Discontinuous finite volume element method of two-dimensional unsaturated soil water movement problem

Abstract

In this paper, a numerical approximation method for the two-dimensional unsaturated soil water movement problem is established by using the discontinuous finite volume method. We prove the optimal error estimate for the fully discrete format. Finally, the reliability of the method is verified by numerical experiments. This method is not only simple to calculate, but also stable and reliable.

1 Introduction

The movement of water in soil is a very complicated problem. This paper mainly studies the water movement in furrow irrigation, that is, the water movement in the trapezoidal region to the soil diffusion on both sides and the infiltration of the underground pipeline into the surrounding soil. Unsaturated soil water movement refers to the movement of water in the soil when the water is not full of pores. It is an important form of fluid movement in porous media. We assume that the soil is homogeneous and isotropic. Let the x-axis be horizontal to the right and the z-axis vertically downward. According to Darcy’s law and the continuity principle, the problem of unsaturated soil water movement can be reduced to the following model (see [1]):

$$ \frac{\partial Q}{\partial t}-\frac{\partial }{\partial x}\biggl(D(Q)\frac{ \partial Q}{\partial x} \biggr)-\frac{\partial }{\partial z}\biggl(D(Q)\frac{ \partial Q}{\partial z}\biggr)+ \frac{\partial K(Q)}{\partial z}=S_{r}. $$

(1)

where $Q(x,z,t)$ is the soil moisture volume water content, $D(Q)$ indicates the diffusion rate of soil water, $K(Q)$ indicates the hydraulic conductivity, $-S_{r}$ is the absorption rate of the root zone, the relationship between $K(Q)$, $D(Q)$ and Q is as follows:

$$ \textstyle\begin{cases} K(Q)=K_{s}(\frac{Q}{Q_{s}})^{2b+3} , \\ D(Q)=-\frac{bK_{s}\psi _{s}}{Q_{s}}(\frac{Q}{Q_{s}})^{b+2}, \quad Q_{r}\leq Q(z,t)\leq Q_{s}, \end{cases} $$

(2)

where $Q_{s}$ is the soil water saturated water content, $Q_{r}$ is the residual moisture content of the soil moisture, where $0< Q_{s}<1$, the saturated water conductivity $K_{s}$, the soil parameter b and the saturated soil water potential $\psi _{s}$ are all related to the soil structure and are known constants. Therefore, it can be determined that $K(Q)$, $D(Q)$, $\frac{\partial K(Q)}{\partial Q}$, $\frac{\partial K(Q)}{\partial z}$, $\frac{\partial D(Q)}{\partial Q}$ are bounded, that is, there are two constants $K_{1}$, $K_{2}$, such that: $K_{1}\leq K(Q)$, $\frac{\partial K(Q)}{\partial Q}$, $\frac{\partial K(Q)}{\partial z}$, $\frac{\partial D(Q)}{ \partial Q}$, $D(Q)\leq K_{2}$.

The following conditions are given for (1):

• (1) Initial condition: $Q(x,z,0)=Q_{0}$;

• (2) Boundary condition:

  $$\textstyle\begin{cases} Q(0,0,t)=Q_{s}, & t\in [0,T], \\ Q(L,z,t)=Q_{0}, & L\rightarrow \pm \infty , t\in [0,T], \\ Q(x,M,t)=Q_{0}, & M\rightarrow \pm \infty , t\in [0,T], \end{cases} $$

  where $Q_{0}$ represents the initial water content and $Q_{s}$ represents the saturated water content.

According to the literature [2], the solution to the problem is existing and unique. Based on the reliability of this problem and its practical significance in meteorology, agricultural environmental engineering, hydrodynamics, etc., in recent years, many scholars have proposed numerical methods to solve it. The numerical solutions of one-dimensional and two-dimensional soil water movement problems are given by the finite difference method in Ref. [2, 3]. However, because the finite difference method is very sensitive to boundary conditions and soil parameters, the error is large. The authors of Ref. [4, 5] used the finite volume element method to simulate the two-dimensional soil water flow problem and overcome the weakness of the finite difference method. To the best of our knowledge, there is no report on the discontinuous finite volume element method to deal with two-dimensional unsaturated water movement. In this paper, we focus on the mathematical model characteristics of the two-dimensional unsaturated water motion problem, and we mainly discuss the discontinuous finite volume element method of the problem. This method not only inherits the advantages of the format of the finite volume element method, that is, a simple structure, high precision, simple calculation and local conservation between physical quantities, but it also has the characteristics of discontinuous finite element, the finite element space does not need to meet any continuity requirements, the space structure is simple, and there is good locality and parallelism.

This paper is organized as follows, in Sect. 2, we derive a discontinuous finite volume element format for unsaturated soil water movement problems. In Sect. 3, we give some lemmas related to error analysis. In Sect. 4, we obtain the optimal estimate of $L^{2}$-norm and $|\!|\!|\cdot |\!|\!|_{1,h}$-norm. In Sect. 5, numerical experiments are given to verify the validity of the theoretical analysis.

2 Discontinuous finite volume element format for unsaturated soil water movement problems

For convenience, it is assumed that the region $\varOmega \in R^{2}$ is a suitably smooth and sufficiently large bounded region. On the boundary ∂Ω of the region Ω, the initial moisture content $Q_{0}$ remains unchanged. We define the dual partition $T_{h}^{*}$ of $T_{h}$ for the test function space as follows. Let $T_{h}$ be a triangulation of Ω with $\operatorname{diam}(\varOmega )$, where h is the set of all the triangular elements K, $h(K)$ is the side length of the unit $K\in T_{h}$. Let Γ denote the union of the boundaries of the triangle K of $T_{h}$, $\varGamma _{0}=\varGamma \backslash \partial \varOmega$, $h=\max_{K\in T_{h}}{ h(K)}$. We divide each K into three triangles by connecting the barycenter B and three corners of the triangle as shown in Fig. 1. Let $T_{h}^{*}$ consist of all these triangles $T_{j}$.

Figure 1

Original subdivision and dual subdivision

On the original split, we define the following broken Sobolev space:

$$ H^{m}(T_{h})=\bigl\{ v\in L^{2}(\varOmega ):v|_{k}\in H^{m}(K),\forall K\in T _{h} \bigr\} $$

and its norm

$$ \Vert v \Vert _{m,h}=\Biggl(\sum_{i=0}^{m} \vert v \vert _{i,h}^{2}\Biggr)^{1/2},\qquad \vert v \vert _{i,h}=\biggl( \sum_{K\in T_{h}} \vert v \vert _{H^{i}(K)}^{2}\biggr)^{1/2}, $$

where $H^{i}(K)$ is the standard Sobolev space defined on unit K, m is a positive integer.

