On the Convergence of α Schemes with Source Terms for Scalar Convex Conservation Laws

Abstract

In this study, we use an extension of Yang’s convergence criterion [N. Jiang, On the wavewise entropy inequality for high-resolution schemes with source terms II: the fully discrete case] to show the entropy convergence of a class of fully discrete $\alpha$ schemes, now with source terms, for non-homogeneous scalar convex conservation laws in the one-dimensional case. The homogeneous counterparts (HCPs) of these schemes were constructed by S. Osher and S. Chakravarthy in the mid-1980s [A New Class of High Accuracy TVD Schemes for Hyperbolic Conservation Laws (1985), Very High Order Accurate TVD Schemes (1986)], and the entropy convergence of these methods, when $m=2$, was settled by the author [N. Jiang, The Convergence of α Schemes for Conservation Laws II: Fully-Discrete]. For semi-discrete $\alpha$ schemes, with or without source terms, the entropy convergence of these schemes was previously established (for $m=2$) by the author [N. Jiang, The Convergence of α Schemes for Conservation Laws I: Semi-Discrete Case].

1. Introduction

This paper is concerned with the entropy convergence of fully discrete $\alpha$ schemes with source terms that approximate inhomogeneous scalar conservation laws. The homogeneous counterparts (HCPs) of this class of high-order schemes, i.e., $\alpha$ schemes without source terms, were constructed and analyzed by S. Osher and S. Chakravarthy [1,2], who also established the TVD property of this family of schemes. Furthermore, for $0<\alpha \le \frac{1}{2}$ and $m=2,3,\cdots ,8$, $\alpha$ schemes are in conservative form, using $2m+1$ points and having $2m-2$-order of accuracy (except at isolated critical points). In addition, the utility and the effectiveness of these methods in obtaining approximate solutions to nonlinear hyperbolic conservation laws were demonstrated in [1,2,3]. When $m=2$, the author [4] showed that the numerical solutions to S. Osher and S. Chakravarthy’s original $\alpha$ schemes of a fully discrete case indeed converge to the entropy solution. The goal of this study is to extend this result to the non-homogeneous case. In other words, we show the entropy consistence of fully discrete $\alpha$ schemes with source terms for $m=2$. The technique of this research is based on an extension of [5] Yang’s WEI framework [6], which was established for fully discrete schemes that approximate homogeneous scalar conservation laws, to handle non-homogeneous cases.

We would like to mention that in the literature, not many convergence results for systems or/and multidimensional conservation laws have been verified. Nonetheless, in practice, once the convergence of the scalar case has been established, using nonlinear field-by-field decomposition extension to systems generally works well; see [1,3,7,8], for example. Osher and Chakravarthy also showed that the extensions of $\alpha$ schemes to systems have many of the same properties as in the scalar case [2]. Two-dimensional schemes that are based on dimension-by-dimension high-resolution differencing, even for complex configurations with very strong shocks, also perform well; see [7,9,10], for example. Consequently, the analysis of the entropy convergence of $\alpha$ schemes with source terms is very important and relevant from both the theoretical and practical points of view. These schemes can be embedded in or integrated into the design and development of state-of-the-art schemes that can handle modern computational needs for systems and multidimensional cases.

In this study, we consider numerical approximations to scalar conservation laws with source terms:

$$\left\{\begin{array}{c}{u}_t+f{(u)}_x=q(u),\hfill \\ u(x,0)=u_0(x),\hfill \end{array}\right.$$

(1)

where $f\in C^1(\mathbb{R})$ is convex and $u_0\in BV(\mathbb{R})$. Here, $BV$ stands for the subspace of $L_{loc}^1$ consisting of functions with bounded total variation. Next, to introduce the numerical methods, we let $\lambda =\frac{\tau }{h}$, where h and $\tau$ are spatial and temporal step sizes, respectively. Let $u_k^n=u(x_k,t_n)$ be the nodal values of piecewise constant mesh function $u_h(x,t)$ approximating the solution of (1). Now, the numerical schemes that we are interested in admit the conservative form

$$u_k^{n+1}=H(u_{k-p}^n,\cdots ,u_{k+p}^n;\lambda )=u_k^n-\lambda (g_{k+\frac{1}{2}}^n-g_{k-\frac{1}{2}}^n)+\tau q(u_k^n),$$

(2)

where the numerical flux, g, is given by

$$g_{k+\frac{1}{2}}^n=g_{k+\frac{1}{2}}[u_k^n],$$

(3)

and

$$g_{k+\frac{1}{2}}[v]=g(v_{k-p+1},v_{k-p+2},\cdots ,v_k,\cdots ,v_{k+p}),$$

(4)

for any data $\{v_j\}$. Also, function g is Lipschitz continuous with respect to its $2p$ arguments and is consistent with the conservation law in the sense that

$$g(u,u,\cdots ,u)\equiv f(u).$$

(5)

The homogeneous problems that correspond to (1) are

$$\left\{\begin{array}{c}{w}_t+f{(w)}_x=0,\hfill \\ w(x,0)=w_0(x).\hfill \end{array}\right.$$

(6)

With the numerical flux given by (4), we call the corresponding schemes

$${\widehat{u}}_j^{n+1}=\widehat{H}(u_{j-p}^n,\cdots ,u_{j+p}^n;\lambda )=u_j^n-\lambda (g_{j+\frac{1}{2}}^n-g_{j-\frac{1}{2}}^n)$$

(7)

that are consistent with the problems of (6) the homogeneous counterparts (HCPs) of schemes (2)–(5).

The paper is organized as follows: In Section 2, we review the notions of the extremum paths [6], the definition of $\alpha$ schemes with source terms in the case of $m=2$; then, we establish the extremum traceableness of $\mathit{\epsilon }$-scaled forms [5], which is necessary for analyzing the entropy convergence of the $\alpha$ schemes with source terms that are given in the next section. In Section 3, we present a simplified version of the extension of Yang’s convergence criterion [5], an important entropy estimate, and then the main convergence result. Section 4 presents the concluding remarks; finally, in Abbreviations, we list some abbreviations that are used throughout the paper.

2. Extremum Traceableness of $\mathit{\epsilon }$-Scaled Forms

2.1. Extremum Paths

In this subsection, we introduce the $\alpha$ schemes with source terms, whose HCPs were constructed by Chakravarthy and Osher [1,2], and the notions of Yang’s extremum paths [6].

To improve readability, throughout the rest of the paper, we use the following shorthand notation: $f_k^n:=f(u_k^n)$, $u_{k\pm \frac{1}{2}}^n:=\Delta u_{k\pm \frac{1}{2}}^n=\pm (u_{k\pm 1}^n-u_k^n)$, and $f_{k\pm \frac{1}{2}}^n:=\Delta f_{k\pm \frac{1}{2}}^n=\pm (f_{k\pm 1}^n-f_k^n)$. Also, whenever there is no ambiguity in the context, we employ the following simplified notation: $u^k:=u_k^{n+1}$, $u_k:=u_k^n$, $f_k:=f_k^n$, and $f_{k\pm \frac{1}{2}}^{\pm }:={(f_{k\pm \frac{1}{2}}^n)}^{\pm }$, where k and n are the spatial and temporal indexes, respectively.

Let $g_{k+\frac{1}{2}}^E:=g^E(u_k^n,u_{k+1}^n)$ be the flux of an E-scheme [11] that is characterized by

$$\mathrm{sgn}(u_{k+1}^n-u_k^n)[g_{k+\frac{1}{2}}^E-f(u)]\le 0,$$

(8)

for all u between $u_k^n$ and $u_{k+1}^n$. Then, the flux differences are defined by

$$f_{k+\frac{1}{2}}^{+}=f_{k+1}-g_{k+\frac{1}{2}}^E,f_{k+\frac{1}{2}}^{-}=g_{k+\frac{1}{2}}^E-f_k.$$

(9)

At time level $t=t^n$, for all k, we define a series of local CFL numbers

$${\nu }_{k+\frac{1}{2}}^{+}=\frac{\lambda f_{k+\frac{1}{2}}^{+}}{u_{k+\frac{1}{2}}},{\nu }_{k+\frac{1}{2}}^{-}=\frac{\lambda f_{k+\frac{1}{2}}^{-}}{u_{k+\frac{1}{2}}}.$$

(10)

Clearly, we have ${\nu }_{k+\frac{1}{2}}^{+}\ge 0$ and ${\nu }_{k+\frac{1}{2}}^{-}\le 0$. For convenience, we also set the ratios

$$r_k^{+}=\frac{f_{k-\frac{1}{2}}^{+}}{f_{k+\frac{1}{2}}^{+}},r_k^{-}=\frac{f_{k+\frac{1}{2}}^{-}}{f_{k-\frac{1}{2}}^{-}}.$$

(11)

We recall that the “minmod” operator is defined by

$$\mathrm{minmod}(x,y)=\left\{\begin{array}{c}x,\mathrm{if}|x|\le |y|\mathrm{and}xy>0,\hfill \\ y,\mathrm{if}|x|>|y|\mathrm{and}xy>0,\hfill \\ 0,\mathrm{if}xy\le 0,\hfill \end{array}\right.$$

(12)

which can be converted, when divided by x, to a monotonically increasing function

$$\varphi (r)=max(0,min(1,r))=\left\{\begin{array}{c}1,\mathrm{if}r\ge 1,\\ r,\mathrm{if}0\le r\le 1,\\ 0,\mathrm{if}r\le 0,\end{array}\right.$$

(13)

where $r=\frac{y}{x}$. Clearly, $\varphi (r)$ has a symmetry property,

$$\frac{\varphi (r)}{r}=\varphi (\frac{1}{r}).$$

(14)

In the sequel, using this symmetry property, we can recast a scheme of form (2)–(5) into an increment form. For $0<\alpha \le \frac{1}{2}$ and $m=2$, an $\alpha$ scheme [1,2], now with a source term, is given by

$$u^k=u_k-\lambda (g_{k+\frac{1}{2}}-g_{k-\frac{1}{2}})+\tau q(u_k),$$

(15)

where

$$\begin{array}{ccc}\hfill g_{k+\frac{1}{2}}& =& g_{k+\frac{1}{2}}^E-\alpha {(f_{k+\frac{3}{2}}^{-})}^{(1)}-(\frac{1}{2}-\alpha ){(f_{k+\frac{1}{2}}^{-})}^{(0)}\hfill \\ \hfill & +& (\frac{1}{2}-\alpha ){(f_{k+\frac{1}{2}}^{+})}^{(0)}+\alpha {(f_{k-\frac{1}{2}}^{+})}^{(-1)}.\hfill \end{array}$$

(16)

The superscripts shown over $f_{}^{\pm }$ denote flux-limited values of $f_{}^{\pm }$ and are computed as follows:

$$\begin{array}{ccc}\hfill {(f_{k+\frac{3}{2}}^{-})}^{(1)}& =& \mathrm{minmod}[f_{k+\frac{3}{2}}^{-},b{f}_{k+\frac{1}{2}}^{-}]\hfill \\ \hfill & =& \mathrm{minmod}[\frac{1}{b}\frac{f_{k+\frac{3}{2}}^{-}}{f_{k+\frac{1}{2}}^{-}},1]b{f}_{k+\frac{1}{2}}^{-}\hfill \\ \hfill & =& \varphi (\frac{r_{k+1}^{-}}{b})b{f}_{k+\frac{1}{2}}^{-},\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill {(f_{k+\frac{1}{2}}^{-})}^{(0)}& =& \mathrm{minmod}[f_{k+\frac{1}{2}}^{-},b{f}_{k+\frac{3}{2}}^{-}]\hfill \\ \hfill & =& \mathrm{minmod}[1,b\frac{f_{k+\frac{3}{2}}^{-}}{f_{k+\frac{1}{2}}^{-}}]f_{k+\frac{1}{2}}^{-}\hfill \\ \hfill & =& \varphi (b{r}_{k+1}^{-})f_{k+\frac{1}{2}}^{-},\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill {(f_{k+\frac{1}{2}}^{+})}^{(0)}& =& \mathrm{minmod}[f_{k+\frac{1}{2}}^{+},b{f}_{k-\frac{1}{2}}^{+}]\hfill \\ \hfill & =& \mathrm{minmod}[1,b\frac{f_{k-\frac{1}{2}}^{+}}{f_{k+\frac{1}{2}}^{+}}]f_{k+\frac{1}{2}}^{+}\hfill \\ \hfill & =& \varphi (b{r}_k^{+})f_{k+\frac{1}{2}}^{+},\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill {(f_{k-\frac{1}{2}}^{+})}^{(-1)}& =& \mathrm{minmod}[f_{k-\frac{1}{2}}^{+},b{f}_{k+\frac{1}{2}}^{+}]\hfill \\ \hfill & =& \mathrm{minmod}[\frac{1}{b}\frac{f_{k-\frac{1}{2}}^{+}}{f_{k+\frac{1}{2}}^{+}},1]b{f}_{k+\frac{1}{2}}^{+}\hfill \\ \hfill & =& \varphi (\frac{r_k^{+}}{b})b{f}_{k+\frac{1}{2}}^{+}.\hfill \end{array}$$

(20)

In the above, b is a “compression” parameter [2] chosen in the range

$$0<b\le 1+\frac{1}{2\alpha }=b_{max}.$$

We shall assume for the remainder of the paper that the local CFL numbers satisfy $|{\nu }_{k+\frac{1}{2}}^{\pm }|\le 1$ for all $k\in \mathbb{Z}$.

