Multiscale Polynomial-Based High-Order Central High Resolution Schemes

Abstract

Polynomial-based high order central high resolution schemes with semi-discrete forms are integrated with multiresolution-based adapted cells/grids. To preserve the positivity condition on non-uniform cells/grids, the corresponding formulations are studied, redesigned or developed. Two general approaches can be used for polynomial-based reconstructions: (a) direct interpolation by a polynomial, (b) proper combination of different polynomials to construct a new polynomial with desired features. Based on these approaches, three polynomial-based reconstructions are considered: (i) parabolic polynomials interpolating average solutions of three successive cells; (ii) piece-wise parabolic methods (PPMs) obtained with two different local stencils; (iii) central-WENO schemes [based on the results of approach (i)]. For the first approach, the corresponding features, stability conditions, formulations and nonlinear limiters are studied and updated. For the second approach, for more localized stencils, new independent variables (e.g., first and second spatial derivatives) are introduced by adding new conservation laws. Two PPM-based central schemes are formulated and a new limiter and a new updating procedure are introduced. For the third approach, the average-interpolating parabolic polynomial [in approach (i)] is used in the framework of the central-WENO formulation. Third and fourth order formulations are provided on non-uniform grids/cells. Finally some numerical examples are presented to verify the results and to assess effectiveness and robustness of the three approaches.

Appendices

Appendix A

Let us assume a scalar conservation law $u_t + F(u)_x = 0$. In the following, fully-discrete and semi-discrete forms of third-order central high resolution schemes will be provided on non-uniform grids with non-centered cell-centers. At first, the fully-discrete form is derived and accordingly the semi-discrete form will be evaluated in the limit state $\varDelta t \rightarrow 0$ [106].

1.1 A.1 The Fully-Discrete Form

For deriving the fully-discrete form, the three stages of reconstruction-evolution-projection will be followed [114].

1.1.1 A.1.1 The Reconstruction Stage

A piece-wise polynomial is assumed to be in cell $I_j$ and time $t^n:= n \varDelta t$ as: $P_j (x,t^n)=A_j+B_j (x-\bar{x}_j) + \frac{1}{2} C_j (x-\bar{x}_j)^2$, where $\bar{x}_j = \left( x_{j+1/2} + x_{j-1/2} \right) /2$.

1.1.2 A.1.2 The Evolution Stage

It is assumed that the cell center $x_j$ is not located in the middle of the cell $I_j$. The location of $x_j$ can be determined from cell edges $x_{j \pm 1/2}$ as:  $x_{j+1/2} := x_j + p_j \varDelta x_j$ and $x_{j-1/2} := x_j - \left( 1-p_j \right) \varDelta x_j$. Spatial locations $x_{j \pm 1/2,l}^n$ and $x_{j \pm 1/2,r}^n$ are also defined as: $x_{j \pm 1/2,l}^n := x_{j \pm 1/2}^n - a_{j \pm 1/2}^n \varDelta t$ and $x_{j \pm 1/2,r}^n := x_{j \pm 1/2}^n + a_{j \pm 1/2}^n \varDelta t$. The parameter $a_{j+1/2}^n$ shows the upper bound of propagating speed of a possible discontinuity at the cell edge $x_{j+1/2}$ [114].

The evaluation stage (from $t^n$ to $t^{n+1}$) can be performed by the integration of $u_t + F(u)_x = 0$, over spatio-temporal domains  $\left[ x_{j+1/2,l}^n,x_{j+1/2,r}^n\right] \times \left[ t^n,t^{n+1}\right]$,  $\left[ x_{j-1/2,r}^n,x_{j+1/2,l}^n\right] \times \left[ t^n,t^{n+1}\right]$ and  $\left[ x_{j-1/2,l}^n,x_{j-1/2,r}^n\right] \times \left[ t^n,t^{n+1}\right]$. The first and the third domains are around cell edges and contain non-smooth solutions (with possible discontinuities) and the second one involves a smooth response. By integrating over above-mentioned three spatio-temporal volumes, evolved solutions can be expresses as follows.