We define a finite dimensional trial function space $U_{h}$ with the original partition $T_{h}$:

$$ U_{h}=\bigl\{ u_{h}\in L^{2}(\varOmega ):u_{h}|_{K}\in P_{1}(K),\forall K \in T_{h}\bigr\} . $$

Define the finite dimensional test function space on the dual partition $T_{h}^{*}$:

$$ V_{h}=\bigl\{ v_{h}\in L^{2}(\varOmega ):v_{h}|_{T}\in P_{0}(T),\forall T \in T_{h}^{*}\bigr\} , $$

where $P_{l}$ denotes a polynomial with degree less than or equal to l ($l=0,1$) defined on $K(T)$.

Let $U(h)=U_{h}+H^{2}(\varOmega )\cap H_{0}^{1}(\varOmega )$. Define a mapping $\gamma _{h}:U(h)\rightarrow V_{h}$ as

$$ \gamma _{h} v_{h}|_{T}= \frac{1}{h_{e}} \int _{e}v_{h}|_{T} \,ds, \quad \forall T\in T_{h}^{*} , $$

where $h_{e}$ is the length of the boundary e of the unit K.

In order to facilitate theoretical analysis, take $Q_{0}=0$, and let $F(Q)=\mathbf{p}\cdot K(Q)$, we have

$$ \frac{\partial K(Q)}{\partial z}=\nabla \cdot F(Q), \quad \mathbf{p}=(0,1). $$

So (1) can be written as

$$ \frac{\partial Q}{\partial t}-\frac{\partial }{\partial x}\biggl(D(Q)\frac{ \partial Q}{\partial x} \biggr)-\frac{\partial }{\partial z}\biggl(D(Q)\frac{ \partial Q}{\partial z}\biggr)+\nabla \cdot F(Q)=S_{r}, $$

(3)

Multiplying (3) by $v_{h}\in V_{h}$, integrating on the dual unit, summing over T, and using the Green formula, we obtain

$$ \biggl(\frac{\partial Q}{\partial t},v_{h}\biggr)-\sum _{T\in T_{h}^{*}} \int _{\partial T}D(Q)\nabla Q\cdot \mathbf{n} v_{h} \,ds=\bigl(S _{r}-\nabla \cdot F(Q),v_{h}\bigr), $$

(4)

where n is the unit outward normal vector on ∂T.

So

$$ \begin{aligned} \sum_{T\in T_{h}^{*}} \int _{\partial T}D(Q)\nabla Q\cdot \mathbf{n} v _{h} \,ds &=\sum_{K\in T_{h}}\sum _{i=1}^{3} \int _{P_{i+1}BP_{i}}D(Q) \nabla Q\cdot \mathbf{n} v_{h} \, ds \\ &\quad{} +\sum_{K\in T_{h}} \int _{\partial T}D(Q)\nabla Q\cdot \mathbf{n} v_{h} \,ds, \end{aligned} $$

where $P_{4}=P_{1}$, $P_{5}=P_{2}$, $P_{6}=P_{3}$.

Assume $e=\partial K_{1}\cap \partial K_{2}$, where $K_{1}$, $K_{2}$ are the adjacent two units in $T_{h}$, and let $\mathbf{n}_{1}$ and $\mathbf{n}_{2}$ be unit normal vectors on e pointing exterior to $K_{1}$ and $K_{2}$. For scalar functions p and vector functions q, we define the average $\{\cdot \}$ and the jump $[\cdot ]$ on e, as follows (see [6]).

If $e\in \varGamma _{0}$ and $e\subset \partial K_{1}\cap \partial K_{2}$, then

$$\begin{aligned}& [v]|_{e}=v|_{\partial K_{1}}{\mathbf{n}}_{1}+v|_{\partial K_{2}}{ \mathbf{n}} _{2}, \qquad \{v\}|_{e}=\frac{1}{2}(v|_{\partial K_{1}}+v|_{\partial K_{2}}), \\& [{\mathbf{w}}]|_{e}=\mathbf{w}|_{\partial K_{1}}\cdot {\mathbf{n}} _{1}+\mathbf{w}|_{\partial K_{2}}\cdot {\mathbf{n}}_{2}, \qquad \{\mathbf{w}\}|_{e}=\frac{1}{2}( \mathbf{w}|_{\partial K_{1}}+ \mathbf{w}|_{\partial K_{2}}). \end{aligned}$$

If $e\in \varGamma \backslash \varGamma _{0}$, and $e\subset \partial K$, then

$$\begin{aligned}& [v]|_{e}=v|_{\partial K}{\mathbf{n}}_{k}, \qquad \{v\}|_{e}=v|_{\partial K}, \\& [{\mathbf{w}}]|_{e}=\mathbf{w}|_{\partial K}\cdot { \mathbf{n}}_{k}, \qquad \{\mathbf{w}\}|_{e}= \mathbf{w}|_{\partial K}. \end{aligned}$$

According to the above average and the definition of the jump, a straightforward computation gives

$$ \sum_{K\in T_{h}} \int _{\partial K}p\mathbf{w}\cdot {\mathbf{n}}\,ds= \sum _{e\in \varGamma } \int _{e}[p]\cdot \{\mathbf{w}\}\,ds +\sum _{e\in \varGamma _{0}} \int _{e}\{p\}[{\mathbf{w}}]\,ds. $$

(5)

Using (5) and the fact that $[D(Q)\nabla Q\cdot \mathbf{n}v _{h}]| _{e}=0$, $\forall e\in \varGamma _{0}$, we have

$$ \sum_{K\in T_{h}} \int _{\partial K}D(Q)\nabla Q\cdot \mathbf{n} v_{h} \,d s=\sum_{e\in \varGamma } \int _{e}\bigl\{ D(Q)\nabla Q\bigr\} \cdot [ v _{h}] \,ds. $$

Define the following bilinear form:

$$ \begin{aligned} A(Q_{h};Q_{h},\gamma _{h} v_{h})&=-\sum_{K\in T_{h}} \sum^{3}_{i=1} \int _{P_{i+1}BP_{i}}D(Q_{h})\nabla Q_{h}\cdot {\mathbf{n}} \gamma _{h}v_{h}\,ds \\ &\quad{} -\sum_{e\in \varGamma } \int _{e}\bigl\{ D(Q_{h})\nabla Q_{h}\bigr\} [\gamma _{h}v_{h}]\,ds \\ &\quad{} -\sum_{e\in \varGamma } \int _{e}\bigl\{ D(Q_{h})\nabla v_{h}\bigr\} [\gamma _{h}Q_{h}]\,ds \\ &\quad{} +\alpha \sum_{e\in \varGamma }[\gamma _{h}Q_{h}] [\gamma _{h}v_{h}], \end{aligned} $$

where α is a real constant (see [7]).