By rewriting the expression

$$[(\frac{1}{2}-\alpha )\varphi (b{r}_k^{+})+\alpha b\varphi (\frac{r_k^{+}}{b})]{\nu }_{k+\frac{1}{2}}^{+}{u}_{k+\frac{1}{2}}$$

as

$$[(\frac{1}{2}-\alpha )\frac{1}{r_k^{+}}\varphi (b{r}_k^{+})+\alpha b\frac{1}{r_k^{+}}\varphi (\frac{r_k^{+}}{b})]{\nu }_{k-\frac{1}{2}}^{+}{u}_{k-\frac{1}{2}};$$

and

$$[\alpha b\varphi (\frac{r_k^{-}}{b})+(\frac{1}{2}-\alpha )\varphi (b{r}_k^{-})]{\nu }_{k-\frac{1}{2}}^{-}{u}_{k-\frac{1}{2}}$$

as

$$[\alpha b\frac{1}{r_k^{-}}\varphi (\frac{r_k^{-}}{b})+(\frac{1}{2}-\alpha )\frac{1}{r_k^{-}}\varphi (b{r}_k^{-})]{\nu }_{k+\frac{1}{2}}^{-}{u}_{k+\frac{1}{2}},$$

it is easy to see that an $\alpha$ scheme of form (15)–(16) can be written in an increment form [4] as

$$u^k=u_k-C_{k-\frac{1}{2}}{u}_{k-\frac{1}{2}}+D_{k+\frac{1}{2}}{u}_{k+\frac{1}{2}}+\tau q(u_k),$$

(21)

with

$$\begin{array}{ccc}\hfill C_{k-\frac{1}{2}}& =& {\nu }_{k-\frac{1}{2}}^{+}[(\frac{1}{2}-\alpha )\frac{1}{r_k^{+}}\varphi (b{r}_k^{+})+\alpha b\frac{1}{r_k^{+}}\varphi (\frac{r_k^{+}}{b})\hfill \\ \hfill & +& 1-(\frac{1}{2}-\alpha )\varphi (b{r}_{k-1}^{+})-\alpha b\varphi (\frac{r_{k-1}^{+}}{b})],\hfill \end{array}$$

(22)

and

$$\begin{array}{ccc}\hfill D_{k+\frac{1}{2}}& =& -{\nu }_{k+\frac{1}{2}}^{-}[1-\alpha b\varphi (\frac{r_{k+1}^{-}}{b})-(\frac{1}{2}-\alpha )\varphi (b{r}_{k+1}^{-})\hfill \\ \hfill & +& \alpha b\frac{1}{r_k^{-}}\varphi (\frac{r_k^{-}}{b})+(\frac{1}{2}-\alpha )\frac{1}{r_k^{-}}\varphi (b{r}_k^{-})].\hfill \end{array}$$

(23)

The HCPs of schemes (21)–(23) are given by

$${\widehat{u}}_k^{n+1}=\widehat{H}(u_{k-p}^n,\cdots ,u_{k+p}^n;\lambda )=u_k^n-C_{k-\frac{1}{2}}{u}_{k-\frac{1}{2}}^n+D_{k+\frac{1}{2}}{u}_{k+\frac{1}{2}}^n.$$

(24)

A scheme of form (21)–(23) provides a convenient way of checking the extremum traceability of a scheme of form (25) and the TVD property of the associated HCP (7) [10,12].

For convenience, let $\mathrm{Y}$ be the set of all sequences of numbers in $(0,1)$ with a zero limit. We use bold-faced letters to represent the sequences in $\mathrm{Y}$ and use the corresponding light-faced ones with subscripts to represent the terms in such a sequence.

Let $\mathit{\epsilon }\in \mathrm{Y}$, i.e., $\mathit{\epsilon }={\{\epsilon \}}_{l=0}^{\infty }$ and ${lim}_{l\to \infty }{\epsilon }_l=0$. We call the following scheme the $\mathit{\epsilon }$-scaled form [5] of schemes (2)–(5):

$$u_k^{n+1}=H_{\mathit{\epsilon }}(u_{k-p}^n,\cdots ,u_{k+p}^n;\lambda )=u_k^n-\lambda [g_{k+\frac{1}{2}}(u_k^n)-g_{k-\frac{1}{2}}(u_k^n)]+{\epsilon }_l\tau q(u_k^n),$$

(25)

where step size $\mathit{\tau }\in \mathrm{Y}$ and step size $\mathbf{h}\in \mathrm{Y}$.

The concept of discrete extremum paths was introduced by Yang (see Definition 6.3 [13] and Definition 2.13 [6]) for both semi-discrete and fully discrete cases. For the purpose of keeping this paper reasonably self-contained, we re-state Yang’s definitions for the fully discrete case in order.

Let us consider a numerical solution u defined on set of grid points $X:=\{(x_j,t_n):j\in \mathbb{Z},n\in {\mathbb{Z}}^{+}\}.$ A finite set of successive grid points $\{x_q,\cdots ,x_r\}$, with $r\ge q$, is said to be the stencil of a spatial maximum or simply an $\overline{E}$-stencil of u at time $t_n$, provided that $u_q^n=\cdots =u_r^n,u_{q-1}^n<u_q^n$, and $u_{r+1}^n<u_r^n$. Notions of $\underline{E}$-stencils for minima and E-stencils for general extrema are defined similarly. Throughout the paper, we refer to [6] for the definitions, lemmas, and theorems that we quote in the context.

Definition 1 

(see Definition 2.13 [6]). A nonempty subset of X denoted by ${\overline{E}}_{t_n,t_m},n\le m$, is called a ridge of numerical solution u from $t_n$ to $t_m$ if the following apply:

(i) For all ν, $n\le \nu \le m$, set

$$P_{\overline{E}}(\nu ):=\{x_j:(x_j,t_{\nu })\in {\overline{E}}_{t_n,t_m}\}=\{x_{q^{\nu }},\cdots ,x_{r^{\nu }}\}$$

is not empty and is an $\overline{E}$-stencil of u at $t_{\nu }$;

(ii) For all ν, $n\le \nu \le m-1$,

$$P_{\overline{E}}(\nu )\cup P_{\overline{E}}(\nu +1)=\{x_j:min(q^{\nu },q^{\nu +1})\le j\le max(r^{\nu },r^{\nu +1})\}.$$

Set $P_{\overline{E}}(\nu )$ is called the x-projection of ${\overline{E}}_{t_n,t_m}$ at $t_{\nu }$. The value of u along the ridge is denoted by $V_{\overline{E}}(\nu ):V_{\overline{E}}(\nu )=u_j^{\nu }$ for $q^{\nu }\le j\le r^{\nu }$.

If for all $\nu$, $n\le \nu \le m$, the $\overline{E}$-stencil in item (i) of the definition is replaced with an $\underline{E}$-stencil, then the set is called a trough of u from $t_n$ to $t_m$ and is denoted by ${\underline{E}}_{t_n,t_m}$. The related notions $P_{\underline{E}}(\nu )$ and $V_{\underline{E}}(\nu )$ are defined similarly. Ridges and troughs are also called extremum paths. When we do not distinguish between ridges and troughs, we use $E_{t_n,t_m}$, $P_E(\nu )$, or $V_E(\nu )$ for either type. We write

$$E_{t_n,t_m}^1<(\le )E_{t_n,t_m}^2,\mathrm{if}max{P}_{E^1}(\nu )<(\le )max{P}_{E^2}(\nu )\mathrm{for}n\le \nu \le m.$$

Definition 2 

(see Definition 2.14 [6]). A scheme is said to be extremum-traceable if there exists a positive constant $c\ge 1$ such that for each numerical solution u of the scheme and each integer $N>0$, there exists a finite or infinite collection of extremum paths ${\{E_{t_0,t_N}^l\}}_{l=l_1}^{l_2}$ with the following properties:

(i) ${\{P_{E^l}(N)\}}_{l=l_1}^{l_2}$ is precisely the set of E-stencils of $u_j^n$ at time $t_N$ arranged in ascending spatial coordinates.

(ii) If $E_{t_0,t_N}^l$ is a ridge (trough), then $V_{E_l}(n)$ is a non-increasing (non-decreasing) function of n.

(iii) Let $P_{E^l}(n)=\{x_{q^l(n)},\cdots ,x_{r^l(n)}\}$ for $1\le n\le N$. If $P_{E^l}(n)\cap P_{E^l}(n+1)=\varnothing$, then

$$|u_{q^l(n+1)}^n-u_{r^l(n)}^n|\le c|V_{E^l}(n+1)-V_{E^l}(n)|when{q}^l(n+1)>r^l(n),$$

$$|u_{r^l(n+1)}^n-u_{q^l(n)}^n|\le c|V_{E^l}(n+1)-V_{E^l}(n)|when{q}^l(n)>r^l(n+1).$$

(iv) If $l_2>l_1$, then $E_{t_0,t_N}^l<E_{t_0,t_N}^{l+1}$ for $l_1\le l\le l_2-1$.

2.2. Extremum Traceableness of the $\mathit{\epsilon }$-Scaled Form

In the next section, we use an extension of Yang’s convergence criterion [5,6] to show the entropy convergence of schemes (15)–(16). For this purpose, we need to verify that $\mathit{\epsilon }$-scaled form (25) is extremum-traceable. The proof of Theorem 1 is similar to the proof of Theorem 2.3 [12]. For this reason, we omit the proof.

Theorem 1 

(see Theorem 2.3 [12]). The conditions for ε-scaled form (25) to be extremum-traceable for sufficiently small ε and ${\epsilon }_l<\epsilon$ are the following inequalities:

$$0\le C_{k+\frac{1}{2}},0\le D_{k+\frac{1}{2}},0\le C_{k+\frac{1}{2}}+D_{k+\frac{1}{2}}\le 1, for all k;$$

(26)

and there is a positive constant μ such that if $u_k$ is a space extremum, then

$$max\{C_{k\pm \frac{1}{2}},C_{k\pm \frac{3}{2}},D_{k\pm \frac{1}{2}},D_{k+\frac{3}{2}}\}\le \frac{\mu }{4}<\frac{1}{4},$$

(27)

where $C_{k+\frac{1}{2}}$ and $D_{k+\frac{1}{2}}$ are given by (22)–(23).

