Phi }$ as well as the unconditional stability for the FVE solution vectors ${\mathit{W}}^n$ in Theorem 3, we can attest

$${\displaystyle \parallel {\mathit{W}}_d^n\parallel = \parallel \mathbf{\Phi }{\mathbf{\Phi }}^T{\mathit{W}}^n\parallel ⩽ \parallel \mathbf{\Phi }{\mathbf{\Phi }}^T{\parallel }_{2,2}\parallel {\mathit{W}}^n\parallel ⩽ c\parallel {\mathit{W}}^n\parallel ⩽ c.}$$

(21)

Noticing that ${\parallel \mathit{\xi }\parallel }_1⩽c$, based on (21), we may derive

$$\parallel \nabla {\varpi }_d^n{\parallel }_0 = \parallel \nabla (\mathit{\xi }·{\mathit{W}}_d^n){\parallel }_0 ⩽ {\parallel \nabla \mathit{\xi }\parallel }_0\parallel {\mathit{W}}_d^n\parallel ⩽ c\parallel {\mathit{W}}_d^n\parallel ⩽ c,1⩽n⩽L.$$

(22)

(ii)

Under the case of $L⩽n⩽N-1$, noticing that, from [28] (Lemma 1.3.7), we can obtain that ${\parallel \mathit{\xi }\parallel }_1⩽c$ and ${\parallel \mathit{D}\parallel }_{2,2}⩽c$, from the first equation for (20), we can attest

$$\begin{array}{ccc}\hfill {\displaystyle \parallel {\mathit{W}}_d^{n+1}-{\mathit{W}}_d^n\parallel }& ⩽& \parallel {\mathit{W}}_d^n-{\mathit{W}}_d^{n-1}\parallel + \Delta t^2\parallel {\mathit{D}}_H^2{\parallel }_{2,2}\parallel {\mathit{W}}_d^n\parallel + \Delta t^2\parallel {\mathit{F}}^n\parallel \hfill \\ & ⩽& {\displaystyle \parallel {\mathit{W}}_d^n-{\mathit{W}}_d^{n-1}\parallel + c\Delta t^2(\parallel {\mathit{W}}_d^n\parallel + \parallel {\mathit{F}}^n\parallel ).}\hfill \end{array}$$

(23)

Summing (23) from L to n (where $n⩽N-1$) and utilizing (16), we get

$$\begin{array}{c}{\displaystyle \parallel {\mathit{W}}_d^{n+1}\parallel -\parallel {\mathit{W}}_d^n\parallel ⩽\parallel {\mathit{W}}_d^{n+1}-{\mathit{W}}_d^n\parallel }\hfill \\ {\displaystyle ⩽c\Delta t+c\Delta t^2\sum _{i=L}^n(\parallel {\mathit{W}}_d^i\parallel + \parallel {\mathit{F}}^i\parallel ),L⩽n⩽N-1.}\hfill \end{array}$$

(24)

Summing (24) from L until n (where $n⩽N-1$), we get

$$\parallel {\mathit{W}}_d^n\parallel ⩽ \parallel {\mathit{W}}_d^L\parallel + cn\Delta t + cn\Delta t^2\sum _{i=L}^{n-1}(\parallel {\mathit{W}}_d^i\parallel + \parallel {\mathit{F}}^i\parallel )⩽c+c\Delta t\sum _{i=L}^{n-1}\parallel {\mathit{W}}_d^i\parallel ,$$

(25)

where $n=L+1,L+2,\dots ,N.$ To apply Gronwall’s inequality to (25) produces

$${\displaystyle \parallel {\mathit{W}}_d^n\parallel ⩽cexp(c\Delta t(n-L))⩽c,n=L+1,L+2,\dots ,N.}$$

(26)

Employing ${\parallel \mathit{\xi }\parallel }_1⩽c$ and (26), we can attest

$$\parallel \nabla {\varpi }_d^n{\parallel }_0 ⩽ \parallel \nabla (\mathit{\xi }·{\mathit{W}}_d^n){\parallel }_0 ⩽ {\parallel \nabla \mathit{\xi }\parallel }_0·\parallel {\mathit{W}}_d^n\parallel ⩽c,n=L+1,L+2,\dots ,N.$$

(27)

Thereupon, (18) is procured by combining (22) with (27).