The semi-discrete discontinuous finite volume element format of problem (1) is to find $Q_{h}\in U_{h}$, such that

$$ \textstyle\begin{cases} \mbox{(a)}\quad (\frac{\partial Q_{h}}{\partial t},\gamma _{h}v_{h})+A(Q_{h};Q_{h}, \gamma _{h}v_{h})=(S_{r}-\nabla \cdot F(Q),\gamma _{h}v_{h}), \quad \forall v_{h}\in U_{h}, \\ \mbox{(b)}\quad Q_{h}(0)=0. \end{cases} $$

(6)

Since Q is the solution to the problem (1), and $[\gamma _{h}Q]|_{e}=0$, the true solution Q satisfies

$$ \textstyle\begin{cases} \mbox{(a)}\quad (\frac{\partial Q}{\partial t},\gamma _{h}v_{h})+A(Q;Q,\gamma _{h}v _{h})=(S_{r}-\nabla \cdot F(Q),\gamma _{h}v_{h}), \quad \forall v_{h}\in U_{h}, \\ \mbox{(b)}\quad Q(0)=0. \end{cases} $$

(7)

Next, we present the fully discrete discontinuous finite volume element method. Take Δt as the time step, recorded as $\Delta t=\frac{T}{N}$, $N>0$, and N is a positive integer, $t^{j}=j\Delta t$,when $t=t^{j}$. If we use the backward difference quotient $\partial _{t}Q_{h}^{j}=\frac{Q_{h}^{j}-Q_{h}^{j-1}}{\Delta t}$ to approximate the differential quotient in the semi-discrete scheme, then we get the backward Euler fully discrete discontinuous finite volume element format: find $Q_{h}^{j}\in U_{h}$, ($j=1,\ldots ,N$), $\forall v_{h}\in U_{h}$, satisfying

$$ \textstyle\begin{cases} \mbox{(a)}\quad (\frac{Q_{h}^{j}-Q_{h}^{j-1}}{\Delta t},\gamma _{h}v_{h})+A(Q_{h} ^{j};Q_{h}^{j},\gamma _{h}v_{h})=(S_{r}^{j}-\nabla \cdot F(Q_{h}^{j}), \gamma _{h}v_{h}), \\ \mbox{(b)}\quad Q_{h}(0)=0. \end{cases} $$

(8)

3 Some lemmas

We define a norm $|\!|\!|\cdot |\!|\!|$ for $U_{h}$ as follows:

$$ |\!|\!|u |\!|\!|^{2}_{1,h}= \vert u \vert ^{2}_{1,h}+\sum_{e\in \varGamma }[\gamma _{h} u]^{2}_{e}+\sum _{K\in T_{h}}h_{k}^{2} \vert u \vert ^{2}_{2,K},\quad \forall u \in U(h), $$

where $|u|^{2}_{1,h}=\sum_{K\in T_{h}}|u|^{2}_{1,K}$.

The following trace inequality can be found in [8]. If e is the edge of unit K with length $h_{e}$, then

$$ \Vert u \Vert _{e}^{2}\leq C\bigl(h_{e}^{-1} \Vert u \Vert _{K}^{2}+h_{e} \vert u \vert ^{2}_{1,K}\bigr), \quad \forall u\in H^{2}(K). $$

For $\forall Q\in H^{1}(\varOmega )$, introduce the original equation solution Q to geta Ritz projection $R_{h}(t): H^{1}(\varOmega )\rightarrow U_{h}$ which satisfies

$$ A(Q;Q,\gamma _{h}v_{h})= A(Q;R_{h}Q, \gamma _{h}v_{h}), \quad \forall v_{h}\in V_{h}. $$

(9)

According to the relevant theory of the Ritz projection $R_{h}$ (see [3]), the following interpolation properties are obtained:

$$ \textstyle\begin{cases} \mbox{(a)}\quad \Vert Q-R_{h}Q \Vert \leq Ch^{2} \Vert Q \Vert _{3} , \\ \mbox{(b)}\quad \Vert (Q-R_{h}Q)_{t} \Vert \leq Ch^{2} \Vert Q \Vert _{1,3,2} , \\ \mbox{(c)}\quad |\!|\!|Q-R_{h}Q |\!|\!|_{1,h}\leq Ch \Vert Q \Vert _{2} , \\ \mbox{(d)}\quad |\!|\!|(Q-R_{h}Q)_{t} |\!|\!|_{1,h}\leq Ch \Vert Q \Vert _{1,2,2}. \end{cases} $$

(10)

Lemma 1

(see [9])

The operator $\gamma _{h}$is self-adjoint with respect to the $L^{2}$-inner product, that is,

$$ (u_{h},\gamma _{h}v_{h})=(v_{h}, \gamma _{h}u_{h}), \quad \forall u_{h},v_{h} \in U_{h}, $$

and if we define

$$ |\!|\!|u_{h} |\!|\!|_{0}=(u_{h}, \gamma _{h}u_{h})^{1/2}, $$

then $|\!|\!|\cdot |\!|\!|_{0}$and $\Vert \cdot \Vert$are equivalent, and

$$ \Vert \gamma _{h}u_{h} \Vert = \Vert u_{h} \Vert , \quad \forall u_{h}\in U_{h}. $$

Lemma 2

(see [9])

The operator $\gamma _{h}$satisfies the following properties:

$$\begin{aligned}& \int _{K}(w_{h}-\gamma _{h}w_{h})\,dx=0, \quad \forall w_{h}\in U(h), \forall K\in T_{h}, \\& \int _{e}(w_{h}-\gamma _{h}w_{h})\,ds=0, \quad \forall w_{h}\in U(h), \forall e\in \partial K, \forall K\in T_{h}, \\& [w_{h}]=0\quad \Longrightarrow \quad [\gamma _{h}w_{h}]=0, \quad \forall w_{h} \in U(h), \\& \Vert \gamma _{h}w_{h}-w_{h} \Vert _{0,K}\leq Ch_{K} \vert w_{h} \vert _{1,K},\quad \forall w _{h}\in U(h), \forall K\in T_{h}. \end{aligned}$$

Lemma 3

(see [10])

There is a normal numberβthat is independent ofh, such that

$$ \beta |\!|\!|u_{h} |\!|\!|^{2}_{1,h}\leq A(q;u_{h},\gamma _{h}u_{h}), \quad \forall u_{h}\in U_{h} . $$

(11)

Lemma 4

(see [6])

For $\forall u_{h},v_{h}\in U_{h}$, there exists a positive constantCindependent ofh, and we have

$$ A(q;u_{h},\gamma _{h} v_{h})\leq C |\!|\!|u_{h} |\!|\!|_{1,h} |\!|\!|v_{h} |\!|\!|_{1,h}. $$

(12)

Lemma 5

(see [2, 5, 6])

There is a constant C independent of h such that

$$\begin{aligned}& \bigl\vert A(q;u_{h},\gamma _{h} v_{h})-A(q;v_{h},\gamma _{h} u_{h}) \bigr\vert \leq Ch |\!|\!|u _{h} |\!|\!|_{1,h} |\!|\!|v_{h} |\!|\!|_{1,h}, \quad \forall u_{h}\in U_{h}. \end{aligned}$$