The inequalities of (26) are Harten’s [10] sufficient conditions for the HCPs of schemes (15)–(16) to be TVD. In terms of the local CFL numbers, the conclusion of Theorem 1 can be stated as follows.

Corollary 1. 

The conditions for ε-scaled form (25) to be extremum-traceable for sufficiently small ε and ${\epsilon }_l<\epsilon$ are the following inequalities:

$${\nu }_{k+\frac{1}{2}}^{+}-{\nu }_{k+\frac{1}{2}}^{-}\le \frac{4\alpha }{1+4\alpha },$$

(28)

for all k and for $0<\alpha \le \frac{1}{2}$; when $u_k$ is an extremum,

$$max({\nu }_{k-\frac{1}{2}}^{+},-{\nu }_{k+\frac{1}{2}}^{-})<\frac{\alpha }{2},$$

(29)

and

$$max\{{\nu }_{k+\frac{1}{2}}^{+},{\nu }_{k\pm \frac{3}{2}}^{+},-{\nu }_{k+\frac{3}{2}}^{-},-{\nu }_{k-\frac{1}{2}}^{-}\}\le \frac{\alpha }{2}\frac{4\alpha }{1+4\alpha }.$$

(30)

Proof. 

Indeed, for all k, in reference to (22)–(23), we have $C_{k+\frac{1}{2}}\ge 0$, $D_{k+\frac{1}{2}}\ge 0$, and

$$C_{k+\frac{1}{2}}+D_{k+\frac{1}{2}}\le ({\nu }_{k+\frac{1}{2}}^{+}-{\nu }_{k+\frac{1}{2}}^{-})\frac{4\alpha +1}{4\alpha }\le 1.$$

When $u_k$ is an extremum, $C_{k\pm \frac{1}{2}}$ and $D_{k\pm \frac{1}{2}}$ are reduced to

$$C_{k-\frac{1}{2}}={\nu }_{k-\frac{1}{2}}^{+}[1-(\frac{1}{2}-\alpha )\varphi (b{r}_{k-1}^{+})-\alpha b\varphi (\frac{r_{k-1}^{+}}{b})]\le {\nu }_{k-\frac{1}{2}}^{+},$$

$$D_{k+\frac{1}{2}}=-{\nu }_{k+\frac{1}{2}}^{-}[1-\alpha b\varphi (\frac{r_{k+1}^{-}}{b})-(\frac{1}{2}-\alpha )\varphi (b{r}_{k+1}^{-})]\le -{\nu }_{k+\frac{1}{2}}^{-},$$

$$C_{k+\frac{1}{2}}={\nu }_{k+\frac{1}{2}}^{+}[1+(\frac{1}{2}-\alpha )\frac{1}{r_{k+1}^{+}}\varphi (b{r}_{k+1}^{+})+\alpha b\frac{1}{r_{k+1}^{+}}\varphi (\frac{r_{k+1}^{+}}{b})]\le {\nu }_{k+\frac{1}{2}}^{+}\frac{4\alpha +1}{4\alpha },$$

and

$$D_{k-\frac{1}{2}}=-{\nu }_{k-\frac{1}{2}}^{-}[1+\alpha b\frac{1}{r_{k-1}^{-}}\varphi (\frac{r_{k-1}^{-}}{b})+(\frac{1}{2}-\alpha )\frac{1}{r_{k-1}^{-}}\varphi (b{r}_{k-1}^{-})]\le -{\nu }_{k-\frac{1}{2}}^{-}\frac{4\alpha +1}{4\alpha }.$$

Using the four inequalities above, we arrive at the desired estimations:

$$C_{k\pm \frac{1}{2}}+D_{k\pm \frac{1}{2}}\le ({\nu }_{k\pm \frac{1}{2}}^{+}-{\nu }_{k\pm \frac{1}{2}}^{-})\frac{4\alpha +1}{4\alpha }\le 1,$$

$$\begin{array}{c}\hfill 2C_{k-\frac{1}{2}}+D_{k-\frac{1}{2}}\le 2{\nu }_{k-\frac{1}{2}}^{+}-{\nu }_{k-\frac{1}{2}}^{-}\frac{4\alpha +1}{4\alpha }<1,\end{array}$$

$$\begin{array}{c}\hfill C_{k+\frac{1}{2}}+2D_{k+\frac{1}{2}}\le {\nu }_{k+\frac{1}{2}}^{+}\frac{4\alpha +1}{4\alpha }-2{\nu }_{k+\frac{1}{2}}^{-}<1,\end{array}$$

$$\begin{array}{ccc}& & C_{k+\frac{1}{2}}+D_{k+\frac{1}{2}}+D_{k+\frac{3}{2}}+C_{k-\frac{1}{2}}\hfill \\ & \le & ({\nu }_{k+\frac{1}{2}}^{+}-{\nu }_{k+\frac{3}{2}}^{-})\frac{4\alpha +1}{4\alpha }+{\nu }_{k-\frac{1}{2}}^{+}-{\nu }_{k+\frac{1}{2}}^{-}<1,\hfill \end{array}$$

$$\begin{array}{ccc}& & C_{k-\frac{1}{2}}+D_{k-\frac{1}{2}}+D_{k+\frac{1}{2}}+C_{k-\frac{3}{2}}\hfill \\ & \le & {\nu }_{k-\frac{1}{2}}^{+}-{\nu }_{k-\frac{1}{2}}^{-}\frac{4\alpha +1}{4\alpha }-{\nu }_{k+\frac{1}{2}}^{-}+{\nu }_{k-\frac{3}{2}}^{+}\frac{4\alpha +1}{4\alpha }<1,\hfill \end{array}$$

and so on. □

Notice that ${max}_{0<\alpha \le \frac{1}{2}}\frac{\alpha }{2}\frac{4\alpha }{1+4\alpha }=\frac{1}{6}$; Lemma 1 follows from Corollary 1 when the building block of $\alpha$ schemes belongs to a subfamily of E-fluxes defined by

$$g^E(x,y)=\left\{\begin{array}{c}f(x)\mathrm{if}s\le x\le y,\hfill \\ f(y)\mathrm{if}x\le y\le s,\hfill \end{array}\right.$$

(31)

where s is a sonic point of $f(·)$: $f^{\prime }(s)=0$.

It is clear that both Godunov [14] and Engquist–Osher [15] fluxes,

$$g^{God}(u_j,u_{j+1})=\left\{\begin{array}{cc}\underset{u_j\le w\le u_{j+1}}{min}f(w)\hfill & \mathrm{when}{u}_j\le u_{j+1},\hfill \\ \underset{u_j\ge w\ge u_{j+1}}{max}f(w)\hfill & \mathrm{when}{u}_j\ge u_{j+1},\hfill \end{array}\right.$$

(32)

and

$$g^{EO}(u_j,u_{j+1})={\int }_0^{u_j}max(f^{\prime }(w),0)dw+{\int }_0^{u_{j+1}}min(f^{\prime }(w),0)dw+f(0),$$

(33)

are members of the fluxes given by (31).

Lemma 1. 

ε-scaled form (25), with the building block of an E-flux defined by (31) and ${\epsilon }_l<\epsilon$ for sufficiently small ε, is extremum-traceable provided that

$${\nu }_{k+\frac{1}{2}}^{+}-{\nu }_{k+\frac{1}{2}}^{-}\le \frac{4\alpha }{1+4\alpha }$$

(34)

for all k and for $0<\alpha \le \frac{1}{2}$; when $u_k$ is an extremum, we have

$$\lambda K^{\prime }=\lambda \underset{u_{k-2}\le w\le u_{k+2}}{max}|f^{\prime }(w)|\le \frac{1}{6}.$$

3. The Main Result: The Entropy Convergence of $\mathit{\alpha }$ Schemes with Source Terms

The schemes that we are concerned satisfy the following separation property at the spatial extrema. See E. Tadmor [16] for a similar kind of conditions that were used to check the TVD conditions of the HCPs of schemes (2)–(5).

Assumption 1. 

The numerical fluxes $g_{k+\frac{1}{2}}^n$, $-\infty <k<\infty$, satisfy

$$g_{k+\frac{1}{2}}^n\ge f(u_k^n)\ge g_{k-\frac{1}{2}}^{n}if{u}_k^n\ge u_{k\pm 1}^n,$$

and

$$g_{k+\frac{1}{2}}^n\le f(u_k^n)\le g_{k-\frac{1}{2}}^{n}if{u}_k^n\le u_{k\pm 1}^n.$$

Lemma 2. 

A scheme of form (15)–(16) satisfies Assumption 1.

The proof of Lemma 2 is the same as the one for the associated HCPs of (15)–(16), since the proof does not involve source terms.

In reference to (25), we denote ${\tilde{v}}_j=H_{\mathit{\epsilon }}(v_{j-p},\cdots ,v_{j+p};\lambda )$ and ${\overline{v}}_j=\frac{v_j+{\tilde{v}}_j}{2}$ for any collection of data $\{v_j\}$. Let ${\overline{v}}_{j\pm \frac{1}{2}}^n:=\Delta {\overline{v}}_{j\pm \frac{1}{2}}^n$, ${\tilde{v}}_{j\pm \frac{1}{2}}^n:=\Delta {\tilde{v}}_{j\pm \frac{1}{2}}^n,$ and $f[w;L,R]$ be the linear functions interpolating $f(w)$ at $w=L$ and $w=R$. In this section, we assume that $f^{″}(w)\ge 0$ and $q^{\prime }(w)\ge 0$. Clearly, HCPs (24) of schemes (21)–(23) and thus of schemes (15)–(16) are TVD, since schemes (24) are the original TVD $\alpha$ schemes (for $m=2$) by Osher and Chakravarthy [2].

Definition 3 

(see Definition 3.3 [5]). For ε-scaled form (25), we call an ordered pair of numbers $\{L,R\}$ a rarefying pair if $L<R$ and $f[w;L,R]>f(w)$ when $L<w<R$. We call a collection of data $\Gamma ={\{v_j\}}_{j=I-p}^{J+p}$ an ε-rarefying collection of ε-scaled form (25) to rarefying pair $\{L,R\}$ if, for $\epsilon >0$, the following apply:

(i) $L=v_I\le v_{I+1}\le \cdots \le v_J=R$;

(ii) ${\tilde{v}}_I\le {\tilde{v}}_{I+1}\le \cdots \le {\tilde{v}}_J$, $|L-{\tilde{v}}_I|<\epsilon$, $|R-{\tilde{v}}_J|<\epsilon$;

(iii) Either $v_{I-1}\ge v_I$ or $v_I=v_{I+1}$, and either $v_{J+1}\le v_J$ or $v_{J-1}=v_J$.

Clearly, conditions (i) and (ii) imply that

$${\overline{v}}_I\le {\overline{v}}_{I+1}\le \cdots \le {\overline{v}}_J,|L-{\overline{v}}_I|<\frac{\epsilon }{2}, \mathrm{and} |R-{\overline{v}}_J|<\frac{\epsilon }{2}.$$

We define piecewise constant function $g_{\Gamma }$ associated with $\epsilon$-rarefying collection $\Gamma$ of $\mathit{\epsilon }$-scaled form (25) as follows:

$$g_{\Gamma }(w)=g_{j+\frac{1}{2}}[v]\mathrm{for} w\in ({\overline{v}}_j,{\overline{v}}_{j+1}),I\le j\le J-1.$$

(35)

An $\epsilon$-rarefying collection $\Gamma ={\{v_j\}}_{j=I-2}^{J+2}$ of $\mathit{\epsilon }$-scaled form (25) to pair $\{L,R\}$ that satisfies

$$L=v_{I-2}=v_{I-1}=v_I=v_{I+1}\le \cdots \le v_{J-1}=v_J=v_{J+1}=v_{J+2}=R$$

(36)

is called an ε-normal collection.

