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Bound-preserving schemes for \(P^2\) local discontinuous Galerkin discretizations of KdV-type equations

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Abstract

In this paper, we construct a bound-preserving (BP) local discontinuous Galerkin (LDG) method for the generalized third-order Korteweg–de Vries (KdV) equations. The KdV equations have been widely used in mathematical models in dispersive hydrodynamics. We are interested in a new algorithm that is closer to the real physical bounds of solitary waves and dispersive shock waves (DSWs). We design a BP scheme for \(P^2\) LDG discretizations on nonuniform meshes so that an appropriate choice for the time step, the space size, and the penalty term is allowed to make the cell average of numerical solutions within the scope of global bounds. The third-order strong stability preserving (SSP) Runge–Kutta method is used to preserve the BP property of the discrete scheme. Furthermore, we extend the BP scheme to the two-dimensional Zakharov–Kuznetsov (ZK) equation. Numerical experiments demonstrate a good performance and accuracy of the BP LDG method.

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Acknowledgements

This research was supported by the Natural Science Foundation of Heilongjiang Province (LH2020A015).

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Authors and Affiliations

  1. College of Science, Harbin University of Science and Technology, Harbin, 150080, China

    Hui Bi & Feilong Zhao

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  1. Hui Bi
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  2. Feilong Zhao
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HB: Conceptualization, Methodology, Writing-Review and Editing, Supervision. FZ: Methodology, Writing-Original Draft, Software, Visualization.

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Correspondence to Hui Bi.

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Appendix A: The proof of some conclusions in Sect. 4

Appendix A: The proof of some conclusions in Sect. 4

1.1 The proof of Lemma 4.3

In this appendix, we report the proof of Lemma 4.3.

Suppose \({U_{i,j}^\gamma } \ge 0\) on each cell \({I_{i,j}}\), it is obvious that \(\bar{u}_{i,j}^n + \frac{{3\varepsilon \Delta t}}{{\Delta {y_j}}}\sum _{\gamma = 1}^3 {U_{i,j}^\gamma } \ge 0\). Therefore, we directly estimate the case for \({U_{i,j}^\gamma } < 0\). If \({U_{i,j}^\gamma } < 0\) on each cell \({I_{i,j}}\), we obtain that \(\bar{u}_{i,j}^n + \frac{{3\varepsilon \Delta t}}{{\Delta {y_j}}}\sum _{\gamma = 1}^3 {U_{i,j}^\gamma } \ge 0\) is equivalent to

$$\begin{aligned} \Delta t \le \frac{{\bar{u}_{i,j}^n}}{{3\varepsilon \left( { - \sum _{\gamma = 1}^3 {U_{i,j}^\gamma } /\Delta {y_j}} \right) }}. \end{aligned}$$

Now, we only need to estimate an upper bound of \(- \frac{1}{{\Delta {y_j}}}\sum _{\gamma = 1}^3 {U_{i,j}^\gamma } \).

$$\begin{aligned}&- \frac{{U_{ i ,j}^\gamma }}{{\Delta {y_j}}} = - \sum \limits _{l = 1}^3 {{w_l}\left[ {\frac{{{\psi ^\gamma }\left( {x_i^l} \right) }}{{\Delta y_j^2}}\int _{{J_j}} {{u_h}\left( {x_i^l, \cdot } \right) s{{\left( {{\eta ^j}\left( y \right) } \right) }_y}dy} } \right. } \nonumber \\&\quad \left. { - 3\left( {\frac{1}{{\Delta y_j^2}} + \frac{3}{{\Delta {y_j}\Delta {y_{j + 1}}}}} \right) {\psi ^\gamma }\left( {x_i^l} \right) {{\tilde{u}}_h}\left( {x_i^l,{y_{j + 1/2}}} \right) } \right] + \left( {\frac{1}{{\Delta y_j^2}} + \frac{1}{{\Delta {y_j}\Delta {y_{j + 1}}}}} \right) F_{i,j}^\gamma \nonumber \\&\quad + \left[ {\frac{1}{{\Delta y_j^3}}W_{i,j}^\gamma u_h^ + \left( {{x_i},{y_{j - 1/2}}} \right) + \frac{1}{{\Delta {y_j}\Delta y_{j + 1}^2}}W_{i,j + 1}^\gamma u_h^ - \left( {{x_i},{y_{j + 1/2}}} \right) } \right] \sum \limits _{l = 1}^3 {{w_l}} s\left( {{\eta _l}} \right) \nonumber \\&\quad - \frac{{{\varphi ^\gamma }\left( {{x_{i - 1/2}}} \right) }}{{\Delta {x_i}\Delta y_j^2}}\sum \limits _{l = 1}^3 {s\left( {{\eta _l}} \right) \sum \limits _{{l_1} = 1}^3 {{w_{{l_1}}}{\psi ^l}\left( {y_j^{{l_1}}} \right) \int _{{I_i}} {{u_h}\left( { \cdot ,y_j^{{l_1}}} \right) s\left( {{\xi ^i}\left( x \right) } \right) dx} } } . \end{aligned}$$
(1)