(13)

$$\begin{aligned}& \bigl\vert A(p;u_{h},\gamma _{h} v_{h})-A(q;u_{h},\gamma _{h} v_{h}) \bigr\vert \leq C \vert u_{h} \vert _{1, \infty }\bigl( \Vert p-q \Vert +h |\!|\!|p-q |\!|\!|_{1,h}\bigr)) |\!|\!|v_{h} |\!|\!|_{1,h}. \end{aligned}$$

(14)

Lemma 6

(see [9])

There is a constant C independent of h such that

$$\begin{aligned}& h |\!|\!|u_{h} |\!|\!|_{1,h}\leq C \Vert u_{h} \Vert , \quad \forall u_{h}\in U_{h}. \end{aligned}$$

(15)

$$\begin{aligned}& \Vert u_{h} \Vert \leq C |\!|\!|u_{h} |\!|\!|_{1,h}, \quad \forall u_{h}\in U_{h}. \end{aligned}$$

(16)

Lemma 7

If $Q,Q_{t}\in W^{2,\infty }(\varOmega )\cap H ^{3}(\varOmega )$, there is a constant $C_{0}$, $C_{1}$independent ofhsuch that, for small enough, we have

$$\begin{aligned}& \vert R_{h}Q \vert _{1,\infty }\leq C_{0}, \end{aligned}$$

(17)

$$\begin{aligned}& \vert R_{h}Q_{t} \vert _{1,\infty }\leq C_{1}. \end{aligned}$$

(18)

Proof

Let $Q_{I}\in U_{h}$ be the interpolation of Q, it is well known that

$$ \vert Q-Q_{I} \vert _{s,p,k}\leq Ch^{2-s} \vert Q \vert _{2,p,k} \quad \forall K\in T_{h}, s=0,1, 1\leq p\leq \infty . $$

(19)

By the definition of $|\!|\!|\cdot |\!|\!|_{1,h}$, we have

$$ |\!|\!|Q-Q_{I} |\!|\!|_{1,h}\leq Ch \vert Q \vert _{2}. $$

(20)

Using Lemma 3 and Lemma 4, we have

$$ \begin{aligned} \beta |\!|\!|R_{h}Q-Q_{I} |\!|\!|_{1,h}^{2}&\leq A\bigl(Q;R_{h}Q-Q_{I}, \gamma _{h}(R _{h}Q-Q_{I})\bigr) \\ &=A\bigl(Q;Q-Q_{I},\gamma _{h}(R_{h}Q-Q_{I}) \bigr) \\ &\leq C |\!|\!|Q-Q_{I} |\!|\!|_{1,h}\cdot |\!|\!|R_{h}Q-Q_{I} |\!|\!|_{1,h}. \end{aligned} $$

Using triangular inequalities, inverse inequalities and (19), we obtain

$$ \vert R_{h}Q \vert _{1,\infty }\leq \vert R_{h}Q-Q_{I} \vert _{1,\infty }+ \vert Q_{I}-Q \vert _{1, \infty }+ \vert Q \vert _{1,\infty }. $$

So

$$ |\!|\!|R_{h}Q-Q_{I} |\!|\!|_{1,h}\leq Ch \vert Q \vert _{2}. $$

Therefore

$$ \vert R_{h}Q \vert _{1,\infty }\leq C\bigl( \vert Q \vert _{2}+ \vert Q \vert _{1}\bigr). $$

For (18), the proof is similar to that of (17). □

4 Convergence analysis

Theorem 1

LetQ, $Q_{h}^{n}$be the solutions of (1) and (8), if $Q\in L^{2}(0,T;H^{3}(\varOmega ))$, $Q_{t}\in L^{2}(0,T;L^{2}(\varOmega ))$, $Q_{tt}\in L^{2}(0,T;L^{2}( \varOmega ))$, $Q_{h}(0)=R_{h}Q(0)=0$, then there is a constantCindependent ofhand Δt, such that

$$\begin{aligned}& \bigl\Vert Q\bigl(t^{n}\bigr)-Q_{h}^{n} \bigr\Vert \leq Ch^{2}\biggl( \int _{0}^{t^{n}} \Vert Q_{t} \Vert _{3}^{2}\,dt\biggr)^{ \frac{1}{2}}+C\Delta t \biggl( \int _{0}^{t^{n}} \Vert Q_{tt} \Vert ^{2}\,dt\biggr)^{\frac{1}{2}}, \end{aligned}$$

(21)

$$\begin{aligned}& \begin{aligned}[b] \bigl|\!\bigl|\!\bigl|Q\bigl(t^{n} \bigr)-Q_{h}^{n} \bigr|\!\bigr|\!\bigr|_{1,h}&\leq Ch \biggl( \bigl\Vert Q^{n} \bigr\Vert _{3}+ \bigl\Vert Q^{n} \bigr\Vert _{2}+ \int _{0}^{t^{n}} \Vert Q_{t} \Vert _{3}^{2}\,dt\biggr)^{\frac{1}{2}} \\ &\quad{} +C\Delta t\biggl( \int _{0}^{t^{n}} \Vert Q_{tt} \Vert ^{2}\,dt\biggr)^{\frac{1}{2}}. \end{aligned} \end{aligned}$$

(22)

Proof

Let $Q_{h}^{n}=Q(t^{n})$, $R_{h}Q^{n}=R_{h}{Q(t^{n})}$, $Q _{t}^{n}=\frac{\partial Q^{n}}{\partial t}$, $\rho ^{n}=Q^{n}-R_{h}{Q ^{n}}$, $\theta ^{n}=R_{h}{Q^{n}}-Q_{h}^{n}$. Now we estimate $\Vert \theta ^{n}\Vert$.

Subtracting (8) from (7) gives the error equation

$$ \begin{aligned}[b] &\biggl(Q_{t}^{j}- \frac{Q^{j}-Q^{j-1}}{\triangle t},\gamma _{h}v_{h}\biggr)+A \bigl(Q^{j};Q ^{j},\gamma _{h}v_{h} \bigr)-A\bigl(Q_{h}^{j};Q_{h}^{j}, \gamma _{h}v_{h}\bigr) \\ &\quad =\bigl(\nabla \cdot \bigl(F\bigl(Q_{h}^{j} \bigr)-F\bigl(Q^{j}\bigr)\bigr),\gamma _{h}v_{h} \bigr), \end{aligned} $$

(23)

where

$$ \begin{gathered} A\bigl(Q^{j};Q^{j},\gamma _{h}v_{h}\bigr)-A\bigl(Q_{h}{^{j}};Q_{h}{^{j}}, \gamma _{h}v_{h}\bigr) \\ \quad =A\bigl(Q^{j};\rho ^{j}+\theta ^{j},\gamma _{h}v_{h}\bigr)+A \bigl(Q^{j};Q_{h}^{j}, \gamma _{h}v_{h}\bigr)-A\bigl(Q_{h}^{j};Q_{h}^{j}, \gamma _{h}v_{h}\bigr), \\ Q_{t}^{j}-\frac{Q^{j}-Q^{j-1}}{\triangle t} =Q_{t}^{j}+\partial t \theta ^{j}-\partial tR_{h}Q^{j}. \end{gathered} $$