Theorem 2 

(see Theorem 3.21 [5]). A scheme of form (15)–(16), thus of form (21)–(23), with extremum-traceable ε-scaled form (25) converges for convex conservation laws (1) if, for every rarefying pair {$L,R$} and ε-rarefying collection of ε-scaled form (25) to the pair, the quadrature inequality

$${\int }_L^{R}f[w;L,R]dw-{\int }_{{\overline{v}}_I}^{{\overline{v}}_J}{g}_{\Gamma }(w)dw>\delta$$

(37)

holds for some constant $\delta >0$, provided that ε is sufficiently small.

For the class of $\alpha$ schemes with source terms, the condition of every $\epsilon$-rarefying collection in Theorem 2 can be weakened by every $\epsilon$-normal collection; thus, we can simplify this convergence criterion as the following lemma.

Lemma 3. 

A scheme of form (15)–(16), thus of form (21)–(23), with extremum-traceable ε-scaled form (25) converges for convex conservation laws (1), provided that for each rarefying pair $\{L,R\}$, there is a constant $\delta >0$ such that, for sufficiently small ε, inequality (37) holds for all ε-normal collections of ε-scaled form (25) to rarefying pair $\{L,R\}$.

Proof. 

Let $\Lambda =\{{\kappa }_{P-2},\cdots ,{\kappa }_{Q+2}\}$ be an arbitrary $\epsilon$-rarefying collection of $\mathit{\epsilon }$-scaled form (25) to pair $\{L,R\}$. Without loss of generality, we assume that $|{\epsilon }_l\tau q|<\epsilon$. Let

$$S^{\prime }={\int }_{{\overline{\kappa }}_P}^{{\overline{\kappa }}_Q}{g}_{\Lambda }(w)dw=\sum _{j=P}^{Q-1}({\overline{\kappa }}_{j+1}-{\overline{\kappa }}_j)g_{j+\frac{1}{2}}[\kappa ].$$

(38)

By (i) and (iii) of Definition 3, ${\kappa }_P$ is a minimum or a maximum. In either case, Assumption 1 and condition (ii) of Definition 3 imply that

$$\begin{array}{ccc}\hfill \epsilon >|L-{\tilde{\kappa }}_P|& =& |{\tilde{\kappa }}_P-{\kappa }_P|\ge \lambda |g_{P+\frac{1}{2}}[\kappa ]-g_{P-\frac{1}{2}}[\kappa ]|-|{\epsilon }_l\tau q|\hfill \\ \hfill & \ge & \lambda |g_{P\pm \frac{1}{2}}[\kappa ]-f(L)|-|{\epsilon }_l\tau q|,\hfill \end{array}$$

(39)

or

$$\lambda |g_{P\pm \frac{1}{2}}[\kappa ]-f(L)|\le \epsilon +|{\epsilon }_l\tau q|<2\epsilon .$$

Similarly, we have

$$\epsilon >|R-{\tilde{\kappa }}_Q|\ge \lambda |g_{Q\pm \frac{1}{2}}[\kappa ]-f(R)|-|{\epsilon }_l\tau q|,$$

(40)

or

$$\lambda |g_{Q\pm \frac{1}{2}}[\kappa ]-f(R)|<\epsilon +|{\epsilon }_l\tau q|<2\epsilon .$$

Next, we construct an $\epsilon$-normal collection $\Gamma ={\{v_j\}}_{j=I-2}^{J+2}$ as follows: First, let $I=P-1$ and $J=Q+1$; we also set $v_{I-2}=v_{I-1}=v_I=L$, $v_J=v_{J+1}=v_{J+2}=R$, and $v_j={\kappa }_j$ for $I+1\le j\le J-1$. Then, we have

$$g_{I\pm \frac{1}{2}}[v]=f(L)\mathrm{and}{g}_{J\pm \frac{1}{2}}[v]=f(R),$$

(41)

which imply that

$${\tilde{v}}_I=L+{\epsilon }_l\tau q(L),{\tilde{v}}_J=R+{\epsilon }_l\tau q(R).$$

(42)

Thus, the $\epsilon$-normality of $\Gamma ={\{v_j\}}_{j=I-2}^{J+2}$ is justified by the non-decreasing relation of

$${\tilde{v}}_I\le {\tilde{v}}_{I+1}\le \cdots \le {\tilde{v}}_J.$$

Indeed, we notice that the relationship

$${\tilde{v}}_{I+3}\le {\tilde{v}}_{I+4}\le \cdots \le {\tilde{v}}_{J-4}\le {\tilde{v}}_{J-3},$$

is directly inherited from condition (ii) of the given $\epsilon$-rarefying collection of $\Lambda$,

$${\tilde{\kappa }}_{P+2}\le {\tilde{\kappa }}_{P+3}\le \cdots \le {\tilde{\kappa }}_{Q-3}\le {\tilde{\kappa }}_{Q-2}.$$

Therefore, we only need to verify that

$${\tilde{v}}_I\le {\tilde{v}}_{I+1}\le {\tilde{v}}_{I+2}\le {\tilde{v}}_{I+3},$$

and similarly, we have

$${\tilde{v}}_{J-3}\le {\tilde{v}}_{J-2}\le {\tilde{v}}_{J-1}\le {\tilde{v}}_J.$$

The proof is straightforward, and we omit the details.

Now, let G be the Lipschitz constant of numerical flux g, and $K=max\{|f(L)|,|f(R)|,$$|g|\}+G(R-L)$. If we denote

$$S={\int }_{{\overline{v}}_I}^{{\overline{v}}_J}{g}_{\Gamma }(w)dw=\sum _{j=I}^{J-1}({\overline{v}}_{j+1}-{\overline{v}}_j)g_{j+\frac{1}{2}}[v],$$

(43)

then a priori estimate $|S-S^{\prime }|\le 4K\epsilon$ holds. Let ${\delta }^{\prime }$ be a constant such that for all $\epsilon$-normal collections of $\mathit{\epsilon }$-scaled form (25) to pair $\{L,R\}$, inequality (37) holds for $\delta ={\delta }^{\prime }$. Thus, for $\delta ={\delta }^{\prime }$, inequality (37) also holds for $\epsilon$-normal collection $\Gamma ={\{v_j\}}_{j=I-2}^{J+2}$. Therefore, for $\delta =\frac{{\delta }^{\prime }}{2}$, inequality (37) holds for all $\epsilon$-rarefying collections of $\mathit{\epsilon }$-scaled form (25) to pair $\{L,R\}$ provided that $\epsilon \le \frac{\delta }{4K}$.

It remains for us to show the a priori estimate. First, we notice that ${\overline{\kappa }}_j={\overline{v}}_j$ for $P+2\le j\le Q-3$, and we can verify that for $P+1\le j\le Q-2$, the jth term of S equals the jth term of $S^{\prime }$. Therefore, the terms of the difference

$$S-S^{\prime }=\sum _{j=I}^{J-1}({\overline{v}}_{j+1}-{\overline{v}}_j)g_{j+\frac{1}{2}}[v]-\sum _{j=P}^{Q-1}({\overline{\kappa }}_{j+1}-{\overline{\kappa }}_j)g_{j+\frac{1}{2}}[\kappa ]$$

from $j=P+1$ to $j=Q-2$ are all diminished. Then, for the remaining terms, we use the relationship of $\Lambda$ and $\Gamma$, and (39)–(42) to yield the following estimates:

$$|{\overline{v}}_{I+1}-{\overline{\kappa }}_{I+1}|<\epsilon ,|{\overline{v}}_{J-1}-{\overline{\kappa }}_{J-1}|<\epsilon ,$$

(44)

$$|{\overline{v}}_{I+1}-{\overline{v}}_I|=\frac{\lambda }{2}|g_{I+\frac{3}{2}}-f(L)|=\frac{\lambda }{2}|g_{P+\frac{1}{2}}-f(L)|<\epsilon ,$$

(45)

and

$$|{\overline{v}}_J-{\overline{v}}_{J-1}|=\frac{\lambda }{2}|f(R)-g_{J-\frac{3}{2}}|=\frac{\lambda }{2}|f(R)-g_{Q-\frac{1}{2}}|<\epsilon .$$

(46)

Finally, using the fact that ${\overline{v}}_{I+2}={\overline{\kappa }}_{P+1}$, ${\overline{v}}_{J-2}={\overline{\kappa }}_{Q-1}$, $g_{I+\frac{3}{2}}[v]=g_{P+\frac{1}{2}}[\kappa ]$, $g_{J-\frac{3}{2}}[v]=g_{Q-\frac{1}{2}}[\kappa ]$, and (44)–(46), we derive the desired estimate as follows:

$$\begin{array}{ccc}\hfill |S-S^{\prime }|& =& |({\overline{v}}_{I+1}-{\overline{v}}_I)g_{I+\frac{1}{2}}[v]+({\overline{v}}_J-{\overline{v}}_{J-1})g_{J-\frac{1}{2}}[v]\hfill \\ \hfill & +& ({\overline{v}}_{I+2}-{\overline{v}}_{I+1})g_{I+\frac{3}{2}}[v]-({\overline{\kappa }}_{P+1}-{\overline{\kappa }}_P)g_{P+\frac{1}{2}}[\kappa ]\hfill \\ \hfill & +& ({\overline{v}}_{J-1}-{\overline{v}}_{J-2})g_{J-\frac{3}{2}}[v]-({\overline{\kappa }}_Q-{\overline{\kappa }}_{Q-1})g_{Q-\frac{1}{2}}[\kappa ]|\hfill \\ \hfill & \le & |{\overline{v}}_{I+1}-{\overline{v}}_I||g_{I+\frac{1}{2}}[v]|+|{\overline{v}}_J-{\overline{v}}_{J-1}||g_{J-\frac{1}{2}}[v]|\hfill \\ \hfill & +& |{\overline{v}}_{I+1}-{\overline{\kappa }}_P||g_{I+\frac{3}{2}}[v]|+|{\overline{v}}_{J-1}-{\overline{\kappa }}_Q||g_{Q-\frac{1}{2}}[\kappa ]|\hfill \\ \hfill & <& (\epsilon +\epsilon +\epsilon +\epsilon )K=4K\epsilon ,\hfill \end{array}$$

(47)

and the proof is completed. □

For an $\epsilon$-normal collection $\Gamma ={\{v_j\}}_{j=I-2}^{J+2}$, we denote vertex $(v_j,f(v_j))$ by $V_j$ and the area of convex polygon $V_{j_1}{V}_{j_2}\cdots V_{j_r}$ by $S_{j_1,\cdots ,j_r}$. Let ${\sigma }_{\Gamma }={max}_{I-2\le j\le J+2}|{\nu }_{j\pm \frac{1}{2}}^{\pm }|$, and let

$${\alpha }_j=\left\{\begin{array}{cc}0.5\hfill & \mathrm{if}\Delta v_{j-2}=\Delta v_{j+1}=0,\hfill \\ 1\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

When the building blocks of schemes (15)–(16), thus of schemes (21)–(23), are the subclass of E-schemes with the fluxes defined by (31), we obtain a very important inequality (49), which enables us to prove Theorem 3, the main result of this paper. The proof of Lemma 4 is similar to the one, given by the author [4], for the HCPs of (15)–(16). To see the reason for that, first, we can show the inequality

$${\int }_L^R{g}_{\Gamma }(w)dw\le {\int }_{{\overline{v}}_I}^{{\overline{v}}_J}{g}_{\Gamma }(w)dw+\epsilon$$

(48)

by mimicking the proof for inequality (50) with the following modifications: In the proof of case 1, let c be a constant such that $|g_{\Gamma }(w)|\le c$ for $w\in (R,{\overline{v}}_J)$, and we set $g_{\Gamma }(w)=c$, when $w\in (L,{\overline{v}}_I)$; for case 3, let c be a constant such that $|g_{\Gamma }(w)|\le c$ for $w\in ({\overline{v}}_I,L)$, and we set $g_{\Gamma }(w)=c$, when $w\in ({\overline{v}}_J,R)$. Next, we remark that the derivations of all estimations in the proof of (49) are the same as the ones in the proof for their HCPs; except for $\mathit{\epsilon }$-scaled form (25), we now have extra finite terms of form ${\epsilon }_l\tau q^{\prime }$ (which may be multiplied by a bounded quantity), which all vanish to zero as $l\to \infty$. This means that if $\epsilon$ is small enough and ${\epsilon }_l\tau q^{\prime }<\epsilon$, then the contributions of these terms are negligible; therefore, inequality (49) still holds for $\mathit{\epsilon }$-scaled form (25). For the convenience of the reader, we provide the proof at the end of the paper. Also, we would like to recall that Lemma 5 is Yang’s original result [6].