Step 3.

Estimate the errors for the FVEMDR solutions.

(1)

Under the case of $1⩽n⩽L$, noticing that ${\mathit{W}}_d^n=\mathbf{\Phi }{\mathbf{\Phi }}^T{\mathit{W}}^n$, and using (15), we draw

$$\parallel {\mathit{W}}^n-{\mathit{W}}_d^n\parallel = \parallel {\mathit{W}}^n-\mathbf{\Phi }{\mathbf{\Phi }}^T{\mathit{W}}^n\parallel ⩽ c\sqrt{{\lambda }_{d+1}}.$$

(28)

(2)

Under the case of $L+1⩽n⩽N$, subducing the first equation in (20) through the second equation in (13), setting ${\mathit{ϵ}}^n={\mathit{W}}^n-{\mathit{W}}_d^n$, and noticing that ${\parallel \mathit{D}\parallel }_{2,2}⩽c$, we can attest

$${\displaystyle \parallel {\mathit{ϵ}}^{n+1}-{\mathit{ϵ}}^n\parallel ⩽\Delta t^2\parallel {\mathit{ϵ}}^n{\parallel \parallel \mathit{D}\parallel }_{2,2} + \parallel {\mathit{ϵ}}^n-{\mathit{ϵ}}^{n-1}\parallel ⩽c\Delta t^2\parallel {\mathit{ϵ}}^n\parallel +\parallel {\mathit{ϵ}}^n-{\mathit{ϵ}}^{n-1}\parallel .}$$

(29)

By summing (29) from L to n (where $n⩽N-1)$ and using (16), we draw

$$\parallel {\mathit{ϵ}}^{n+1}\parallel -\parallel {\mathit{ϵ}}^n\parallel ⩽ \parallel {\mathit{ϵ}}^{n+1}-{\mathit{ϵ}}^n\parallel ⩽ c\Delta t\sqrt{{\lambda }_{d+1}}+c\Delta t^2\sum _{i=L}^n\parallel {\mathit{ϵ}}^i\parallel ,L⩽n⩽N-1.$$

(30)

By summing (30) from L to n (where $n⩽N-1)$, we draw

$$\parallel {\mathit{ϵ}}^{n+1}\parallel ⩽ \parallel {\mathit{ϵ}}^L\parallel + cn\Delta t\sqrt{{\lambda }_{d+1}}+cn\Delta t^2\sum _{i=L}^n\parallel {\mathit{ϵ}}^i\parallel ⩽ c\sqrt{{\lambda }_{d+1}}+c\Delta t\sum _{i=L}^n\parallel {\mathit{ϵ}}^i\parallel ,$$

(31)

where $L⩽n⩽N-1.$ Applying the Gronwall inequality to (30), we produce

$$\begin{array}{c}\hfill {\displaystyle \parallel {\mathit{W}}^n-{\mathit{W}}_d^n\parallel ⩽cexp(c\Delta t(k-L))\sqrt{{\lambda }_{d+1}}⩽c\sqrt{{\lambda }_{d+1}},L+1⩽n⩽N.}\end{array}$$

(32)

Using (28) and (32), and ${\parallel \mathit{\xi }\parallel }_1⩽c$, we draw

$${\displaystyle \parallel \nabla ({\varpi }_h^n-{\varpi }_d^n){\parallel }_0 = \parallel \nabla (\mathit{\xi }·({\mathit{W}}^n-{\mathit{W}}_d^n)){\parallel }_0 ⩽ {\parallel \nabla \mathit{\xi }\parallel }_0\parallel {\mathit{W}}^n-{\mathit{W}}_d^n\parallel ⩽c\sqrt{{\lambda }_{d+1}},}$$

(33)

where $1⩽n⩽N.$ Thereupon, (19) are obtained by combining (33) with (11) in Theorem 2. This accomplishes the attestation to Theorem 4. □

Remark 5.