(24)

Using (10), the error equation is equivalent to

$$ \begin{aligned}[b] &\bigl(\partial _{t}\theta ^{j},\gamma _{h}v_{h}\bigr)+A \bigl(Q^{j};\theta ^{j},\gamma _{h}v_{h} \bigr)\\ &\quad =\bigl(\partial _{t} R_{h}{Q^{j}}-Q_{t}^{j}, \gamma _{h}v_{h}\bigr) \\ &\qquad {}+A\bigl(Q_{h}^{j};Q_{h}^{j}, \gamma _{h}v_{h}\bigr)-A\bigl(Q^{j};Q_{h}^{j}, \gamma _{h}v _{h}\bigr)+\bigl(\nabla \cdot \bigl(F \bigl(Q_{h}^{j}\bigr)-F\bigl(Q^{j}\bigr) \bigr),\gamma _{h}v_{h}\bigr). \end{aligned} $$

(25)

Choosing $v_{h}=\theta ^{j}$ in (25), using Lemma 3, we have

$$ \begin{aligned}[b] \bigl(\partial _{t}\theta ^{j},\gamma _{h}\theta ^{j}\bigr)+A \bigl(Q^{j};\theta ^{j}, \gamma _{h}\theta ^{j}\bigr) &=\biggl(\frac{\theta ^{j}-\theta ^{j-1}}{\triangle t}, \gamma _{h} \theta ^{j}\biggr)+A\bigl(Q^{j};\theta ^{j},\gamma _{h}\theta ^{j}\bigr) \\ & \geq \frac{ \Vert \theta ^{j} \Vert ^{2}- \Vert \theta ^{j-1} \Vert ^{2}}{2\triangle t}+ \beta \bigl|\!\bigl|\!\bigl|\theta ^{j-1} \bigr|\!\bigr|\!\bigr|_{1,h}^{2}. \end{aligned} $$

(26)

The right side of Eq. (25) is recorded as $I_{1}$, $I_{2}$, $I_{3}$, and is estimated item by item.

Using the Hölder inequalities and ε inequalities, we get

$$ \vert I_{1} \vert \leq C \bigl|\!\bigl|\!\bigl|\partial _{t} R_{h}{Q^{j}}-Q_{t}^{j} \bigr|\!\bigr|\!\bigr|_{0}^{2}+C \bigl\Vert \theta ^{j} \bigr\Vert ^{2}. $$

Using Lemma 5, the triangular inequalities and the ε inequalities, we have

$$ \begin{aligned} \vert I_{2} \vert &\leq C \bigl\vert Q_{h}^{j} \bigr\vert _{1,\infty }\bigl( \bigl\Vert Q^{j}-Q_{h}^{j} \bigr\Vert +h \bigl|\!\bigl|\!\bigl|Q^{j}-Q _{h}^{j} \bigr|\!\bigr|\!\bigr|_{1,h}\bigr) \bigl|\!\bigl|\!\bigl|\theta ^{j} \bigr|\!\bigr|\!\bigr|_{1,h} \\ & \leq C\bigl( \bigl\Vert \rho ^{j} \bigr\Vert + \bigl\Vert \theta ^{j} \bigr\Vert +h \bigl|\!\bigl|\!\bigl|\rho ^{j} \bigr|\!\bigr|\!\bigr|_{1,h}+h \bigl|\!\bigl|\!\bigl|\theta ^{j} \bigr|\!\bigr|\!\bigr|_{1,h}\bigr) \bigl|\!\bigl|\!\bigl|\theta ^{j} \bigr|\!\bigr|\!\bigr|_{1,h} \\ &\leq C\bigl( \bigl\Vert \rho ^{j} \bigr\Vert ^{2}+ \bigl\Vert \theta ^{j} \bigr\Vert ^{2}+h^{2} \bigl|\!\bigl|\!\bigl|\rho ^{j} \bigr|\!\bigr|\!\bigr|_{1,h} ^{2}\bigr)+\frac{\beta }{2} \bigl|\!\bigl|\!\bigl|\theta ^{j} \bigr|\!\bigr|\!\bigr|_{1,h}^{2}. \end{aligned} $$

Assume $|Q_{h}^{j}|_{1,\infty }\leq C_{0}$, $j=0,1,\ldots ,N$. Proof of it will be given later.

From the Hölder inequalities and the ε inequalities, we have

$$ \vert I_{3} \vert \leq C \bigl\Vert Q^{j}-Q_{h}^{j} \bigr\Vert ^{2}+C \bigl\Vert \theta ^{j} \bigr\Vert ^{2} \leq C \bigl\Vert \rho ^{j} \bigr\Vert ^{2}+ \bigl\Vert \theta ^{j} \bigr\Vert ^{2}. $$

Let $\xi ^{j}=\partial _{t} R_{h}{Q^{j}}-Q_{t}^{j}$ and use the above estimate; (25) is transformed into

$$ \begin{aligned}[b] & \frac{ \Vert \theta ^{j} \Vert ^{2}- \Vert \theta ^{j-1} \Vert ^{2}}{2\triangle t}+\beta \bigl|\!\bigl|\!\bigl|\theta ^{j} \bigr|\!\bigr|\!\bigr|_{1,h}^{2} \\ &\quad \leq C \bigl|\!\bigl|\!\bigl|\xi ^{j} \bigr|\!\bigr|\!\bigr|_{0}^{2}+C \bigl\Vert \theta ^{j} \bigr\Vert ^{2}+C \bigl\Vert \rho ^{j} \bigr\Vert ^{2}+C \bigl\Vert \theta ^{j} \bigr\Vert ^{2}+Ch^{2} \bigl|\!\bigl|\!\bigl|\rho ^{j} \bigr|\!\bigr|\!\bigr|_{1,h}^{2}. \end{aligned} $$

(27)

Multiplying $2\Delta t$ on both sides of type (27), summing over j from 1 to n at both sides of (27) and noting that $\theta ^{0}=0$,

$$ \begin{aligned}[b] & \bigl\Vert \theta ^{n} \bigr\Vert ^{2}+\Delta t\beta \sum_{j=1}^{n} \bigl|\!\bigl|\!\bigl|\theta ^{j} \bigr|\!\bigr|\!\bigr|_{1,h} ^{2} \\ &\quad \leq C\Delta t\sum_{j=1}^{n} \bigl|\!\bigl|\!\bigl|\xi ^{j} \bigr|\!\bigr|\!\bigr|_{0}^{2}+C \Delta t \sum_{j=1}^{n}\bigl( \bigl\Vert \rho ^{j} \bigr\Vert ^{2}+h^{2} \bigl|\!\bigl|\!\bigl|\rho ^{j} \bigr|\!\bigr|\!\bigr|_{1,h}^{2} \bigr)+C \Delta t\sum_{j=1}^{n} \bigl\Vert \theta ^{j} \bigr\Vert ^{2}, \end{aligned} $$