• 1. Integral over the spatio-temporal intervals $\left[ x_{j+1/2,l}^n,x_{j+1/2,r}^n\right] \times \left[ t^n,t^{n+1}\right]$

  $$\begin{aligned} \begin{aligned} \bar{\omega }_{j+\frac{1}{2}}^{n+1} =\,&\frac{1}{x_{j+1/2,r}^n - x_{j+1/2,l}^n} \\&\left[ \int _{x_{j+1/2,l}^n}^{x_{j+1/2}^n} P_j^n (x) dx + \int _{x_{j+1/2}^n}^{x_{j+1/2,r}^n} P_{j+1}^n (x) dx - \int _{t^n}^{t^{n+1}} \left( F\left( u_{j+1/2,r}^n \right) - F\left( u_{j+1/2,l}^n \right) \right) dt \right] \\ = \,&\frac{1}{2} \left( A_j+A_{j+1}\right) \\&+ \frac{1}{4} \left[ \varDelta \text {t} a_{j+\frac{1}{2}}^n \left( B_{j+1}-B_j\right) +2 \left( B_j \varDelta \text {x}_j p_j+B_{j+1} \left( \varDelta \text {x}_j p_j+x_j-x_{j+1}\right) \right) \right] \\&+ \frac{1}{12} \left[ \varDelta \text {t}^2 \left( a_{j+\frac{1}{2}}^n \right) ^2 \left( C_j+C_{j+1}\right) + 3 \varDelta \text {t} a_{j+\frac{1}{2}}^n \left( C_{j+1} \left( \varDelta \text {x}_j p_j+x_j-x_{j+1}\right) -C_j \varDelta \text {x}_j p_j\right) \right. \\&+\, \left. 3 \left( C_j \varDelta \text {x}_j^2 p_j^2+C_{j+1} \left( \varDelta \text {x}_j p_j+x_j-x_{j+1}\right) {}^2\right) \right] \\&- \frac{1}{2 a^n_{j+1/2} \varDelta t} \left[ \int _{t^n}^{t^{n+1}} \left\{ F\left( u \left( x_{j+1/2,r}^n \right) \right) - F\left( u \left( x_{j+1/2,l}^n \right) \right) \right\} dt \right] . \end{aligned} \end{aligned}$$

  (A.1)

• 2. Integral over the spatio-temporal intervals $\left[ x_{j-1/2,r}^n,x_{j+1/2,l}^n\right] \times \left[ t^n,t^{n+1}\right]$

  $$\begin{aligned} \bar{\omega }_j^{n+1}&= \frac{1}{x_{j+1/2,l}^n - x_{j-1/2,r}^n} \left[ \int _{x_{j-1/2,r}^n}^{x_{j+1/2,l}^n} P_j^n (x) dx - \int _{t^n}^{t^{n+1}} \left( F\left( u_{j+1/2,l}^n \right) - F\left( u_{j-1/2,r}^n \right) \right) dt \right] \nonumber \\&= A_j + \frac{1}{2} B_j \left[ \varDelta \text {t} \left( a_{j-\frac{1}{2}}^n-a_{j+\frac{1}{2}}^n \right) +\varDelta \text {x}_j \left( 2 p_j-1\right) \right] \nonumber \\&\quad +\,\frac{1}{6} C_j \left[ \varDelta \text {t}^2 \left( \left( a_{j-\frac{1}{2}}^n \right) ^2+ \left( a_{j+\frac{1}{2}}^n \right) ^2 \right) -\varDelta \text {t} a_{j-\frac{1}{2}}^n \left( \varDelta \text {t} a_{j+\frac{1}{2}}^n +\varDelta \text {x}_j \left( 2-3 p_j\right) \right) \right. \nonumber \\&\quad \left. +\, a_{j+\frac{1}{2}}^n \varDelta \text {t} \varDelta \text {x}_j \left( 1-3 p_j\right) +\varDelta \text {x}_j^2 \left( 3 p_j^2-3 p_j+1\right) \right] \nonumber \\&\quad - \frac{1}{\varDelta x_j - \left( a^n_{j-1/2} + a^n_{j+1/2} \right) \varDelta t} \left[ \int _{t^n}^{t^{n+1}} \left\{ F\left( u \left( x_{j+1/2,l}^n \right) \right) - F\left( u \left( x_{j-1/2,r}^n \right) \right) \right\} dt \right] . \end{aligned}$$