First of all, we estimate the integral term on cells \(J_j\) in (1) as follows.

$$\begin{aligned}&- \frac{1}{{\Delta y_j^2}}\sum \limits _{l = 1}^3 {{w_l}{\psi ^\gamma }\left( {x_i^l} \right) \int _{{J_j}} {{u_h}\left( {x_i^l, \cdot } \right) s{{\left( {{\eta ^j}\left( y \right) } \right) }_y}dy} } \nonumber \\&\quad \le \frac{2}{{\Delta y_j^3}}\sum \limits _{l = 1}^3 {{w_l}\left| {{\psi ^\gamma }\left( {x_i^l} \right) } \right| \left| {\int _{{J_j}} {{u_h}\left( {x_i^l, \cdot } \right) \left( {3 - 15{\eta ^j}\left( y \right) } \right) dy} } \right| } \nonumber \\&\quad \le \frac{{36}}{{\Delta y_j^3}}\sum \limits _{l = 1}^3 {{w_l}\left| {{\psi ^\gamma }\left( {x_i^l} \right) } \right| \left| {\int _{{J_j}} {{u_h}\left( {x_i^l, \cdot } \right) dy} } \right| } \le \frac{{48\sqrt{15} }}{{\Delta {x_i}\Delta y_j^2}}\sum \limits _{l = 1}^3 {{w_l}\bar{u}_{l,j}^n} \le \frac{{48\sqrt{15} }}{{\Delta x\Delta {y^2}}}\bar{u}_{i,j}^n. \end{aligned}$$
(2)

Notice that the Lagrange interpolation polynomial at three points \(x_i^\gamma \) is taken as \({\varphi ^\gamma }(x)\). Therefore, the penultimate step in (2) follows that \( - \frac{{4\sqrt{15} }}{{3\Delta {x_i}}} \le {\psi ^\gamma }\left( {x_i^l} \right) \le \frac{{4\sqrt{15} }}{{3\Delta {x_i}}}\). Through some simple calculations, we have

$$\begin{aligned} - \frac{2}{3} \le {\varphi ^\gamma }\left( {{x_{i \pm 1/2}}} \right) \le \frac{5}{6}\left( {1 + \frac{{\sqrt{15} }}{5}} \right) , \quad - \frac{3}{2} \le s\left( {{\eta _\gamma }} \right) \le 3\left( {1 + \frac{{\sqrt{15} }}{5}} \right) . \end{aligned}$$

The estimate of the integral term on cells \(I_i\) in (1) is given by

$$\begin{aligned}&- \frac{{{\varphi ^\gamma }\left( {{x_{i - 1/2}}} \right) }}{{\Delta {x_i}\Delta y_j^2}}\sum \limits _{l = 1}^3 {s\left( {{\eta _l}} \right) \sum \limits _{{l_1} = 1}^3 {{w_{{l_1}}}{\psi ^l}\left( {y_j^{{l_1}}} \right) \int _{{I_i}} {{u_h}\left( { \cdot ,y_j^{{l_1}}} \right) s\left( {{\xi ^i}\left( x \right) } \right) dx} } } \nonumber \\&\quad \le \frac{{16\left( {15 + 4\sqrt{15} } \right) }}{{5\Delta {x_i}\Delta y_j^3}}\sum \limits _{l = 1}^3 {\sum \limits _{{l_1} = 1}^3 {{w_{{l_1}}}\left| {\int _{{I_i}} {{u_h}\left( { \cdot ,y_j^{{l_1}}} \right) dx} } \right| } } \nonumber \\&\quad = \frac{{16\left( {15 + 4\sqrt{15} } \right) }}{{5\Delta y_j^3}}\sum \limits _{l = 1}^3 {\sum \limits _{{l_1} = 1}^3 {{w_{{l_1}}}\bar{u}_{i,{l_1}}^n} } \le \frac{{48\left( {15 + 4\sqrt{15} } \right) }}{{5\Delta {y^3}}}\bar{u}_{i,j}^n. \end{aligned}$$
(3)