Lemma 4. 

Let $\Gamma ={\{v_j\}}_{j=I-2}^{J+2}$ be an ε-normal collection of ε-scaled form (25) to rarefying pair $\{L,R\}$. Then, the numerical solutions of ε-scaled form (25) for convex conservation laws (1) satisfy, for sufficiently small ε and ${\sigma }_{\Gamma }$, the following inequality:

$${\int }_L^R(f[w;L,R]-g_{\Gamma })dw\ge S_{I,I+1,\cdots ,J}-\sum _{j=I+1}^{J-1}{\alpha }_j{S}_{j-1,j,j+1}.$$

(49)

Lemma 5 

(see Lemma 3.7 [6]). We have

$$S_{I,I+1,\cdots ,J}-\sum _{j=I+1}^{J-1}{S}_{j-1,j,j+1}\ge S_{I,i,i+1,J}-(S_{I,i,i+1}+S_{i,i+1,J})$$

for $I<i<J-1$.

Let $\sigma =\lambda {max}_w|f^{\prime }(w)|$. When $m=2$, for the class of fully discrete $\alpha$ schemes with source terms, equipped with the above lemmas, we obtain the following entropy convergence result.

Theorem 3. 

The numerical solutions of schemes (15)–(16) to convex problems (1) converge to the entropy solution provided that $g^E(·,·)$ is a numerical flux given by (31) and that σ and ε are sufficiently small.

Proof. 

First, by Lemma 1, $\mathit{\epsilon }$-scaled form (25) is extremum-traceable for sufficiently small $\sigma$ and $\epsilon$. Next, for each $\epsilon$-normal collection $\Gamma ={\{v_i\}}_{i=I-2}^{J+2}$ of (25) to rarefying pair $\{L,R\}$, we claim that

$${\int }_{{\overline{v}}_I}^{{\overline{v}}_J}{g}_{\Gamma }(w)dw\le {\int }_L^R{g}_{\Gamma }(w)dw+\epsilon .$$

(50)

Indeed, for the given $\epsilon$-normal collection $\Gamma ={\{v_i\}}_{i=I-2}^{J+2}$, we have ${\overline{v}}_I=L+\frac{{\epsilon }_l\tau }{2}q(L)$ and ${\overline{v}}_J=R+\frac{{\epsilon }_l\tau }{2}q(R)$. Also, we recall that $q^{\prime }(w)\ge 0$ and

$$g_{\Gamma }(w)=g_{j+\frac{1}{2}}[v]\mathrm{for}w\in ({\overline{v}}_j,{\overline{v}}_{j+1}),I\le j\le J-1.$$

Case 1. If $q(L)\ge 0$, then $q(R)\ge 0$ as well. Let c be a constant such that $|g_{\Gamma }(w)|\le c$ for $w\in (R,{\overline{v}}_J)$ and set $g_{\Gamma }(w)=-c$, when $w\in (L,{\overline{v}}_I)$. We obtain

$${\int }_{{\overline{v}}_I}^{{\overline{v}}_J}{g}_{\Gamma }(w)dw=\{{\int }_{{\overline{v}}_I}^L+{\int }_L^R+{\int }_R^{{\overline{v}}_J}\}{g}_{\Gamma }(w)dw\le c{\epsilon }_l\tau \frac{q(L)+q(R)}{2}+{\int }_L^R{g}_{\Gamma }(w)dw.$$

Case 2. If $q(L)\le 0$ and $q(R)\ge 0$, we let c be a constant such that $|g_{\Gamma }(w)|\le c$ for $w\in (R,{\overline{v}}_J)\cup ({\overline{v}}_I,L)$. Now, we have

$${\int }_{{\overline{v}}_I}^{{\overline{v}}_J}{g}_{\Gamma }(w)dw=\{{\int }_{{\overline{v}}_I}^L+{\int }_L^R+{\int }_R^{{\overline{v}}_J}\}{g}_{\Gamma }(w)dw\le c{\epsilon }_l\tau \frac{-q(L)+q(R)}{2}+{\int }_L^R{g}_{\Gamma }(w)dw.$$

Case 3. If $q(L)\le 0$ and $q(R)\le 0$, we let c be a constant such that $|g_{\Gamma }(w)|\le c$ for $w\in ({\overline{v}}_I,L)$ and set $g_{\Gamma }(w)=-c$, when $w\in ({\overline{v}}_J,R)$. We obtain

$${\int }_{{\overline{v}}_I}^{{\overline{v}}_J}{g}_{\Gamma }(w)dw=\{{\int }_{{\overline{v}}_I}^L+{\int }_L^R+{\int }_R^{{\overline{v}}_J}\}{g}_{\Gamma }(w)dw\le c{\epsilon }_l\tau \frac{-q(L)-q(R)}{2}+{\int }_L^R{g}_{\Gamma }(w)dw.$$

In all three cases, without loss of generality, for the given $\epsilon >0$, we let $c{\epsilon }_l\tau \frac{|q(L)|+|q(R)|}{2}<\epsilon$ for all l. Thus, as claimed, the following inequality holds:

$${\int }_{{\overline{v}}_I}^{{\overline{v}}_J}{g}_{\Gamma }(w)dw\le {\int }_L^R{g}_{\Gamma }(w)dw+\epsilon .$$

Now, we set

$$d_1(\Gamma )=\underset{I\le i\le J}{max}min(v_i-L,R-v_i).$$

Since $J-I$ is finite, $d_1(\Gamma )=min(v_j-L,R-v_j)$ for some j between I and J. We then let

$$d_2(\Gamma )=\underset{I\le i\le J,i\ne j}{max}min(v_i-L,R-v_i).$$

We also have $d_2(\Gamma )=min(v_k-L,R-v_k)$ for some $k\ne j$ between I and J. Clearly, we can choose j and k so that $|j-k|=1$.

To complete the proof, we argue by contradiction. Hence, we assume that for certain convex f, a scheme of form (15)–(16), thus of form (21)–(23), does not converge. By Lemma 3 and inequality (50), there is a rarefying pair $\{L,R\}$ such that for each $\delta >0$, ${\delta }^{\prime }=\delta /2$, and $\epsilon =\delta /2$, there is an $\epsilon$-normal collection $\Gamma ={\{v_j\}}_{j=I-2}^{J+2}$ of $\mathit{\epsilon }$-scaled form (25) to the pair that satisfies

$${\int }_L^R\{f[w;L,R]-g_{\Gamma }(w)\}dw\le {\delta }^{\prime }+\epsilon =\delta .$$

It follows that there is a sequence of $\epsilon$-normal collections ${\{{\Gamma }_{\nu }\}}_{\nu =1}^{\infty }$, where ${\Gamma }_{\nu }={\{v_j^{\nu }\}}_{j=I^{\nu }-2}^{J^{\nu }+2}$, such that

$$\underset{\nu \to \infty }{lim}{\int }_L^R\{f[w;L,R]-g_{{\Gamma }_{\nu }}(w)\}\le 0.$$

(51)

The following three cases exhaust all possibilities.

Case 1. $lim{sup}_{\nu \to \infty }{d}_2({\Gamma }_{\nu })>0.$ Let us set $\rho =1/2lim{sup}_{\nu \to \infty }{d}_2({\Gamma }_{\nu }).$ Then, there are a subsequence of $\epsilon$-normal collections, still denoted by ${\{{\Gamma }_{\nu }\}}_{\nu =1}^{\infty }$, and a corresponding sequence of integers ${\{i(\nu )\}}_{\nu =1}^{\infty }$ such that

$$L+\rho \le v_{i(\nu )}^{\nu }\le v_{i(\nu )+1}^{\nu }\le R-\rho ,$$

and ${sup}_{\nu }{\sigma }_{{\Gamma }_{\nu }}\le \sigma$. For simplicity, we fix $\nu$ and drop it from the notation. Let us set $\gamma =f[\frac{L+R}{2};L,R]-f(\frac{L+R}{2})$. It is a positive constant, since $\{L,R\}$ is a rarefying pair. By applying Lemmas 4 and 5, we have

$$\begin{array}{ccc}& & {\int }_L^R\{f[w;L,R]-g_{{\Gamma }_{\nu }}(w)\}dw\ge S_{I,i,i+1,J}-(S_{I,i,i+1}+S_{i,i+1,J})\hfill \\ \hfill & =& 1/2\{(v_i-v_I)(f[v_{i+1};L,R]-f(v_{i+1}))+(v_J-v_{i+1})(f[v_i;L,R]-f(v_i))\}>\eta ,\hfill \end{array}$$

(52)

if $\eta =2{\rho }^2\gamma /(R-L).$ This contradicts (51).

Case 2. $lim{sup}_{\nu \to \infty }{d}_1({\Gamma }_{\nu })>lim{sup}_{\nu \to \infty }{d}_2({\Gamma }_{\nu })=0.$ Let us set $\rho =1/2lim{sup}_{\nu \to \infty }{d}_1({\Gamma }_{\nu }).$ Then, there are a subsequence of the $\epsilon$-normal collections, still denoted by ${\{{\Gamma }_{\nu }\}}_{\nu =1}^{\infty }$, and a corresponding sequence of integers ${\{i^{\nu }\}}_{\nu =1}^{\infty }$ such that ${lim}_{\nu \to \infty }{v}_{i^{\nu }-1}^{\nu }=L,{lim}_{\nu \to \infty }{v}_{i^{\nu }+1}^{\nu }=R$, and ${lim}_{\nu \to \infty }{v}_{i^{\nu }}^{\nu }=v\in [L+\rho ,R-\rho ]$. We then have

$${\int }_L^R(f[w;L,R]-g_{{\Gamma }_{\nu }}(w))dw\to {\int }_L^R(f[w;L,R]-g_{\Gamma }(w))dw,$$

where $\Gamma$ is the following $\epsilon$-normal collection: $I=0,J=4,v_{-2}=v_{-1}=v_0=v_1=L,v_2=v,$ and $v_3=v_4=v_5=v_6=R$. By Lemma 4, we have

$${\int }_L^R(f[w;L,R]-g_{\Gamma }(w))dw\ge S_{1,2,3}-{\alpha }_2{S}_{1,2,3}=1/2S_{1,2,3}>0$$

for ${\alpha }_2=1/2$ since ${\Delta }_{+}{v}_0={\Delta }_{+}{v}_3=0.$ This contradicts (51).