Although the errors of Theorem 4 contain one more term $\sqrt{{\lambda }_{d+1}}$ than those of Theorem 2, the increased term may act as a criterion to choose the number of POD basic vectors d while computing the FVEMDR solutions. As a matter of fact, as long as the chosen number of POD basic vectors d meets $\sqrt{{\lambda }_{d+1}}⩽(\Delta t^2+h^2)$, the precision for the FVEMDR solutions would not be impacted by the POD-based dimension reduction. A large number of numeric examples (such as those given in [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]) indicate that the eigenvalues ${\lambda }_i$ $(1⩽i⩽r)$ fall rapidly into 0. Generally, as $d=5\sim 7$, they are already very small to satisfy $\sqrt{{\lambda }_{d+1}}⩽(\Delta t^2+h^2)$. The brightest point of the FVEMDR formulation is that if the FVEMDR solution ${\varpi }_d^{n_0+1}$ for Problem 6 at temporal node $t_{n_0+1}$ could not satisfy the needed accuracy, but the FVEMDR solutions ${\varpi }_d^n$ at the foregoing temporal nodes $t_n$ $(n⩽n_0)$ still satisfy the precision requirements, and we redraw a new group of solution coefficient vectors ${\mathit{W}}_d^{n_0-L+1},{\mathit{W}}_d^{n_0-L+2},\dots ,{\mathit{W}}_d^{n_0}$ to make up a new matrix $\mathit{B}=({\mathit{W}}_d^{n_0-L+1},{\mathit{W}}_d^{n_0-L+2},\dots ,{\mathit{W}}_d^{n_0},{\tilde{\mathit{W}}}_d^{n_0+1})$ (where ${\tilde{\mathit{W}}}_d^{n_0+1}=[{\mathit{W}}_d^{n_0}-{\mathit{W}}_d^{n_0-1}]/\Delta t$) and reconstruct a new POD basic vector set and a new FVEMDR formulation to compute the FVEMDR solutions that satisfy the precision requirements. Thereupon, we may compute the FVEMDR solutions at all temporal nodes that satisfy the accuracy requirements, which is impossible to achieve in the FVE formulation.

4. Two Numerical Examples

For the sake of convenience of comparison, in this section, we employ two numerical examples to test the validity of the obtained theory results and exhibit the advantages to the FVEMDR formulation of the hyperbolic equation that possesses the analytic solutions. It generally has no analytic solution.

4.1. The First Numerical Example

In the 2D hyperbolic equation, we take $\overline{\Omega }=[0,1]\times [0,1]$, the initial functions ${\varpi }_0(x,y)={\varpi }_1(x,y)=0$, and the course function $f(x,y,t)=2(1-{\pi }^2{t}^2)sin\pi xsin\pi y$. Thereupon, Problem 1 has an analytical solution $\varpi (x,y,t)=t^{2}sin\pi xsin\pi y$.

The triangular division ${\mathfrak{F}}_h$ is formed by the equilateral right-angled triangle, whose two right-angled sides are parallel to the x-axis or y-axis, and the lengths of both right-angle sides are $1/100$. The dual partition ${\mathfrak{F}}_h^{\ast }$ is selected into the barycenter dual partition, i.e., the barycenter for the right-angled triangle $E\in {\mathfrak{F}}_h$ acts as the node of the dual partition. Thus, if $\Delta t=1/100$ and $\sqrt{{\lambda }_{d+1}}=O\left({10}^{-4}\right)$, via Theorems 2 and 4, we get the result that the theoretical errors of the FVE and FVEMDR solutions can reach $O\left({10}^{-4}\right)$ under the $L^2$ norm.