(28)

where

$$ \begin{aligned}[b] \sum_{j=1}^{n} \bigl|\!\bigl|\!\bigl|\xi ^{j} \bigr|\!\bigr|\!\bigr|_{0}^{2}& \leq \frac{1}{\Delta t} \sum_{j=1}^{n} \int _{t^{j-1}}^{t^{j}} \Vert Q_{t}-R_{h}Q_{t} \Vert ^{2}\,dt+ \Delta t\sum_{j=1}^{n} \int _{t^{j-1}}^{t^{j}} \Vert Q_{tt} \Vert ^{2}\,dt \\ &\leq C\frac{h^{4}}{\Delta t} \int _{0}^{t^{n}} \Vert Q_{t} \Vert _{3}^{2}\,dt+C \Delta t \int _{0}^{t^{n}} \Vert Q_{tt} \Vert ^{2}\,dt. \end{aligned} $$

(29)

Taking (10) and the discrete Gronwall lemma,

$$ \begin{aligned} & \bigl\Vert \theta ^{n} \bigr\Vert ^{2}+\Delta t\beta \sum_{j=1}^{n} \bigl|\!\bigl|\!\bigl|\theta ^{j} \bigr|\!\bigr|\!\bigr|_{1,h} ^{2} \\ &\quad \leq Ch^{4} \int _{0}^{t^{n}} \Vert Q_{t} \Vert _{3}^{2}\,dt+C\Delta t^{2} \int _{0}^{t ^{n}} \Vert Q_{tt} \Vert ^{2}\,dt+Ch^{4}\bigl( \bigl\Vert Q^{n} \bigr\Vert _{3}^{2}+ \bigl\Vert Q^{n} \bigr\Vert _{2}^{2}\bigr) \\ &\quad \leq Ch^{4} \int _{0}^{t^{n}} \Vert Q_{t} \Vert _{3}^{2}\,dt+C\Delta t^{2} \int _{0}^{t ^{n}} \Vert Q_{tt} \Vert ^{2}\,dt, \end{aligned} $$

which is

$$ \bigl\Vert \theta ^{n} \bigr\Vert \leq Ch^{2} \biggl( \int _{0}^{t^{n}} \Vert Q_{t} \Vert _{3}^{2}\,dt\biggr)^{ \frac{1}{2}}+\Delta t \biggl( \int _{0}^{t^{n}} \Vert Q_{tt} \Vert ^{2}\,dt\biggr)^{\frac{1}{2}}. $$

(30)

Combining (10) with the triangle inequality, we obtain the desired result (21).

The following proves $|Q_{h}^{j}|_{1,\infty }\leq C_{0}$, $j=0,1, \ldots ,N$.

Assume $C_{0}=1+\sup_{[0,T]}|R_{h}Q|_{1,\infty }$, actually, when $t=0$, $Q_{h}^{0}=R_{h}Q^{0}=0$, obviously

$$ \bigl\vert Q_{h}^{0} \bigr\vert _{1,\infty } \leq \sup_{[0,T]} \vert R_{h}Q_{t} \vert _{1,\infty }< C_{0}. $$

Let us assume that $j=0,1,\ldots ,k-1$, $|Q_{h}^{j}|_{1, \infty }\leq C_{0}$ is established. So when $\Delta t=O(h)$ and h is full,

$$ \begin{aligned} \bigl\vert Q_{h}^{j} \bigr\vert _{1,\infty }&\leq \bigl\vert Q_{h}^{j}-R_{h}Q_{t}^{j} \bigr\vert _{1,\infty }+ \bigl\vert R _{h}Q^{j} \bigr\vert _{1,\infty } \\ &\leq C \vert \ln h \vert ^{\frac{1}{2}} \bigl\vert Q_{h}^{j}-R_{h}Q^{j} \bigr\vert _{1,\infty }+ \bigl\vert R_{h}Q ^{j} \bigr\vert _{1,\infty } \\ &\leq C_{0}. \end{aligned} $$

Next we estimate $|\!|\!|\theta ^{n}|\!|\!|_{1,h}$.

By letting $v_{h}=\partial _{t}\theta ^{j}$ in (24), we have

$$ \begin{aligned}[b] &\bigl(\partial _{t}\theta ^{j},\gamma _{h}\partial _{t}\theta ^{j}\bigr)+A\bigl(Q^{j}; \theta ^{j}, \gamma _{h}\partial _{t}\theta ^{j}\bigr) \\ &\quad =\bigl(\partial _{t} R_{h}{Q^{j}}-Q_{t}^{j}, \gamma _{h}\partial _{t}\theta ^{j}\bigr) +A\bigl(Q_{h}^{j};Q_{h}^{j}, \gamma _{h}\partial _{t}\theta ^{j}\bigr)-A \bigl(Q ^{j};Q_{h}^{j},\gamma _{h}\partial _{t}\theta ^{j}\bigr) \\ &\qquad {}+\bigl(\nabla \cdot \bigl(F\bigl(Q_{h}^{j} \bigr)-F\bigl(Q^{j}\bigr)\bigr),\gamma _{h}\partial _{t}\theta ^{j}\bigr). \end{aligned} $$

(31)

The left end item is obtained by Lemma 1

$$\begin{aligned}& \bigl(\partial _{t}\theta ^{j},\gamma _{h}\partial _{t}\theta ^{j}\bigr)= \bigl|\!\bigl|\!\bigl|\partial _{t}\theta ^{j} \bigr|\!\bigr|\!\bigr|_{0}^{2}, \\& \begin{aligned} A\bigl(Q^{j};\theta ^{j}, \gamma _{h}\partial _{t}\theta ^{j}\bigr)&= \frac{1}{2\Delta t}\bigl[A\bigl(Q^{j};\theta ^{j}+ \theta ^{j-1},\gamma _{h}\bigl(\theta ^{j}- \theta ^{j-1}\bigr)\bigr) \\ &\quad{} +A\bigl(Q^{j};\theta ^{j}-\theta ^{j-1},\gamma _{h}\bigl(\theta ^{j}-\theta ^{j-1}\bigr)\bigr)\bigr] \\ & \geq \frac{1}{2\Delta t}A\bigl(Q^{j};\theta ^{j}+ \theta ^{j-1},\gamma _{h}\bigl(\theta ^{j}- \theta ^{j-1}\bigr)\bigr) \\ & = \frac{1}{2\Delta t}\bigl[A\bigl(Q^{j};\theta ^{j},\gamma _{h}\theta ^{j}\bigr)-A \bigl(Q^{j}; \theta ^{j-1},\gamma _{h}\theta ^{j-1}\bigr)\bigr] \\ &\quad{} -\frac{1}{2}\bigl[A\bigl(Q^{j};\partial _{t}\theta ^{j},\gamma _{h}\theta ^{j}\bigr)-A\bigl(Q ^{j};\theta ^{j}, \gamma _{h}\partial _{t}\theta ^{j}\bigr) \bigr]. \end{aligned} \end{aligned}$$