Case 3. $lim{sup}_{\nu \to \infty }{d}_1({\Gamma }_{\nu })=0$. Then, there exists a sequence of integers $\{i^{\nu }\}$ with $I^{\nu }+1\le i^{\nu }<J^{\nu }-1$ such that ${lim}_{\nu \to \infty }{v}_{i^{\nu }}^{\nu }=L,{lim}_{\nu \to \infty }{v}_{i^{\nu }+1}^{\nu }=R$. We then have

$${\int }_L^R(f[w;L,R]-g_{{\Gamma }_{\nu }}(w))dw\to {\int }_L^R(f[w;L,R]-g_{\Gamma }(w))dw,$$

where $\Gamma$ is the following $\epsilon$-normal collection: $I=0,J=3,v_{-2}=v_{-1}=v_0=v_1=L,v_2=v_3=v_4=v_5=R$. In this case, numerical flux $g_{\Gamma }(w)$ becomes E-flux $g^E(L,R)$. Hence, we have

$${\int }_L^R(f[w;L,R]-g_{\Gamma }(w))dw\ge {\int }_L^R(f[w;L,R]-f(w))dw.$$

The right-hand side of the inequality is a positive constant, since $\{L,R\}$ is a rarefying pair. This contradicts (51) again. We have thus completed the proof of Theorem 3. □

Finally, we finish this section by presenting the proof of Lemma 4.

Proof of Lemma 4. 

In the following, we keep the same notation, $f_{j+\frac{1}{2}}^{\pm }$ and $r_j^{\pm }$ for $\{v_j\}$ instead of $\{u_j\}$. We also use

$$f_{j+\frac{1}{2}}^{\prime }:=\frac{f(v_{j+1})-f(v_j)}{v_{j+1}-v_j}$$

(53)

to denote the divided difference.

To justify inequality (49), it suffices to show the following inequality:

$${\int }_L^R{g}_{{}_{\Gamma }}(w)dw-\sum _{j=I}^{J-1}{\int }_{v_j}^{v_{j+1}}f[w;v_j,v_{j+1}]dw\le \sum _{j=I+1}^{J-1}{\alpha }_j{S}_{j-1,j,j+1}.$$

(54)

Without loss of generality, let $v_s$ be a sonic point ($f^{\prime }(v_s)=0$) such that $v_k\le v_s\le v_{k+1}$ for some integer k with $I\le k\le J-1$. Then, for any $g^E(·,·)$ given by (31), we have

$$f_{j+\frac{1}{2}}^{+}=0,\mathrm{for} I\le j\le k-1;$$

$$f_{j+\frac{1}{2}}^{+}=f_{j+\frac{1}{2}}^{\prime }{v}_{j+\frac{1}{2}},\mathrm{for} J-1\ge j\ge k+1;$$

$$f_{j+\frac{1}{2}}^{-}=0,\mathrm{for} J-1\ge j\ge k+1;$$

and

$$f_{j+\frac{1}{2}}^{-}=f_{j+\frac{1}{2}}^{\prime }{v}_{j+\frac{1}{2}},\mathrm{for} I\le j\le k-1.$$

To enhance readability, we further simplify the notation. We denote $f_j:=f(v_j)$, $f_s:=f(v_s)$, $v_{j\pm \frac{1}{2}}:=\Delta v_{j\pm \frac{1}{2}}=\pm (v_{j\pm 1}-v_j)$, $v_{s+\frac{1}{2}}:=v_{k+1}-v_s$, $v_{s-\frac{1}{2}}:=v_s-v_k$, ${\overline{v}}_{j\pm \frac{1}{2}}:=\Delta {\overline{v}}_{j\pm \frac{1}{2}}$, ${\tilde{v}}_{j\pm \frac{1}{2}}:=\Delta {\tilde{v}}_{j\pm \frac{1}{2}}$, $f_{s+\frac{1}{2}}^{\prime }:=(f_{k+1}-f_s)/v_{s+\frac{1}{2}}$, and $f_{s-\frac{1}{2}}^{\prime }:=(f_s-f_k)/v_{s-\frac{1}{2}}$. Then, by (35) and (48), we have

$$\begin{array}{ccc}\hfill \mathrm{LHS} \mathrm{of} (54)& =& {\int }_L^R{g}_{{}_{\Gamma }}(w)dw-\sum _{j=I}^{J-1}{\int }_{v_j}^{v_{j+1}}f[w;v_j,v_{j+1}]dw\hfill \\ & \le & {\int }_{{\overline{v}}_I}^{{\overline{v}}_J}{g}_{{}_{\Gamma }}(w)dw-\sum _{j=I}^{J-1}{\int }_{v_j}^{v_{j+1}}f[w;v_j,v_{j+1}]dw+\epsilon \hfill \\ & =& \sum _{j=I}^{J-1}{g}_{j+\frac{1}{2}}{\overline{v}}_{j+\frac{1}{2}}-\sum _{j=I}^{J-1}{\int }_{v_j}^{v_{j+1}}f[w;v_j,v_{j+1}]dw+\epsilon \hfill \\ & =& \sum _{j=I}^{J-1}{g}_{j+\frac{1}{2}}\frac{v_{j+\frac{1}{2}}+{\tilde{v}}_{j+\frac{1}{2}}}{2}-\sum _{j=I}^{J-1}\frac{f_j+f_{j+1}}{2}{v}_{j+\frac{1}{2}}+\epsilon \hfill \\ & =& \frac{1}{2}(P_{(j\le k-2)}+P_{k-1}+P_k+P_{k+1}+P_{(j\ge k+2)})+\epsilon ,\hfill \end{array}$$

(55)

and the definitions of $P_{(j\le k-2)},P_{k-1},P_k,P_{k+1}$, and $P_{(j\ge k+2)}$ will be given shortly. Let us recall that the numerical flux is defined by

$$\begin{array}{ccc}\hfill g_{j+\frac{1}{2}}& =& g_{j+\frac{1}{2}}^E-\alpha {(f_{j+\frac{3}{2}}^{-})}^{(1)}-(\frac{1}{2}-\alpha ){(f_{j+\frac{1}{2}}^{-})}^{(0)}(\frac{1}{2}-\alpha ){(f_{j+\frac{1}{2}}^{+})}^{(0)}+\alpha {(f_{j-\frac{1}{2}}^{+})}^{(-1)},\hfill \end{array}$$

and with the help of increment form (21)–(23), we have

$${\tilde{v}}_{j+\frac{1}{2}}=v_{j+\frac{1}{2}}-(C_{j+\frac{1}{2}}+D_{j+\frac{1}{2}})v_{j+\frac{1}{2}}+D_{j+\frac{3}{2}}{v}_{j+\frac{3}{2}}+C_{j-\frac{1}{2}}{v}_{j-\frac{1}{2}}+{\epsilon }_l\tau q^{\prime }{v}_{j+\frac{1}{2}}\ge 0.$$

Let

$$d_j:=-(\frac{1}{2}+2\alpha )f_{j+\frac{1}{2}}^{\prime }{v}_{j+\frac{1}{2}}^2-(\frac{1}{2}-\alpha )f_{j-\frac{1}{2}}^{\prime }{v}_{j-\frac{1}{2}}{v}_{j+\frac{1}{2}},$$

$$T:=[f_{k-1}-\alpha f_{k-\frac{1}{2}}^{\prime }{v}_{k-\frac{1}{2}}-(\frac{1}{2}-\alpha )f_{k-\frac{3}{2}}^{\prime }{v}_{k-\frac{3}{2}}]D_{k-\frac{1}{2}}{v}_{k-\frac{1}{2}},$$

and

$$T_{(k\le k-2)}:=-\alpha f_{k-\frac{1}{2}}^{\prime }{v}_{k-\frac{1}{2}}^2+T.$$

With these definitions and the convexity of the flux in mind, we derive the following estimates:

$$\begin{array}{ccc}\hfill & & P_{(j\le k-2)}:=\sum _{j=I}^{k-2}\{g_{j+\frac{1}{2}}[v_{j+\frac{1}{2}}+{\tilde{v}}_{j+\frac{1}{2}}]-(f_j+f_{j+1})v_{j+\frac{1}{2}}\}\hfill \\ & =& \sum _{j=I}^{k-2}\{[g_{j+\frac{1}{2}}^E-\alpha {(f_{j+\frac{3}{2}}^{-})}^{(1)}-(\frac{1}{2}-\alpha ){(f_{j+\frac{1}{2}}^{-})}^{(0)}][2v_{j+\frac{1}{2}}-D_{j+\frac{1}{2}}{v}_{j+\frac{1}{2}}\hfill \\ & & +D_{j+\frac{3}{2}}{v}_{j+\frac{3}{2}}+{\epsilon }_l\tau q^{\prime }{v}_{j+\frac{1}{2}}]-(f_j+f_{j+1})v_{j+\frac{1}{2}}\}\hfill \\ & \le & \sum _{j=I}^{k-2}\{[g_{j+\frac{1}{2}}^E-\alpha f_{j+\frac{3}{2}}^{-}-(\frac{1}{2}-\alpha )f_{j+\frac{1}{2}}^{-}][2v_{j+\frac{1}{2}}-D_{j+\frac{1}{2}}{v}_{j+\frac{1}{2}}\hfill \\ & & +D_{j+\frac{3}{2}}{v}_{j+\frac{3}{2}}+{\epsilon }_l\tau q^{\prime }{v}_{j+\frac{1}{2}}]-(f_j+f_{j+1})v_{j+\frac{1}{2}}\}\hfill \\ & =& \sum _{j=I}^{k-2}\{[2f_{j+1}-(f_j+f_{j+1})+{\epsilon }_l\tau q^{\prime }]v_{j+\frac{1}{2}}-2\alpha f_{j+\frac{3}{2}}^{\prime }{v}_{j+\frac{3}{2}}{v}_{j+\frac{1}{2}}-(1-2\alpha )f_{j+\frac{1}{2}}^{\prime }{v}_{j+\frac{1}{2}}^2\}\hfill \\ & & -\sum _{j=I}^{k-2}[f_{j+1}{v}_{j+\frac{1}{2}}-\alpha f_{j+\frac{3}{2}}^{\prime }{v}_{j+\frac{3}{2}}{v}_{j+\frac{1}{2}}-(\frac{1}{2}-\alpha )f_{j+\frac{1}{2}}^{\prime }{v}_{j+\frac{1}{2}}^2]D_{j+\frac{1}{2}}\hfill \\ & & +\sum _{j=I}^{k-2}[f_{j+1}{v}_{j+\frac{3}{2}}-\alpha f_{j+\frac{3}{2}}^{\prime }{v}_{j+\frac{3}{2}}^2-(\frac{1}{2}-\alpha )f_{j+\frac{1}{2}}^{\prime }{v}_{j+\frac{1}{2}}{v}_{j+\frac{3}{2}}]D_{j+\frac{3}{2}}\hfill \\ & =& \sum _{j=I}^{k-2}[-2\alpha f_{j+\frac{3}{2}}^{\prime }{v}_{j+\frac{3}{2}}{v}_{j+\frac{1}{2}}+2\alpha f_{j+\frac{1}{2}}^{\prime }{v}_{j+\frac{1}{2}}^2+{\epsilon }_l\tau q^{\prime }{v}_{j+\frac{1}{2}}]+\sum _{j=I}^{k-2}[-f_{j+1}{v}_{j+\frac{1}{2}}\hfill \\ & & +\alpha f_{j+\frac{3}{2}}^{\prime }{v}_{j+\frac{3}{2}}{v}_{j+\frac{1}{2}}+(\frac{1}{2}-\alpha )f_{j+\frac{1}{2}}^{\prime }{v}_{j+\frac{1}{2}}^2+f_j{v}_{j+\frac{1}{2}}\hfill \\ & & -\alpha f_{j+\frac{1}{2}}^{\prime }{v}_{j+\frac{1}{2}}^2-(\frac{1}{2}-\alpha )f_{j-\frac{1}{2}}^{\prime }{v}_{j-\frac{1}{2}}{v}_{j+\frac{1}{2}}]D_{j+\frac{1}{2}}+T\hfill \\ & \le & \sum _{j=I}^{k-2}[\alpha f_{j+\frac{3}{2}}^{\prime }{(v_{j+\frac{3}{2}}-v_{j+\frac{1}{2}})}^2+{\epsilon }_l\tau q^{\prime }{v}_{j+\frac{1}{2}}]-\alpha f_{k-\frac{1}{2}}^{\prime }{v}_{k-\frac{1}{2}}^2+\sum _{j=I}^{k-2}[-(\frac{1}{2}+2\alpha )f_{j+\frac{1}{2}}^{\prime }{v}_{j+\frac{1}{2}}^2\hfill \\ & & +\alpha f_{j+\frac{3}{2}}^{\prime }{v}_{j+\frac{3}{2}}{v}_{j+\frac{1}{2}}-(\frac{1}{2}-\alpha )f_{j-\frac{1}{2}}^{\prime }{v}_{j-\frac{1}{2}}{v}_{j+\frac{1}{2}}]D_{j+\frac{1}{2}}+T\hfill \\ & \le & \sum _{j=I}^{k-2}[-(\frac{1}{2}+2\alpha )f_{j+\frac{1}{2}}^{\prime }{v}_{j+\frac{1}{2}}^2+\alpha f_{j+\frac{3}{2}}^{\prime }{v}_{j+\frac{3}{2}}{v}_{j+\frac{1}{2}}\hfill \\ & & -(\frac{1}{2}-\alpha )f_{j-\frac{1}{2}}^{\prime }{v}_{j-\frac{1}{2}}{v}_{j+\frac{1}{2}}]D_{j+\frac{1}{2}}+T_{(j\le k-2)}+c\epsilon \hfill \\ & \le & \sum _{j=I}^{k-2}{d}_j{D}_{j+\frac{1}{2}}+T_{(j\le k-2)}+c\epsilon ,\hfill \end{array}$$