The FVEMDR solutions can be computed through the following flow-chart.

(1)

As a matter of experience, compute the initial 20 ($L=20$) FVE solution coefficient vectors ${\mathit{W}}^1,{\mathit{W}}^2,$$\dots ,$${\mathit{W}}^{20}$ through Problem 5 and create a matrix $\mathit{B}=({\mathit{W}}^1,{\mathit{W}}^2,\dots ,$${\mathit{W}}^{20},{\tilde{\mathit{U}}}^{21})$, where ${\tilde{\mathit{W}}}^{21}=({\mathit{W}}^{20}-{\mathit{W}}^{19})/\Delta t$.

(2)

Compute a series of orthonormal eigenvectors ${\widehat{\mathit{\phi }}}_j$ $(1⩽j⩽21)$ associated with the eigenvalues ${\lambda }_1⩾{\lambda }_2⩾\dots ⩾{\lambda }_{21}⩾0$ of the matrix ${\mathit{B}}^T\mathit{B}$.

(3)

By calculation, we obtain the result that $\sqrt{{\lambda }_7}⩽3.514\times {10}^{-4}$. Accordingly, it is only required to select the first 6 standard orthogonal eigenvectors ${\widehat{\mathit{\phi }}}_i$ $(1⩽i⩽6)$ to yield the POD basic vectors $\mathbf{\Phi }=({\mathit{\phi }}_1,{\mathit{\phi }}_2,\dots ,$${\mathit{\phi }}_6)$ through the formulas ${\mathit{\phi }}_j=\mathit{B}{\widehat{\mathit{\phi }}}_j/\sqrt{{\lambda }_j}$ $(1⩽j⩽6)$.

(4)

Input $\mathbf{\Phi }$ into the FVEMDR formulation and compute the FVEMDR solutions ${\varpi }_d^n$ when $n=100$, 150, 200, and 250 (i.e., $t=1.0$, $1.5$, $2.0$, and $2.5$), and exhibited in Figure 2a, Figure 3a, Figure 4a on ThinkPad E530 laptop, and Figure 5a separately.

To illustrate that the FVEMDR model is better than the FVE model, the FVE solutions ${\varpi }_h^n$ when $n=100$, 150, 200, and 250 (i.e., $t=1.0$, $1.5$, $2.0$, and $2.5$) were also computed on the same laptop, and are exhibited in Figure 2b, Figure 3b, Figure 4b and Figure 5a separately.

For ease of comparison, the analytical solutions when $n=100$, 150, 200, and 250 (i.e., $t=1.0$, $1.5$, $2.0$, and $2.5$) are also shown in Figure 2c, Figure 3c, Figure 4c and Figure 5c separately.

By contrasting per set of photos in Figure 2, Figure 3, Figure 4 and Figure 5, it can be clearly observed that the FVEMDR solutions are almost completely consistent with the analytical solutions when $t=1.0$, $1.5$, $2.0$, and $2.5$, and the FVE solutions are also nearly the same as the analytical solutions. Nevertheless, each round of iteration of the FVE model involves 10,000 unknowns, but each round of iteration of the FVEMDR model involves only 6 unknowns. Accordingly, compared with the FVE formulation, the FVEMDR formulation can significantly lessen the unknowns, thereby reducing the computational load, slowing down the cumulation for the truncation errors, and enhancing the calculation efficiency.

To further exhibit the benefits to the FVEMDR model, we also documented the CPU runtime as well as the producing errors for computing the FVE and FVEMDR solutions when $t=1.0$, $1.5$, $2.0$, and $2.5$ through Problems 5 and 6 on ThinkPad E530 laptop separately, as cataloged in

Table 1. The CPU running time and the errors of computing FVE and FVEMDR solutions.