  (A.2)

• 3. Integral over the spatio-temporal intervals $\left[ x_{j-1/2,l}^n,x_{j-1/2,r}^n\right] \times \left[ t^n,t^{n+1}\right]$ This is can be evaluated similar to the case with the volume $\left[ x_{j+1/2,l}^n,x_{j+1/2,r}^n\right] \times \left[ t^n,t^{n+1}\right]$.

1.1.3 A.1.3 The Projection Stage

After evaluating the average evolved solutions $\bar{\omega }_{j+\frac{1}{2}}^{n+1}$, $\bar{\omega }_{j}^{n+1}$ and $\bar{\omega }_{j-\frac{1}{2}}^{n+1}$ (from Eqs. (A.1) and (A.2)), a third order average interpolating piece-wise function with the non-oscillatory feature can be reconstructed. Such piece-wise functions $\tilde{\omega }_{j}^{n+1}(x)$ and $\tilde{\omega }_{j+\frac{1}{2}}^{n+1}(x)$ are defined as:

$$\begin{aligned} \begin{aligned} \tilde{\omega }_{j+1/2}^{n+1}(x)=&\tilde{A}_{j+1/2}+ \tilde{B}_{j+1/2} (x-x_j) + \frac{1}{2} \tilde{C}_{j+1/2} (x-x_j)^2, \quad x \in (x_{j+1/2,l}^n,x_{j+1/2,r}^n), \\ \tilde{\omega }_{j}^{n+1}(x)=&\bar{\omega }_{j}^{n+1},\quad x \in (x_{j-1/2,r}^n,x_{j+1/2,l}^n). \end{aligned} \end{aligned}$$

(A.3)

The projected solution $\bar{u}_j^{n+1}$ at time $t^{n+1}$ can be obtained by a projection step,

$$\begin{aligned} \begin{aligned} \bar{u}_j^{n+1}&= \frac{1}{\varDelta x_j} \left[ \int _{x_{j-1/2}}^{x_{j-1/2,r}} \tilde{\omega }_{j-1/2}^{n+1}(x) dx + \int _{x_{j-1/2,r}}^{x_{j+1/2,l}} \tilde{\omega }_{j}^{n+1}(x) dx + \int _{x_{j+1/2,l}}^{x_{j+1/2}} \tilde{\omega }_{j+1/2}^{n+1}(x) dx \right] \\&= \bar{\omega }_j^{n+1} \left[ 1-\bar{\lambda } _j \left( a_{j-\frac{1}{2}}^n+a_{j+\frac{1}{2}}^n\right) \right] + \bar{\lambda } _j \left[ \tilde{A}_{j-\frac{1}{2}} a_{j-\frac{1}{2}}^n+\tilde{A}_{j+\frac{1}{2}} a_{j+\frac{1}{2}}^n\right] \\&\quad +\frac{1}{2} \varDelta \text {t} \bar{\lambda }_j \left[ \tilde{B}_{j-\frac{1}{2}} \left( a_{j-\frac{1}{2}}^{n}\right) ^2 -\tilde{B}_{j+\frac{1}{2}} \left( a_{j+\frac{1}{2}}^{n}\right) ^2 \right] \\&\quad +\frac{1}{6} \varDelta \text {t}^2 \bar{\lambda }_j \left[ \tilde{C}_{j-\frac{1}{2}} \left( a_{j-\frac{1}{2}}^{n}\right) ^3 + \tilde{C}_{j+\frac{1}{2}} \left( a_{j+\frac{1}{2}}^{n}\right) ^3 \right] , \end{aligned} \end{aligned}$$

(A.4)

where $\bar{\lambda }_j := \frac{\varDelta t}{\varDelta x_j}$.