(56)

where c is some constant. Similarly, notice that by writing

$$\begin{array}{ccc}& & -2\alpha \sum _{j=k+2}^{J-1}{f}_{j+\frac{1}{2}}^{\prime }{v}_{j+\frac{1}{2}}^2\hfill \\ & & =-\alpha \sum _{j=k+2}^{J-1}{f}_{j+\frac{1}{2}}^{\prime }{v}_{j+\frac{1}{2}}^2-\alpha \sum _{j=k+2}^{J-1}{f}_{j-\frac{1}{2}}^{\prime }{v}_{j-\frac{1}{2}}^2+\alpha f_{k+\frac{3}{2}}^{\prime }{v}_{k+\frac{3}{2}}^2,\hfill \end{array}$$

we obtain

$$\begin{array}{ccc}\hfill & & P_{(j\ge k+2)}:=\sum _{j=k+2}^{J-1}\{g_{j+\frac{1}{2}}[v_{j+\frac{1}{2}}+{\tilde{v}}_{j+\frac{1}{2}}]-(f_j+f_{j+1})v_{j+\frac{1}{2}}\}\hfill \\ \hfill \\ & =& \sum _{j=k+2}^{J-1}\{[g_{j+\frac{1}{2}}^E+(\frac{1}{2}-\alpha ){(f_{j+\frac{1}{2}}^{+})}^{(0)}+\alpha {(f_{j-\frac{1}{2}}^{+})}^{(-1)}][2v_{j+\frac{1}{2}}\hfill \\ & & -C_{j+\frac{1}{2}}{v}_{j+\frac{1}{2}}+C_{j-\frac{1}{2}}{v}_{j-\frac{1}{2}}+{\epsilon }_l\tau q^{\prime }{v}_{j+\frac{1}{2}}]-(f_j+f_{j+1})v_{j+\frac{1}{2}}\}\hfill \\ & \le & \sum _{j=k+2}^{J-1}\{[f_j+(\frac{1}{2}-\alpha )f_{j+\frac{1}{2}}^{+}+\alpha f_{j-\frac{1}{2}}^{+}][2v_{j+\frac{1}{2}}\hfill \\ & & -C_{j+\frac{1}{2}}{v}_{j+\frac{1}{2}}+C_{j-\frac{1}{2}}{v}_{j-\frac{1}{2}}+{\epsilon }_l\tau q^{\prime }{v}_{j+\frac{1}{2}}]-(f_j+f_{j+1})v_{j+\frac{1}{2}}\}\hfill \\ & \le & \sum _{j=k+2}^{J-1}[(\frac{1}{2}+2\alpha )f_{j+\frac{1}{2}}^{\prime }{v}_{j+\frac{1}{2}}^2+(\frac{1}{2}-\alpha )f_{j+\frac{3}{2}}^{\prime }{v}_{j+\frac{1}{2}}{v}_{j+\frac{3}{2}}]C_{j+\frac{1}{2}}+T_{(j\ge k+2)}+c\epsilon \hfill \\ & =& \sum _{j=k+2}^{J-1}{c}_j{C}_{j+\frac{1}{2}}+T_{(j\ge k+2)}+c\epsilon ,\hfill \end{array}$$

(57)

where

$$c_j:=(\frac{1}{2}+2\alpha )f_{j+\frac{1}{2}}^{\prime }{v}_{j+\frac{1}{2}}^2+(\frac{1}{2}-\alpha )f_{j+\frac{3}{2}}^{\prime }{v}_{j+\frac{1}{2}}{v}_{j+\frac{3}{2}},$$

and

$$\begin{array}{ccc}\hfill T_{(j\ge k+2)}:& =& \alpha f_{k+\frac{3}{2}}^{\prime }{v}_{k+\frac{3}{2}}^2+[f_{k+2}{v}_{k+\frac{3}{2}}+(\frac{1}{2}-\alpha )f_{k+\frac{5}{2}}^{\prime }{v}_{k+\frac{5}{2}}{v}_{k+\frac{3}{2}}\hfill \\ & & +\alpha f_{k+\frac{3}{2}}^{\prime }{v}_{k+\frac{3}{2}}^2]C_{k+\frac{3}{2}}.\hfill \end{array}$$

Now, we compute the $(k-1)$th, kth, and $(k+1)$th terms of the RHS of (55) defined by $P_{k-1}$, $P_k$, and $P_{k+1}$, respectively, as follows: Again, we let c be some generic constant.

$$\begin{array}{ccc}\hfill P_{k-1}& :=& [g_{k-\frac{1}{2}}^E+\alpha {(f_{k+\frac{1}{2}}^{-})}^{(1)}-(\frac{1}{2}-\alpha ){(f_{k-\frac{1}{2}}^{-})}^{(0)}][2v_{k-\frac{1}{2}}-D_{k-\frac{1}{2}}{v}_{k-\frac{1}{2}}\hfill \\ & & +D_{k+\frac{1}{2}}{v}_{k+\frac{1}{2}}+{\epsilon }_l\tau q^{\prime }{v}_{k-\frac{1}{2}}]-(f_{k-1}+f_k)v_{k-\frac{1}{2}}\hfill \\ & \le & [f_k-\alpha f_{k+\frac{1}{2}}^{-}-(\frac{1}{2}-\alpha )f_{k-\frac{1}{2}}^{-}][2v_{k-\frac{1}{2}}-D_{k-\frac{1}{2}}{v}_{k-\frac{1}{2}}\hfill \\ & & +D_{k+\frac{1}{2}}{v}_{k+\frac{1}{2}}+{\epsilon }_l\tau q^{\prime }{v}_{k-\frac{1}{2}}]-(f_{k-1}+f_k)v_{k-\frac{1}{2}}\hfill \\ & \le & -2\alpha f_{s-\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}{v}_{k-\frac{1}{2}}+2\alpha f_{k-\frac{1}{2}}^{\prime }{v}_{k-\frac{1}{2}}^2+c{\epsilon }_l\tau q^{\prime }{v}_{k-\frac{1}{2}}+[-f_k{v}_{k-\frac{1}{2}}+\alpha f_{s-\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}{v}_{k-\frac{1}{2}}\hfill \\ & & +(\frac{1}{2}-\alpha )f_{k-\frac{1}{2}}^{\prime }{v}_{k-\frac{1}{2}}^2]D_{k-\frac{1}{2}}+[f_k{v}_{k+\frac{1}{2}}-\alpha f_{s-\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}{v}_{k+\frac{1}{2}}\hfill \\ & & -(\frac{1}{2}-\alpha )f_{k-\frac{1}{2}}^{\prime }{v}_{k-\frac{1}{2}}{v}_{k+\frac{1}{2}}]D_{k+\frac{1}{2}},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill P_k& :=& g_{k+\frac{1}{2}}^E[2v_{k+\frac{1}{2}}-D_{k+\frac{1}{2}}{v}_{k+\frac{1}{2}}-C_{k+\frac{1}{2}}{v}_{k+\frac{1}{2}}+{\epsilon }_l\tau q^{\prime }{v}_{k+\frac{1}{2}}]-(f_k+f_{k+1})v_{k+\frac{1}{2}}\hfill \\ & =& f_s[2v_{k+\frac{1}{2}}-D_{k+\frac{1}{2}}{v}_{k+\frac{1}{2}}-C_{k+\frac{1}{2}}{v}_{k+\frac{1}{2}}+{\epsilon }_l\tau q^{\prime }{v}_{k+\frac{1}{2}}]-(f_k+f_{k+1})v_{k+\frac{1}{2}}\hfill \\ & =& f_{s-\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}{v}_{k+\frac{1}{2}}-f_{s+\frac{1}{2}}^{\prime }{v}_{s+\frac{1}{2}}{v}_{k+\frac{1}{2}}-f_s{v}_{k+\frac{1}{2}}{D}_{k+\frac{1}{2}}-f_s{v}_{k+\frac{1}{2}}{C}_{k+\frac{1}{2}}+f_s{\epsilon }_l\tau q^{\prime }{v}_{k+\frac{1}{2}},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill P_{k+1}& :=& [g_{k+\frac{3}{2}}^E+(\frac{1}{2}-\alpha ){(f_{k+\frac{3}{2}}^{+})}^{(0)}+\alpha {(f_{k+\frac{1}{2}}^{+})}^{(-1)}][2v_{k+\frac{3}{2}}-C_{k+\frac{3}{2}}{v}_{k+\frac{3}{2}}\hfill \\ & & +C_{k+\frac{1}{2}}{v}_{k+\frac{1}{2}}+{\epsilon }_l\tau q^{\prime }{v}_{k+\frac{3}{2}}]-(f_{k+1}+f_{k+2})v_{k+\frac{3}{2}}\hfill \\ & \le & [f_{k+1}+(\frac{1}{2}-\alpha )f_{k+\frac{3}{2}}^{+}+\alpha f_{k+\frac{1}{2}}^{+}][2v_{k+\frac{3}{2}}-C_{k+\frac{3}{2}}{v}_{k+\frac{3}{2}}\hfill \\ & & +C_{k+\frac{1}{2}}{v}_{k+\frac{1}{2}}+{\epsilon }_l\tau q^{\prime }{v}_{k+\frac{3}{2}}]-(f_{k+1}+f_{k+2})v_{k+\frac{3}{2}}\hfill \\ & \le & -2\alpha f_{k+\frac{3}{2}}^{\prime }{v}_{k+\frac{3}{2}}^2+2\alpha f_{s+\frac{1}{2}}^{\prime }{v}_{s+\frac{1}{2}}{v}_{k+\frac{3}{2}}+c{\epsilon }_l\tau q^{\prime }{v}_{k+\frac{3}{2}}+[-f_{k+1}{v}_{k+\frac{3}{2}}\hfill \\ & & -(\frac{1}{2}-\alpha )f_{k+\frac{3}{2}}^{\prime }{v}_{k+\frac{3}{2}}^2-\alpha f_{s+\frac{1}{2}}^{\prime }{v}_{s+\frac{1}{2}}{v}_{k+\frac{3}{2}}]C_{k+\frac{3}{2}}+[f_{k+1}{v}_{k+\frac{1}{2}}\hfill \\ & & +(\frac{1}{2}-\alpha )f_{k+\frac{3}{2}}^{\prime }{v}_{k+\frac{1}{2}}{v}_{k+\frac{3}{2}}+\alpha f_{s+\frac{1}{2}}^{\prime }{v}_{s+\frac{1}{2}}{v}_{k+\frac{1}{2}}]C_{k+\frac{1}{2}}.\hfill \end{array}$$