Similar to (3.16), we can rewrite \({\tilde{u}_h}\left( {x_i^l,{y_{j + 1/2}}} \right) \) as

$$\begin{aligned} {\tilde{u}_h}\left( {x_i^l,{y_{j + 1/2}}} \right) = {u_h}\left( {x_i^l,{y_{j + 1/2}}} \right) = \frac{1}{2}\int _{ - 1}^1 {{u_0}\left( {x_i^l,\eta } \right) z\left( {1,\eta } \right) d\eta }, \end{aligned}$$

where \(z\left( {{\eta _0},\eta } \right) \) is given in (3.16) and \(u_0\) is defined as \({u_0}:S_i^x \times \left[ { - 1,1} \right] \rightarrow \mathbb {R}\).

Then, we estimate the term \(F_{i,j}^\gamma \) as follows.

$$\begin{aligned} F_{i,j}^\gamma&= \frac{{{\varphi ^\gamma }\left( {{x_{i - 1/2}}} \right) }}{{\Delta {x_i}}}\sum \limits _{l = 1}^3 {s\left( {{\eta _l}} \right) {\varphi ^l}\left( {{y_{j + 1/2}}} \right) \sum \limits _{{l_1} = 1}^3 {{w_{{l_1}}}s\left( {{\xi _{{l_1}}}} \right) {{\tilde{u}}_h}\left( {x_i^{{l_1}},{y_{j + 1/2}}} \right) } } \\&\le \!\frac{{\left| {{\varphi ^\gamma }\left( {{x_{i - 1/2}}} \right) } \right| }}{{2\Delta {x_i}}} \!\! \sum \limits _{l = 1}^3 {\left| {s\left( {{\eta _l}} \right) } \right| \! \left| {{\varphi ^l}\! \!\left( {{y_{j + 1/2}}} \right) } \right| \! \sum \limits _{{l_1} = 1}^3 {{w_{{l_1}}}\left| {s\left( {{\xi _{{l_1}}}} \right) } \right| \! \left| {\int _{ - 1}^1 \!\! {{u_0}\left( {x_i^{{l_1}},\eta } \right) \! z\left( {1,\eta } \right) d\eta } } \right| } } \\&\le \frac{{9\left( {31 + 8\sqrt{15} } \right) }}{{2\Delta {x_i}}}\sum \limits _{l = 1}^3 {\sum \limits _{{l_1} = 1}^3 {{w_{{l_1}}}\left| {\int _{ - 1}^1 {{u_0}\left( {x_i^{{l_1}},\eta } \right) d\eta } } \right| } } \\&= \frac{{9\left( {31 + 8\sqrt{15} } \right) }}{{\Delta {x_i}}}\sum \limits _{l = 1}^3 {\sum \limits _{{l_1} = 1}^3 {{w_{{l_1}}}\bar{u}_{{l_1},j}^n} } = \frac{{27\left( {31 + 8\sqrt{15} } \right) }}{{\Delta {x_i}}}\bar{u}_{i,j}^n. \end{aligned}$$

Next, we proceed to estimate the rest of terms in (1).

$$\begin{aligned}&3\left( {\frac{1}{{\Delta y_j^2}} \!+\! \frac{3}{{\Delta {y_j}\Delta {y_{j + 1}}}}} \right) \sum \limits _{l = 1}^3 {{w_l}{\psi ^\gamma }\left( {x_i^l} \right) {{\tilde{u}}_h}\left( {x_i^l,{y_{j + 1/2}}} \right) } \!+\! \left( {\frac{1}{{\Delta y_j^2}} \!+\! \frac{1}{{\Delta {y_j}\Delta {y_{j + 1}}}}} \right) F_{i,j}^\gamma \\&\quad \quad + \left[ {\frac{1}{{\Delta y_j^3}}W_{i,j}^\gamma u_h^ + \left( {{x_i},{y_{j - 1/2}}} \right) + \frac{1}{{\Delta {y_j}\Delta y_{j + 1}^2}}W_{i,j + 1}^\gamma u_h^ - \left( {{x_i},{y_{j + 1/2}}} \right) } \right] \sum \limits _{l = 1}^3 {{w_l}} s\left( {{\eta _l}} \right) \\&\quad \le \frac{3}{2}\left( {\frac{1}{{\Delta y_j^2}} + \frac{3}{{\Delta {y_j}\Delta {y_{j + 1}}}}} \right) \sum \limits _{l = 1}^3 {{w_l}\left| {{\psi ^\gamma }\left( {x_i^l} \right) } \right| \left| {\int _{ - 1}^1 {{u_0}\left( {x_i^l,\eta } \right) z\left( {1,\eta } \right) d\eta } } \right| } \\&\quad \quad + \left( {\frac{1}{{\Delta y_j^2}} + \frac{1}{{\Delta {y_j}\Delta {y_{j + 1}}}}} \right) \frac{{27\left( {31 + 8\sqrt{15} } \right) }}{{\Delta {x_i}}}\bar{u}_{i,j}^n\\&\quad \quad + \left| {{\varphi ^\gamma }\left( {{x_{i - 1/2}}} \right) } \right| \left[ {\int _{ - 1}^1 {{u_0}\left( {{x_i},\eta } \right) \left( {\frac{{\left| {{\alpha _{i,j - 1/2}}} \right| \left| {z\left( { - 1,\eta } \right) } \right| }}{{\Delta y_j^3}} + \frac{{\left| {{\alpha _{i,j + 1/2}}} \right| \left| {z\left( {1,\eta } \right) } \right| }}{{\Delta {y_j}\Delta y_{j + 1}^2}}} \right) d\eta } } \right] , \end{aligned}$$