Therefore, the error equation is transformed:

$$ \begin{aligned}[b] & \bigl|\!\bigl|\!\bigl|\partial _{t}\theta ^{j} \bigr|\!\bigr|\!\bigr|_{0}^{2}+ \frac{1}{2\Delta t}\bigl[A\bigl(Q^{j}; \theta ^{j}, \gamma _{h}\theta ^{j}\bigr)-A\bigl(Q^{j}; \theta ^{j-1},\gamma _{h} \theta ^{j-1}\bigr) \bigr] \\ &\quad \leq \bigl(\partial _{t} R_{h}{Q^{j}}-Q_{t}^{j}, \gamma _{h}\partial _{t} \theta ^{j}\bigr) +A\bigl(Q_{h}^{j};Q_{h}^{j}, \gamma _{h}\partial _{t}\theta ^{j}\bigr)-A \bigl(Q ^{j};Q_{h}^{j},\gamma _{h}\partial _{t}\theta ^{j}\bigr) \\ & \qquad {} + \bigl(\nabla \cdot \bigl(F\bigl(Q_{h}^{j} \bigr)-F\bigl(Q^{j}\bigr)\bigr),\gamma _{h}\partial _{t}\theta ^{j}\bigr) \\ &\qquad {} +\frac{1}{2}\bigl[A\bigl(Q^{j};\partial _{t}\theta ^{j},\gamma _{h}\theta ^{j}\bigr)-A\bigl(Q ^{j};\theta ^{j}, \gamma _{h}\partial _{t}\theta ^{j}\bigr) \bigr] \\ &\quad =J_{1}+J_{2}+J_{3}+J_{4}. \end{aligned} $$

(32)

Let $\xi ^{j}=\partial _{t} R_{h}{Q(t^{j})}-Q_{t}(t^{j})$, similar to the previous estimates

$$\begin{aligned}& \vert J_{1} \vert \leq C \bigl|\!\bigl|\!\bigl|\xi ^{j} \bigr|\!\bigr|\!\bigr|_{0}^{2}+\frac{1}{4} \bigl|\!\bigl|\!\bigl|\partial _{t}\theta ^{j} \bigr|\!\bigr|\!\bigr|_{0}^{2}, \\& \begin{aligned} \vert J_{2} \vert &\leq C \bigl\vert Q_{h}^{j} \bigr\vert _{1,\infty }\bigl( \bigl\Vert Q\bigl(t^{j}\bigr)-Q_{h}^{j} \bigr\Vert +h \bigl|\!\bigl|\!\bigl|\rho ^{j} \bigr|\!\bigr|\!\bigr|_{1,h}+h \bigl|\!\bigl|\!\bigl|\theta ^{j} \bigr|\!\bigr|\!\bigr|_{1,h}\bigr) \bigl|\!\bigl|\!\bigl|\partial _{t}\theta ^{j} \bigr|\!\bigr|\!\bigr|_{1,h} \\ &\leq C \bigl\vert Q_{h}^{j} \bigr\vert _{1,\infty }\bigl( \bigl\Vert \rho ^{j} \bigr\Vert + \bigl\Vert \theta ^{j} \bigr\Vert +h \bigl|\!\bigl|\!\bigl|\rho ^{j} \bigr|\!\bigr|\!\bigr|_{1,h}+h \bigl|\!\bigl|\!\bigl|\theta ^{j} \bigr|\!\bigr|\!\bigr|_{1,h}\bigr)Ch^{-1} \bigl\Vert \partial _{t}\theta ^{j} \bigr\Vert \\ & \leq Ch^{-2} \bigl\Vert \rho ^{j} \bigr\Vert ^{2}+Ch^{-2} \bigl\Vert \theta ^{j} \bigr\Vert ^{2}+C \bigl|\!\bigl|\!\bigl|\rho ^{j} \bigr|\!\bigr|\!\bigr|^{2} _{1,h}+C \bigl|\!\bigl|\!\bigl|\theta ^{j} \bigr|\!\bigr|\!\bigr|^{2}_{1,h}+\frac{1}{4} \bigl|\!\bigl|\!\bigl|\partial _{t}\theta ^{j} \bigr|\!\bigr|\!\bigr|_{0}^{2}, \end{aligned} \\& \vert J_{3} \vert \leq C \bigl\Vert Q\bigl(t^{j} \bigr)-Q_{h}^{j} \bigr\Vert ^{2}+ \frac{1}{4} \bigl|\!\bigl|\!\bigl|\partial _{t} \theta ^{j} \bigr|\!\bigr|\!\bigr|_{0}^{2}. \end{aligned}$$

For $J_{4}$, using Lemma 5, the ε inequality and the boundedness of $D(Q_{h})$, we get

$$ \vert J_{4} \vert \leq Ch \bigl|\!\bigl|\!\bigl|\partial _{t}\theta ^{j} \bigr|\!\bigr|\!\bigr|_{1,h} \bigl|\!\bigl|\!\bigl|\theta ^{j} \bigr|\!\bigr|\!\bigr|_{1,h} \leq C \bigl\Vert \partial _{t}\theta ^{j} \bigr\Vert \bigl|\!\bigl|\!\bigl|\theta ^{j} \bigr|\!\bigr|\!\bigr|_{1,h}\leq C \bigl|\!\bigl|\!\bigl|\theta ^{j} \bigr|\!\bigr|\!\bigr|_{1,h}^{2}+ \frac{1}{4} \bigl|\!\bigl|\!\bigl|\partial _{t}\theta ^{j} \bigr|\!\bigr|\!\bigr|_{0}^{2}. $$

In summary,

$$ \begin{aligned}[b] & A\bigl(Q^{j};\theta ^{j},\gamma _{h}\theta ^{j}\bigr)-A \bigl(Q^{j};\theta ^{j-1}, \gamma _{h}\theta ^{j-1}\bigr) \\ &\quad \leq C\Delta t \bigl|\!\bigl|\!\bigl|\xi ^{j} \bigr|\!\bigr|\!\bigr|_{0}^{2}+C\Delta t \bigl|\!\bigl|\!\bigl|\theta ^{j} \bigr|\!\bigr|\!\bigr|_{1,h} ^{2}+C\Delta t \bigl( \bigl|\!\bigl|\!\bigl|\rho ^{j} \bigr|\!\bigr|\!\bigr|_{1,h}^{2}+h^{-2} \bigl\Vert \rho ^{j} \bigr\Vert ^{2}+ \bigl\Vert \rho ^{j} \bigr\Vert ^{2}\bigr). \end{aligned} $$

(33)