		The FVE Model		The FVEMDR Model
$\mathit{t}$	$\mathit{n}$	$\parallel \mathit{u}\left({\mathit{t}}_{\mathit{n}}\right)-{\mathit{u}}_{\mathit{h}}^{\mathit{n}}{\parallel }_{\mathbf{0}}$	CPU Running Time	$\parallel \mathit{u}\left({\mathit{t}}_{\mathit{n}}\right)-{\mathit{u}}_{\mathit{d}}^{\mathit{n}}{\parallel }_{\mathbf{0}}$	CPU Running Time
$1.0$	100	$3.201\times {10}^{-4}$	6485 s	$2.215\times {10}^{-4}$	97 s
$1.5$	150	$4.816\times {10}^{-4}$	31,765 s	$3.323\times {10}^{-4}$	472 s
$2.0$	200	$7.325\times {10}^{-4}$	64,657 s	$4.436\times {10}^{-4}$	956 s
$2.5$	250	$9.746\times {10}^{-4}$	12,324 s	$5.548\times {10}^{-4}$	1441 s

.

Table 1 indicates that the numerical errors to the FVE and FVEMDR solutions can also attain $O\left({10}^{-4}\right)$ at $t=1$, 5, and 10, which accord with the theoretical errors. Nevertheless, the CPU runtime to compute the FVE solutions is about sixty-eight times that to compute the FVEMDR solutions. However, due to the large number of unknowns in the FVE model, the accumulation for truncation errors occurs rapidly during the calculation process, while there are fewer unknown variables in the FVEMDR model, so the accumulation for the truncation errors during the calculation process is relatively slow. It also shows that the FVEMDR formulation is indeed better than the FVE formulation.

4.2. The Second Numerical Example

In order to further show the advantages for the FVEMDR model, we carry out longer temporal numerical simulations. For this purpose, in Problem 1, we take $\overline{\Omega }=[0,2\pi ]\times [0,2\pi ]$, the initial functions ${\varpi }_0(x,y)=-{\varpi }_1(x,y)=sinxsiny$, and the course function $f(x,y,t)=0$. Thus, Problem 1 has an analytical solution $\varpi (x,y,t)=exp(-t)sinxsiny$.

The triangular partition ${\mathfrak{F}}_h$ is still composed of the equilateral right-angled triangle, whose two right-angled sides are parallel to the x-axis or y-axis, and the lengths of both right-angle sides are $1/100$. The dual partition ${\mathfrak{F}}_h^{\ast }$ is then selected as the barycenter dual partition. If $\Delta t=1/100$ and $\sqrt{{\lambda }_{d+1}}$$=O\left({10}^{-4}\right)$, from Theorems 2 and 4, we get the result that the theory errors for the FVE and FVEMDR solutions can also attain $O\left({10}^{-4}\right)$ under the $L^2$ norm.

Similarly, by the flowchart in Section 4.1 and Problem 6, we also computed the FVEMDR solutions ${\varpi }_d^n$ at $n=1000$, 2000, 3000, and 4000 (i.e., $t=10$, 20, 30, and 40) on ThinkPad E530 laptop, as shown in Figure 6a, Figure 7a, Figure 8a and Figure 9a separately.

To illustrate that the FVEMDR formulation is superior to the FVE formulation, we also computed the FVE solutions ${\varpi }_h^n$ for the hyperbolic equation at $n=1000$, 2000, 3000, and 4000 (namely $t=10$, 20, 30, and 40) by the FVE formulation on the same laptop, as exhibited in Figure 6b, Figure 7b, Figure 8b and Figure 9b separately.

For ease of comparison, the analytical solutions at $t=10$, 20, 30, and 40 (i.e., $n=1000$, 2000, 3000, and 4000) are exhibited in Figure 6c, Figure 7c, Figure 8c and Figure 9c separately.