1.2 A.2 The Semi-discrete Form

Based on the fully-discrete form (Eq. (A.4)), the semi-discrete form can be obtained as:

$$\begin{aligned} \begin{aligned} \frac{d}{dt} \bar{u}_j(t) = \underset{\varDelta t \rightarrow 0}{\text {lim}} \frac{\bar{u}_j^{n+1} - \bar{u}_j^{n}}{ \varDelta t}. \end{aligned} \end{aligned}$$

(A.5)

This yields to:

$$\begin{aligned} \frac{d}{dt} \bar{u}_j(t)&= \underset{\varDelta t \rightarrow 0}{\text {lim}} \left\{ \frac{1}{\varDelta \text {t}} \left( \bar{\omega }_j^{n+1} - \bar{u}_j^n\right) - \frac{ 1 }{ \varDelta \text {x}_j} \left( a_{j-\frac{1}{2}}^n + a_{j+\frac{1}{2}}^n \right) \bar{\omega }_j^{n+1}+\frac{1}{\varDelta \text {x}_j} \left( \tilde{A}_{j-\frac{1}{2}} a_{j-\frac{1}{2}}^n+\tilde{A}_{j+\frac{1}{2}} a_{j+\frac{1}{2}}^n \right) \right. \nonumber \\&\quad \left. +~\frac{1}{2} \bar{\lambda }_j \left[ \tilde{B}_{j-\frac{1}{2}} \left( a_{j-\frac{1}{n}}^{2} \right) ^2 - \tilde{B}_{j+\frac{1}{2}} \left( a_{j+\frac{1}{n}}^{2} \right) ^2 \right] + \frac{1}{6} \varDelta \text {t} \bar{\lambda }_j \left[ \tilde{C}_{j-\frac{1}{2}} \left( a_{j-\frac{1}{2}}^{n} \right) ^3\right. \right. \nonumber \\&\quad \left. \left. +~\tilde{C}_{j+\frac{1}{2}} \left( a_{j+\frac{1}{2}}^{n} \right) ^3 \right] \right\} . \end{aligned}$$

(A.6)

Since $\varDelta t \rightarrow 0$, the widths of all the Riemann fans approach zero, then:

$$\begin{aligned} \begin{aligned} \tilde{A}_{j-\frac{1}{2}} \rightarrow \bar{\omega }_{j-1/2}^{n+1}, \quad \tilde{A}_{j+\frac{1}{2}} \rightarrow \bar{\omega }_{j+1/2}^{n+1}, \end{aligned} \end{aligned}$$

(A.7)

and from the parabolic polynomial $P_j (x,t^n)$ in the reconstruction stage (Sect. A.1.1):

$$\begin{aligned} \begin{aligned} u(x_{j+1/2,r}^n,t)&\rightarrow P_{j+1}(x_{j+1/2},t) \\&= A_{j+1} - B_{j+1} \left( (1-p_{j+1}) \varDelta x_{j+1} \right) + \frac{1}{2} C_{j+1} ((1-p_{j+1}) \varDelta x_{j+1})^2 =: u_{j+1/2}^R, \\ u(x_{j+1/2,l}^n,t)&\rightarrow P_{j}(x_{j+1/2},t) \\&= A_{j} + B_{j} \left( p_{j} \varDelta x_{j} \right) + \frac{1}{2} C_{j+1} (p_{j} \varDelta x_{j})^2 =: u_{j+1/2}^L. \end{aligned} \end{aligned}$$

(A.8)

Substituting Eqs. (A.1), (A.2), (A.7) and (A.8) into Eq. (A.6), this equation leads to:

$$\begin{aligned} \begin{aligned} \frac{d}{dt} \bar{u}_j(t)&= - \frac{1}{2 \varDelta x_j} \left[ F\left( u_{j+1/2}^R(t) \right) + F\left( u_{j+1/2}^L(t) \right) - F\left( u_{j-1/2}^R(t) \right) - F\left( u_{j-1/2}^L(t) \right) \right] \\&\quad + \frac{a_{j+1/2}^n}{2 \varDelta x_j} \left[ u_{j+1/2}^R(t) - u_{j+1/2}^L(t) \right] - \frac{a_{j-1/2}^n}{2 \varDelta x_j} \left[ u_{j-1/2}^R(t) - u_{j-1/2}^L(t) \right] . \end{aligned} \end{aligned}$$