Summing over j from 1 to n at both sides of (32) and, noting that $\theta ^{0}=0$, we have

$$ \begin{aligned} \beta \bigl|\!\bigl|\!\bigl|\theta ^{n} \bigr|\!\bigr|\!\bigr|_{1,h}^{2} &\leq C\Delta t\sum _{j=1}^{n} \bigl|\!\bigl|\!\bigl|\xi ^{j} \bigr|\!\bigr|\!\bigr|_{0}^{2}+C\Delta t\sum _{j=1}^{n} \bigl|\!\bigl|\!\bigl|\theta ^{j} \bigr|\!\bigr|\!\bigr|_{1,h} ^{2} \\ &\quad{} +C\Delta t\sum_{j=1}^{n} \bigl( \bigl|\!\bigl|\!\bigl|\rho ^{j} \bigr|\!\bigr|\!\bigr|_{1,h} ^{2}+h^{-2} \bigl\Vert \rho ^{j} \bigr\Vert ^{2}+ \bigl\Vert \rho ^{j} \bigr\Vert ^{2}\bigr). \end{aligned} $$

It follows from (10) and the Gronwall lemma that

$$ \begin{aligned} & \bigl|\!\bigl|\!\bigl|\theta ^{n} \bigr|\!\bigr|\!\bigr|_{1,h}^{2}+\Delta t\beta \sum _{j=1}^{n} \bigl|\!\bigl|\!\bigl|\theta ^{j} \bigr|\!\bigr|\!\bigr|_{1,h}^{2} \\ &\quad \leq Ch^{4} \int _{0}^{t^{n}} \Vert Q_{t} \Vert _{3}^{2}\,dt+C\Delta t^{2} \int _{0}^{t^{n}} \Vert Q_{tt} \Vert ^{2}\,dt+Ch^{2}\bigl( \bigl\Vert Q^{n} \bigr\Vert _{3}^{2}+ \bigl\Vert Q^{n} \bigr\Vert _{2} ^{2}\bigr). \end{aligned} $$

So

$$ \bigl|\!\bigl|\!\bigl|\theta ^{n} \bigr|\!\bigr|\!\bigr|_{1,h}\leq Ch \biggl( \bigl\Vert Q^{n} \bigr\Vert _{3}+ \bigl\Vert Q^{n} \bigr\Vert _{2}+ \int _{0}^{t^{n}} \Vert Q_{t} \Vert _{3}^{2}\,dt\biggr)^{\frac{1}{2}}+C\Delta t \biggl( \int _{0} ^{t^{n}} \Vert Q_{tt} \Vert ^{2}\,dt\biggr)^{\frac{1}{2}}. $$

(34)

Finally, the conclusions are proved by (10), (34) and the triangular inequalities. □

5 Numerical experiments

5.1 Experiment 1

First consider the following questions:

$$ \textstyle\begin{cases} u_{t}-\nabla \cdot \nabla u = t \sin x \sin y , & (x,y,t)\in [0,\pi ]\times [0,\pi ]\times [0,1), \\ u(x,y,0)=0, & (x,y)\in \varOmega , \\ u(x,y,t)=0, & (x,y)\in \partial \varOmega . \end{cases} $$

(35)

where Ω is our solution area, $T=1$, let h be the space step, t be the time step, and the numerical solution be $u_{h}$. The calculation results are shown in Table 1. We use Matlab to calculate the numerical solution.

Table 1 $t=1$

5.2 Experiment 2

In order to verify the correctness of the theoretical analysis results, consider the following questions:

$$ \textstyle\begin{cases} u_{t}-\nabla \cdot (A(u)\nabla u))+\nabla \cdot F(u)=f(x) , & (x,y,t)\in [0,1]\times [0,1]\times [0,1), \\ u(x,y,0)=0, & (x,y)\in \varOmega , \\ u(x,y,t)=0, & (x,y)\in \partial \varOmega . \end{cases} $$

(36)

Take a true solution $u=t \sin (pi x)\sin (pi y)$, $F(u)= \mathbf{p}\cdot K(u)$, $\mathbf{p}=(0,1)$, $K(u)=cu^{10}$. Obviously, the true solution satisfies $0\leq u\leq 1$, the solution interval is $\varOmega =(0,1)\times (0,1)$, and the time interval is $[0, 1]$. h is the space step, Δt is the time step and f is the source and sink item. We take $A(u)$, $K(u)$ with different values, calculate the numerical solution $u_{h}$, and give the corresponding $L^{2}$ mode error and $H^{1}$ mode error and the corresponding error order.

Take $A(u)=u+1$, $K(u)=cu^{10}$, the results are shown in Tables 2–4.

Table 2 $c=\frac{1}{40}$ $t=1$

When we take $A(u)=u+1$, $K(u)=cu^{10}$, we find that: when the nonlinear property of the nonlinear term A(u)is not strong and the nonlinear property of the nonlinear term K(u) is strong, if the convection term is not dominant, as shown in Table 2, the error orders of $L^{2}$-mode and $H^{1}$-mode obtained are approximately equal to 2 and 1, respectively, consistent with the theoretical analysis; if the convection term is dominant, as shown in Tables 3 and 4, the discontinuous finite volume element method is far from the expected result.

Table 3 $c=1$ $t=1$

Table 4 $c=2$ $t=1$

Take $A(u)=u^{8}+1$, $K(u)=c\cdot u^{10}$, the results are shown in Tables 5–7.

Table 5 $c=\frac{1}{40}$ $t=1$

Take $A(u)=u^{9}+1$, $K(u)=c\cdot u^{10}$, the results are shown in Tables 8–10.

When we take $A(u)=u^{8}+1$ and $A(u)=u^{9}+1$, $K(u)=cu^{10}$, we find that: when the nonlinear property of the nonlinear term A(u) and K(u) is strong, if the convection term is not dominant, as shown in Tables 5 and 8, the error orders of the $L^{2}$-mode and $H^{1}$-mode obtained are approximately equal to 2 and 1, respectively, which are consistent with the theoretical analysis; if the convection term is dominant, from Tables 6 and 7 and 9 and 10, the discontinuous finite volume element method is far from the theoretical analysis.

Table 6 $c=1$ $t=1$

Table 7 $c=2$ $t=1$

Table 8 $c=\frac{1}{40}$ $t=1$

Table 9 $c=1$ $t=1$

Table 10 $c=2$ $t=1$

6 Conclusion

In this paper, we mainly apply discontinuous finite volume element method to study two-dimensional unsaturated soil water movement. Firstly, we give semi-discrete discontinuous finite volume element scheme. Secondly, the convergence analysis is carried out on the basis of the fully discrete scheme. It is proved that the $L^{2}$-modulus estimation and the $H^{1}$-modulus estimation of the scheme reach 2 and 1, respectively. Finally, the validity of the theoretical analysis is verified by numerical experiments. Through numerical experiments, it is found that: when the convection term of the nonlinear problem is not dominant, the discontinuous finite volume element method can be considered to deal with such problems.

This paper is the research result of the 2019 Zaozhuang University Research Fund Project “Numerical Simulation of Discontinuous Finite Volume Element Method for Unsaturated Soil Water Flow Problem”.