which follows the upper bound of \(F_{i,j}^\gamma \) and

$$\begin{aligned} \sum \limits _{l = 1}^3 {{w_l}s\left( {{\eta _l}} \right) \approx \int _{ - 1}^1 {s\left( \eta \right) d\eta } } = 2. \end{aligned}$$

Finally, the result of estimating the rest terms is given by

$$\begin{aligned}&3\left( {\frac{1}{{\Delta y_j^2}} \!+\! \frac{3}{{\Delta {y_j}\Delta {y_{j + 1}}}}} \right) \sum \limits _{l = 1}^3 {{w_l}{\psi ^\gamma }\left( {x_i^l} \right) {{\tilde{u}}_h}\left( {x_i^l,{y_{j + 1/2}}} \right) } \!+\! \left( {\frac{1}{{\Delta y_j^2}} \!+\! \frac{1}{{\Delta {y_j}\Delta {y_{j + 1}}}}} \right) F_{i,j}^\gamma \nonumber \\&\quad \quad + \left[ {\frac{1}{{\Delta y_j^3}}W_{i,j}^\gamma u_h^ + \left( {{x_i},{y_{j - 1/2}}} \right) + \frac{1}{{\Delta {y_j}\Delta y_{j + 1}^2}}W_{i,j + 1}^\gamma u_h^ - \left( {{x_i},{y_{j + 1/2}}} \right) } \right] \sum \limits _{l = 1}^3 {{w_l}} s\left( {{\eta _l}} \right) \nonumber \\&\quad \le \frac{{72\sqrt{15} }}{{\Delta x\Delta {y^2}}}\sum \limits _{l = 1}^3 {{w_l}\left| {\int _{ - 1}^1 {{u_0}\left( {x_i^l,\eta } \right) d\eta } } \right| } + \frac{{54\left( {31 + 8\sqrt{15} } \right) }}{{\Delta x\Delta {y^2}}}\bar{u}_{i,j}^n \nonumber \\&\quad \quad + \frac{{3\left( {5 + \sqrt{15} } \right) {{\max }_{i,j}}\left| {{\alpha _{i,j - 1/2}}} \right| }}{{\Delta {y^3}}}\left| {\int _{ - 1}^1 {{u_0}\left( {{x_i},\eta } \right) d\eta } } \right| \nonumber \\&\quad = \frac{{144\sqrt{15} }}{{\Delta x\Delta {y^2}}}\sum \limits _{l = 1}^3 {{w_l}\bar{u}_{l,j}^n} + \frac{{54\left( {31 + 8\sqrt{15} } \right) }}{{\Delta x\Delta {y^2}}}\bar{u}_{i,j}^n + \frac{{6\left( {5 + \sqrt{15} } \right) {{\max }_{i,j}}\left| {{\alpha _{i,j - 1/2}}} \right| }}{{\Delta {y^3}}}\bar{u}_{\mathcal{L},j}^n \nonumber \\&\quad = \frac{{18\left( {93 + 32\sqrt{15} } \right) + 3\left( {5 + \sqrt{15} } \right) \tau {{\max }_{i,j}}\left| {{\alpha _{i,j - 1/2}}} \right| }}{{\Delta x\Delta {y^2}}}\bar{u}_{i,j}^n, \end{aligned}$$
(4)