(A.9)

The semi-discrete form (A.9) can then be rewritten as:

$$\begin{aligned} \begin{aligned} \frac{d}{dt} \bar{u}_j(t) + \frac{F^*_{j+1/2}-F^*_{j-1/2}}{\varDelta x_j} =0, \end{aligned} \end{aligned}$$

(A.10)

where $F^*_{j \pm 1/2} := F^* \left( u_{j \pm 1/2} \right)$ and

$$\begin{aligned} \begin{aligned} F^*_{j \pm 1/2} := \frac{F\left( u_{j \pm 1/2}^R(t) \right) + F\left( u_{j \pm 1/2}^L(t) \right) }{2} - \frac{a_{j \pm 1/2}^n}{2} \left[ u_{j \pm 1/2}^R(t) - u_{j \pm 1/2}^L(t) \right] . \end{aligned} \end{aligned}$$

(A.11)

Appendix B

In this Appendix, properties 1 through 4 are derived which are mentioned in Sect. 4.1.

It is clear that features (1) and (2) are satisfied. Having the same shape means if cell averages are locally monotone, $q_j(x)$ is also monotone in those cells; e.g.: if  $\bar{u}_{j-1} \le \bar{u}_j \le \bar{u}_{j+1}$ then $q_j(x)$ is increasing and vice-versa. Also, in an extremum cell, the function $q_j(x)$ has an extremum. Aforementioned properties will be confirmed in the following. In all calculations, it is assumed that $0.5<a<2$ and $0.5<b<2$. This is because of the multiresolution-based grid-adaptation and also the post-processing stage of grid adaptations, presented in Sect. 4.

1.1 B.1 The Same Shape Feature

The first derivative of $q_j(x)$ ($q'_j(x)$) can be evaluated as:

$$\begin{aligned} \begin{aligned} q'_j(x)&= \frac{ \left[ \varDelta x_j \left( 2 a+1\right) + 6 \left( x - \bar{x}_j \right) \right] }{2 \varDelta x_j (a+b+1)} \frac{2 \varDelta u_j^+}{\left( 1+b \right) \varDelta x_j } + \frac{ \left[ \varDelta x_j \left( 2 b+1\right) - 6 \left( x - \bar{x}_j \right) \right] }{2 \varDelta x_j (a+b+1)} \frac{2 \varDelta u_j^- }{\left( 1+a \right) \varDelta x_j} . \end{aligned} \end{aligned}$$

(B.1)

This equation can be written as:

$$\begin{aligned} \begin{aligned} q'_j(x)= \left( g_1 (x)+g_2 (x) \right) / \left( \varDelta x_j^2 (a+1)(b+1)(a+b+1) \right) , \end{aligned} \end{aligned}$$

(B.2)

where

$$\begin{aligned} \begin{aligned} g_1 (x) = (a+1) \left[ \varDelta x_j \left( 2 a+1\right) + 6 \left( x - \bar{x}_j \right) \right] \varDelta u_j^+, \\ g_2(x) = (b+1) \left[ \varDelta x_j \left( 2 b+1\right) - 6 \left( x - \bar{x}_j \right) \right] \varDelta u_j^-. \end{aligned} \end{aligned}$$

(B.3)

In the following, $q_j(x)$ with monotone variations (increasing and decreasing cell averages) and the same shape feature around extremum points will be studied.