Next, we combine the estimates of $T_{(j\le k-2)}$, $T_{(j\ge k+2)},P_{k-1},P_k$, and $P_{k+1}$ into one estimate. But first, we let

$$d_{k-1}=-(\frac{1}{2}+2\alpha )f_{k-\frac{1}{2}}^{\prime }{v}_{k-\frac{1}{2}}^2-(\frac{1}{2}-\alpha )f_{k-\frac{3}{2}}^{\prime }{v}_{k-\frac{3}{2}}{v}_{k-\frac{1}{2}}+\alpha f_{s-\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}{v}_{k-\frac{1}{2}},$$

$$d_k=-(\frac{1}{2}-\alpha )f_{k-\frac{1}{2}}^{\prime }{v}_{k-\frac{1}{2}}{v}_{k+\frac{1}{2}}-(1+\alpha )f_{s-\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}{v}_{k+\frac{1}{2}},$$

$$c_k=(1+\alpha )f_{s+\frac{1}{2}}^{\prime }{v}_{s+\frac{1}{2}}{v}_{k+\frac{1}{2}}+(\frac{1}{2}-\alpha )f_{k+\frac{3}{2}}^{\prime }{v}_{k+\frac{1}{2}}{v}_{k+\frac{3}{2}},$$

and

$$c_{k+1}=(\frac{1}{2}+2\alpha )f_{k+\frac{3}{2}}^{\prime }{v}_{k+\frac{3}{2}}^2+(\frac{1}{2}-\alpha )f_{k+\frac{5}{2}}^{\prime }{v}_{k+\frac{3}{2}}{v}_{k+\frac{5}{2}}-\alpha f_{s+\frac{1}{2}}^{\prime }{v}_{s+\frac{1}{2}}{v}_{k+\frac{3}{2}}.$$

Using the fact that $v_{k+\frac{1}{2}}=v_{s+\frac{1}{2}}+v_{s-\frac{1}{2}}$, $f_{s-\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}{v}_{k+\frac{1}{2}}-f_{s-\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}^2=f_{s-\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}{v}_{s+\frac{1}{2}}$, and $-f_{s+\frac{1}{2}}^{\prime }{v}_{s+\frac{1}{2}}{v}_{k+\frac{1}{2}}+f_{s+\frac{1}{2}}^{\prime }{v}_{s+\frac{1}{2}}^2=-f_{s+\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}{v}_{s+\frac{1}{2}}$, we have

$$\begin{array}{ccc}\hfill & & T(k):=T_{(j\le k-2)}+T_{(j\ge k+2)}+P_{k-1}+P_k+P_{k+1}\hfill \\ & \le & -\alpha f_{k-\frac{1}{2}}^{\prime }{v}_{k-\frac{1}{2}}^2+[f_{k-1}-\alpha f_{k-\frac{1}{2}}^{\prime }{v}_{k-\frac{1}{2}}-(\frac{1}{2}-\alpha )f_{k-\frac{3}{2}}^{\prime }{v}_{k-\frac{3}{2}}]D_{k-\frac{1}{2}}{v}_{k-\frac{1}{2}}\hfill \\ & & +\alpha f_{k+\frac{3}{2}}^{\prime }{v}_{k+\frac{3}{2}}^2+[f_{k+2}{v}_{k+\frac{3}{2}}+(\frac{1}{2}-\alpha )f_{k+\frac{5}{2}}^{\prime }{v}_{k+\frac{3}{2}}{v}_{k+\frac{5}{2}}+\alpha f_{k+\frac{3}{2}}^{\prime }{v}_{k+\frac{3}{2}}^2]C_{k+\frac{3}{2}}\hfill \\ & & -2\alpha f_{s-\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}{v}_{k-\frac{1}{2}}+2\alpha f_{k-\frac{1}{2}}^{\prime }{v}_{k-\frac{1}{2}}^2+[-f_k{v}_{k-\frac{1}{2}}+\alpha f_{s-\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}{v}_{k-\frac{1}{2}}\hfill \\ & & +(\frac{1}{2}-\alpha )f_{k-\frac{1}{2}}^{\prime }{v}_{k-\frac{1}{2}}^2]D_{k-\frac{1}{2}}+[f_k{v}_{k+\frac{1}{2}}-\alpha f_{s-\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}{v}_{k+\frac{1}{2}}\hfill \\ & & -(\frac{1}{2}-\alpha )f_{k-\frac{1}{2}}^{\prime }{v}_{k-\frac{1}{2}}{v}_{k+\frac{1}{2}}]D_{k+\frac{1}{2}}\hfill \\ & & +f_{s-\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}{v}_{k+\frac{1}{2}}-f_{s+\frac{1}{2}}^{\prime }{v}_{s+\frac{1}{2}}{v}_{k+\frac{1}{2}}-f_s{v}_{k+\frac{1}{2}}{D}_{k+\frac{1}{2}}-f_s{v}_{k+\frac{1}{2}}{C}_{k+\frac{1}{2}}\hfill \\ & & -2\alpha f_{k+\frac{3}{2}}^{\prime }{v}_{k+\frac{3}{2}}^2+2\alpha f_{s+\frac{1}{2}}^{\prime }{v}_{s+\frac{1}{2}}{v}_{k+\frac{3}{2}}\hfill \\ & & +[-f_{k+1}{v}_{k+\frac{3}{2}}-(\frac{1}{2}-\alpha )f_{k+\frac{3}{2}}^{\prime }{v}_{k+\frac{3}{2}}^2-\alpha f_{s+\frac{1}{2}}^{\prime }{v}_{s+\frac{1}{2}}{v}_{k+\frac{3}{2}}]C_{k+\frac{3}{2}}\hfill \\ & & +[f_{k+1}{v}_{k+\frac{1}{2}}+(\frac{1}{2}-\alpha )f_{k+\frac{3}{2}}^{\prime }{v}_{k+\frac{1}{2}}{v}_{k+\frac{3}{2}}+\alpha f_{s+\frac{1}{2}}^{\prime }{v}_{s+\frac{1}{2}}{v}_{k+\frac{1}{2}}]C_{k+\frac{1}{2}}+c\epsilon \hfill \\ & =& T_1(k)+T_2(k)\hfill \\ & \le & T_2(k),\hfill \end{array}$$

(58)

where

$$\begin{array}{ccc}\hfill T_1(k)& :=& \alpha f_{k-\frac{1}{2}}^{\prime }{v}_{k-\frac{1}{2}}^2-2\alpha f_{s-\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}{v}_{k-\frac{1}{2}}+f_{s-\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}{v}_{k+\frac{1}{2}}\hfill \\ & & -f_{s+\frac{1}{2}}^{\prime }{v}_{s+\frac{1}{2}}{v}_{k+\frac{1}{2}}-\alpha f_{k+\frac{3}{2}}^{\prime }{v}_{k+\frac{3}{2}}^2+2\alpha f_{s+\frac{1}{2}}^{\prime }{v}_{s+\frac{1}{2}}{v}_{k+\frac{3}{2}}\hfill \\ & =& \alpha f_{k-\frac{1}{2}}^{\prime }{v}_{k-\frac{1}{2}}^2+\alpha f_{s-\frac{1}{2}}^{\prime }{(v_{s-\frac{1}{2}}-v_{k-\frac{1}{2}})}^2-\alpha f_{s-\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}^2-\alpha f_{s-\frac{1}{2}}^{\prime }{v}_{k-\frac{1}{2}}^2\hfill \\ & & +f_{s-\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}{v}_{k+\frac{1}{2}}-f_{s+\frac{1}{2}}^{\prime }{v}_{s+\frac{1}{2}}{v}_{k+\frac{1}{2}}-\alpha f_{k+\frac{3}{2}}^{\prime }{v}_{k+\frac{3}{2}}^2+2\alpha f_{s+\frac{1}{2}}^{\prime }{v}_{s+\frac{1}{2}}{v}_{k+\frac{3}{2}}\hfill \\ & \le & \alpha (f_{k-\frac{1}{2}}^{\prime }-f_{s-\frac{1}{2}}^{\prime })v_{k-\frac{1}{2}}^2+(1-\alpha )f_{s-\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}^2+f_{s-\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}{v}_{s+\frac{1}{2}}\hfill \\ & & -\alpha f_{s+\frac{1}{2}}^{\prime }{(v_{s+\frac{1}{2}}-v_{k+\frac{3}{2}})}^2-(1-\alpha )f_{s+\frac{1}{2}}^{\prime }{v}_{s+\frac{1}{2}}^2-f_{s+\frac{1}{2}}^{\prime }{v}_{s-\frac{1}{2}}{v}_{s+\frac{1}{2}}\hfill \\ & \le & 0,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill T_2(k)& :=& d_{k-1}{D}_{k-\frac{1}{2}}+d_k{D}_{k+\frac{1}{2}}+c_k{C}_{k+\frac{1}{2}}+c_{k+1}{C}_{k+\frac{3}{2}}+c\epsilon .\hfill \end{array}$$

Finally, using (55)–(58), we have

$$\begin{array}{ccc}\hfill \mathrm{LHS} \mathrm{of} (54)& \le & \sum _{j=I}^{J-1}{g}_{j+\frac{1}{2}}{\overline{v}}_{j+\frac{1}{2}}-\sum _{j=I}^{J-1}{\int }_{v_j}^{v_{j+1}}f[w;v_j,v_{j+1}]dw+\epsilon \hfill \\ & =& \frac{1}{2}(P_{(j\le k-2)}+P_{k-1}+P_k+P_{k+1}+P_{(j\ge k+2)})+\epsilon \hfill \\ & \le & \frac{1}{2}\{\sum _{j=I}^{k-2}{d}_j{D}_{j+\frac{1}{2}}+\sum _{j=k+2}^{J-1}{c}_j{C}_{j+\frac{1}{2}}+T_2(k)\}+c\epsilon \hfill \\ & =& \frac{1}{2}\{\sum _{j=I}^k{d}_j{D}_{j+\frac{1}{2}}+\sum _{j=k}^{J-1}{c}_j{C}_{j+\frac{1}{2}}\}+c\epsilon .\hfill \end{array}$$

Clearly, for sufficiently small ${\sigma }_{\Gamma }$ and $\epsilon$, it is feasible that

$$\mathrm{LHS} \mathrm{of} (54)\le \frac{1}{2}\sum _{j=I+1}^{J-1}{S}_{j-1,j,j+1}\le \sum _{j=I+1}^{J-1}{\alpha }_j{S}_{j-1,j,j+1}.$$

Thus, we have completed the proof of Lemma 4. □

4. Concluding Remarks

In this paper, we establish the entropy convergence of $\alpha$ schemes with source terms for scalar convex conservation laws when $0<\alpha \le \frac{1}{2}$ and $m=2$. These (five-point grid bandwidth) schemes, except at isolated critical points, are second-order-accurate in space. Nevertheless, they are only first-order-accurate in time. Further research could consider the case of $\alpha$ schemes for $m\ge 3$ or their extensions with higher-order accuracy in time and could consider other methods with higher-order accuracy in space and time. The author is currently working on the convergence analysis of a non-homogeneous and nonlinear SSP Runge–Kutta method, which indeed has second-order accuracy both in time and (except at isolated critical points) space, and we hope to report this new result soon. The homogeneous counterpart of this method is one of the original SSP Runge–Kutta methods constructed by Shu [17], and Shu and Osher [18], of which the entropy convergence has been shown by the author [19].