By comparing per set of photos in Figure 6, Figure 7, Figure 8 and Figure 9, it may be easily observed that the FVEMDR solutions are very closed to the analytical solutions at $t=10$, 20, 30, and 40, even if the FVE solutions are also closed to the analytical solutions at $t=10$ and 20, but when $t=30$ and 40, due to the accumulation of truncation error during computation, the FVE solutions deviate from the analytical solutions. In particular, each iteration of the FVE formulation has about $394, 784$ unknowns, but each iteration of the FVEMDR formulation possesses only 6 unknowns. Accordingly, the FVEMDR formulation may greatly lessen unknowns.

To further illustrate the benefit for the FVEMDR formulation, we documented the CPU operating time as well as the yielding errors for computing the FVE and FVEMDR solutions when $t=10$, 20, 30, and 40 on ThinkPad E530 laptop, listed in

Table 2. The CPU running time and the yielded errors to compute FVE and FVEMDR solutions.

t	FVE Solutions	FVEMDR Solutions	FVE Method	FVEMDR Method
	$\parallel \varpi \left({\mathit{t}}_{\mathit{n}}\right)-{\varpi }_{\mathit{h}}^{\mathit{n}}{\parallel }_{\mathbf{0}}$	$\parallel \varpi \left({\mathit{t}}_{\mathit{n}}\right)-{\varpi }_{\mathit{d}}^{\mathit{n}}{\parallel }_{\mathbf{0}}$	CPU Runtime	CPU Runtime
10	3.3153$\times {10}^{-4}$	3.4352$\times {10}^{-4}$	1213.262 s	24.265 s
20	5.5367$\times {10}^{-4}$	3.5674$\times {10}^{-4}$	2426.523 s	48.531 s
30	7.7581$\times {10}^{-4}$	3.6985$\times {10}^{-4}$	4853.051 s	97.061 s
40	9.8795$\times {10}^{-4}$	3.8198$\times {10}^{-4}$	9706.103 s	194.121 s

, too.

The data in Table 2 also indicate that the numeric errors for the FVE and FVEMDR solutions attain $O\left({10}^{-4}\right)$ at $t=10$, 20, 30, and 40, which also accorded with the obtained theoretic errors. Nevertheless, the CPU operating time required to compute the FVE solutions is approximately fifty times that needed for the FVEMDR solutions. Accordingly, the FVEMDR formulation is greatly superior to the FVE formulation. From the second example, we have also seen that the after a long period of calculation, the FVEMDR solutions obtained by the FVEMDR model still converge to the analytical solutions. Accordingly, the FVEMDR model is indeed very effective compared to the hyperbolic equation.

5. Conclusions and Prospects

So far, we have built a new FVEMDR formulation for the 2D hyperbolic equation, and theoretically proven the existence, stability, and error estimations of the FVEMDR solutions, strictly. We have also adopted two groups of numeric tests to attest to the validity of the obtained theoretic conclusions and show the effectiveness of the FVEMDR formulation. The FVEMDR formulation for the hyperbolic equation is first created in this article, which is thoroughly distinguished from the existing dimension reduction formulations, comprising those in [17,18,19]. At the moment, no one else has performed such a study. Accordingly, this study is original.

Although we have only presented the FVEMDR formulation of the 2D hyperbolic equation herein, the methods and ideas of this study can be extended to the three-dimensional hyperbolic equation, and more complex non-stationary PDEs, even to the issues in actual engineering projects. Accordingly, it has highly widespread application prospects.

Although we have only adopted the central difference scheme with second-order temporal accuracy to discretize the temporal derivative, in order to attain the higher-order temporal precision, the temporal derivative may adopt some higher-order methods, for example, the space-time FVE method, multi-step methods (e.g., higher-order Adams–Bashforth or Adams–Moulton schemes), Runge–Kutta methods (including strong-stability-preserving variants for hyperbolic problems), or implicit–explicit approaches for stiff systems. These methods will be very meaningful and will be studied in the future.

Although we have only discussed the Dirichlet first boundary problem for the hyperbolic equation, the methods and ideas of this study can also be extended to other boundary problems, such as Neumann, Robin, or inflow/outflow conditions. These problems will be very meaningful and will also be studied in the future.