Fig. 29

Different variation patterns over three successive cells $\left\{ I_j \right\}$

Case I: Increasing cell averages For the monotone increasing case, i.e.: $\varDelta u_j^ \pm \ge 0$, two variation patterns are considerable: convex (Fig. 29a) and concave increasing cases (Fig. 29b):

• 1. The convex increasing: In this case (Fig. 29a) $a \ge 1$ and  $b \le 1$ and $\varDelta u_j^+ > \varDelta u_j^-$: due to a proper cell adaptation and convex variation of cell averages. It should be shown that for $x \in I_j$, we have  $q'_j(x) > 0$. Since both $g_1(x)$ and  $g_2(x)$ are linear, $g_1(x)+g_2(x)$ is also linear. So, if $q'_j(x)$ at points $x_{i \pm 1/2}$ are positive, then $q'_j(x)$ is also positive for $x \in I_j$:

  • (a) Controlling of $q'_j(x_{j - 1/2})$: Since $x_{j-1/2}-x_j = - \varDelta x_j/2$, $a \ge 1$ and $b \le 1$, then $g_1(x_{j-1/2}) \ge 0$ and $g_2(x_{j-1/2}) \ge 0$; therefore $q'_j(x_{j - 1/2}) \ge 0$.

  • (b) Controlling of $q'_j(x_{j + 1/2})$: Since $x_{j+1/2}-x_j = \varDelta x_j/2$, $a \ge 1$ and $b \le 1$, then $g_1(x_{j+1/2}) \ge 0$, $g_2(x_{j+1/2}) \le 0$ and $g_1(x_{j+1/2}) \ge \left| g_2(x_{j+1/2}) \right|$; therefore $q'_j(x_{j + 1/2}) \ge 0$.

  So, for the convex increasing of $\left\{ \bar{u}_i: ~ i=j-1,j,j+1 \right\}$, $q_j(x)$ will remain monotone increasing.

• 2. The concave increasing: For this case (Fig. 29b), we have: $b \ge 1$, $a \le 1$ and $\varDelta u_j^+ < \varDelta u_j^-$:

  • (a) At the point $x_{j-1/2}$, it is easy to show that: $g_1(x_{j-1/2}) \le 0$, $g_2(x_{j-1/2}) \ge 0$ and $| g_2(x_{j-1/2}) | \ge | g_1(x_{j-1/2}) |$; hence $q'_j(x_{j-1/2}) \ge 0$,

  • (b) At the point $x_{j+1/2}$: $g_1(x_{j+1/2}) \ge 0$ and $g_2(x_{j+1/2}) \ge 0$; so, $q'_j(x_{j+1/2}) \ge 0$.

  Hence, $q'_j(x) \ge 0$ for $x \in I_j$.

Case II: Decreasing cell averages In this case, for a monotone decreasing we have $\varDelta u_j^ \pm \le 0$. And again convex (Fig. 29c) and concave decreasing (Fig. 29d) cases are possible:

• 1. The convex decreasing: For this case (Fig. 29c), we have: $b \ge 1$, $a \le 1$ and $\left| \varDelta u_j^- \right| \ge \left| \varDelta u_j^+ \right|$:

  • (a) At the point $x_{j-1/2}$, it is easy to show that: $g_1(x_{j-1/2}) \ge 0$, $g_2(x_{j-1/2}) \le 0$ and  $| g_2(x_{j-1/2}) | \ge | g_1(x_{j-1/2}) |$; hence $q_j'(x_{j-1/2}) \le 0$.

  • (b) At the point $x_{j+1/2}$: $g_1(x_{j+1/2}) \le 0$ nd $g_2(x_{j+1/2}) \le 0$; so $q'_j(x_{j+1/2}) \le 0$.

  In this regard, $q'_j(x) \le 0$ for $x \in I_j$.

• 2. The concave decreasing: For this case (Fig. 29d), we have: $a \ge 1$, $b \le 1$ and $\left| \varDelta u_j^- \right| \le \left| \varDelta u_j^+ \right|$; therefore:

  • (a) At the point $x_{j-1/2}$: $g_1(x_{j-1/2}) \le 0$ and $g_2(x_{j-1/2}) \le 0$; so: $q'_j(x_{j-1/2}) \le 0$,

  • (b) At the point $x_{j+1/2}$: $g_1(x_{j+1/2}) \le 0$, $g_2(x_{j+1/2}) \ge 0$ and $| g_1(x_{j+1/2}) | \ge | g_2(x_{j+1/2}) |$; hence: $q'_j(x_{j+1/2}) \le 0$.

  So, in general: $q'_j(x) \le 0$ for $x \in I_j$.

Case III: Extrema points In these points, due to the cell adaptation, $a \rightarrow 1$ and $b \rightarrow 1$; therefor,therefore it is easy to show that:

$$\begin{aligned} \begin{aligned} q'_j(x) = \frac{\left( u'\right) _j^- \left[ \varDelta x_j-2 (x-x_j) \right] + \left( u'\right) _j^+ \left[ \varDelta x_j + 2 (x-x_j) \right] }{2 \varDelta x_j}. \end{aligned} \end{aligned}$$

(B.4)

Hence in the cell edges $x_{j \pm 1/2}$, the function $q'_j(x)$ becomes: $q'_j(x_{j-1/2}) = \left( u' \right) _j^-$ and $q' _j(x_{j+1/2}) = \left( u' \right) _j^+$. Therefore $q'_j(x_{j-1/2}).q' _j(x_{j+1/2}) = (\bar{u}_j-\bar{u}_{j-1}) (\bar{u}_{j+1}-\bar{u}_{j}) / \left[ \frac{1}{4} (1+a) (1+b) \varDelta x_j^2 \right]$:  $q_j(x)$ has the same shape of cell-averages $\left\{ \bar{u}_i \right\}$.

1.2 B.2 The Fourth Property

• 1. The maximum cell: At the two edges of $I_j$, the fourth property (in Sect. 4) is controlled:

  • (a) Controlling at $x=x_{j-1/2}$: The density of an adapted grid increases by approaching to the maximum point $x_j$ from the left side. As a result: $a \ge 1$, $b \ge 1$, $\varDelta u_j^- \ge 0$ and $\varDelta u_j^+ \le 0$; therefore:

    $$\begin{aligned} \begin{aligned} \delta _{j-1/2}^{(max)} := q(x_{j-1/2})- \frac{\bar{u}_{j}+a \bar{u}_{j-1}}{1+a} = \frac{(b+1) \left( a^2+ (b+1) (a-1)\right) \varDelta u_j^- - a (a+1) \varDelta u_j^+}{(a+1) (b+1) (a+b+1)}. \end{aligned} \end{aligned}$$

    (B.5)

    It is clear that $\delta _{j-1/2}^{(max)} > 0$.

  • (b) Controlling at $x=x_{j+1/2}$: For this case, it is obvious that: $a \ge 1$, $b \ge 1$, $\varDelta u_j^- \ge 0$ and $\varDelta u_j^+ \le 0$. So,

    $$\begin{aligned} \begin{aligned} \delta _{j+1/2}^{(max)} := q(x_{j+1/2})- \frac{\bar{u}_{j+1}+b \bar{u}_{j}}{1+b} = \frac{b \left( -(a+1) \varDelta u^+_j + (b+1) \varDelta u^-_j \right) }{(a+1) (b+1) (a+b+1)}. \end{aligned} \end{aligned}$$

    (B.6)

    It is clear that $\delta _{j+1/2}^{(max)} > 0$.

• 2. The minimum cell: Controlling at $x=x_{j-1/2}$ and $x=x_{j+1/2}$. It is straightforward to show that:

  $$\begin{aligned} \begin{aligned} \delta _{j-1/2}^{(min)} := q(x_{j-1/2})- \frac{\bar{u}_{j}+a \bar{u}_{j-1}}{1+a} = - \delta _{j-1/2}^{(max)}, \\ \delta _{j+1/2}^{(min)} := q(x_{j+1/2})- \frac{\bar{u}_{j+1}+b \bar{u}_{j}}{1+b} = - \delta _{j+1/2}^{(max)}. \end{aligned} \end{aligned}$$

  (B.7)

  It is clear that $\delta _{j-1/2}^{(min)} < 0$ and $\delta _{j+1/2}^{(min)} < 0$.