where \(\bar{u}_{i,j}^n\) can be approximated as \(2\bar{u}_{\mathcal{L},j}^n\) by one-point Gaussian quadrature. According to (2), (3) and (4), we have

$$\begin{aligned}&- \frac{1}{{\Delta {y_j}}}\sum \limits _{\gamma = 1}^3 {U_{*,j}^\gamma } \\&\quad \le \frac{{90\left( {279 + 104\sqrt{15} } \right) + 9\tau \left[ {16\left( {15 + 4\sqrt{15} } \right) + 5\left( {5 + \sqrt{15} } \right) {{\max }_{i,j}}\left| {{\alpha _{i,j - 1/2}}} \right| } \right] }}{{5\Delta x\Delta {y^2}}}\bar{u}_{i,j}^n, \end{aligned}$$

and thus can obtain the conclusion.

1.2 The proof of Lemma 4.4

In this appendix, we report the proof of Lemma 4.4.

1.3 The estimate of terms \(U_{i + 1,j}^\gamma \)

First of all, we consider the lower bound of \(F_{i + 1,j}^\gamma \) as follows.

$$\begin{aligned} F_{i + 1,j}^\gamma&= \frac{{{\varphi ^\gamma }\left( {{x_{i + 1/2}}} \right) }}{{\Delta {x_{i + 1}}}}\sum \limits _{l = 1}^3 {s\left( {{\eta _l}} \right) {\varphi ^l}\left( {{y_{j + 1/2}}} \right) \sum \limits _{{l_1} = 1}^3 {{w_{{l_1}}}s\left( {{\xi _{{l_1}}}} \right) {{\tilde{u}}_h}\left( {x_{i + 1}^{{l_1}},{y_{j + 1/2}}} \right) } } \\&= \frac{{{\varphi ^\gamma }\left( {{x_{i + 1/2}}} \right) }}{{2\Delta {x_{i + 1}}}}\sum \limits _{l = 1}^3 {s\left( {{\eta _l}} \right) {\varphi ^l}\left( {{y_{j + 1/2}}} \right) \sum \limits _{{l_1} = 1}^3 {{w_{{l_1}}}s\left( {{\xi _{{l_1}}}} \right) \int _{ - 1}^1 \! \! \!{{u_1}\left( {x_{i + 1}^{{l_1}},\eta } \right) z\left( {1,\eta } \right) d\eta } } } \\&\ge \frac{1}{2}\sum \limits _{l = 1}^3 {\sum \limits _{{l_1} = 1}^3 {{w_{{l_1}}}\int _{ - 1}^1 {{u_1}\left( {x_{i + 1}^{{l_1}},\eta } \right) d\eta \mathop {\min }\limits _{\gamma ,l,{l_1} \in \left\{ {1,2,3} \right\} } } } } \left\{ {\mathop {\min }\limits _{i,j} \tilde{Z}_1} \right\} \\&= \sum \limits _{l = 1}^3 {\sum \limits _{{l_1} = 1}^3 {{w_{{l_1}}}\bar{u}_{{l_1},j}^n} } \mathop {\min }\limits _{\gamma ,l,{l_1} \in \left\{ {1,2,3} \right\} } \left\{ {\mathop {\min }\limits _{i,j} \tilde{Z}_1} \right\} = 3\bar{u}_{i + 1,j}^n\mathop {\min }\limits _{\gamma ,l,{l_1} \in \left\{ {1,2,3} \right\} } \left\{ {\mathop {\min }\limits _{i,j} \tilde{Z}_1} \right\} , \end{aligned}$$

where

$${\tilde{Z}_1} = \frac{{s\left( {{\xi _{{l_1}}}} \right) {\varphi ^\gamma }\left( {{x_{i + 1/2}}} \right) s\left( {{\eta _l}} \right) {\varphi ^l}\left( {{y_{j + 1/2}}} \right) }}{{\Delta {x_{i + 1}}}}z\left( {1,\eta } \right) .$$

Then, the estimate of the integral term in \(U_{i + 1,j}^\gamma \) is given by

$$\begin{aligned}&- \frac{{{\varphi ^\gamma }\left( {{x_{i + 1/2}}} \right) }}{{\Delta {x_{i + 1}}\Delta {y_j}}}\sum \limits _{l = 1}^3 {s\left( {{\eta _l}} \right) \sum \limits _{{l_1} = 1}^3 {{w_{{l_1}}}{\psi ^l}\left( {y_j^{{l_1}}} \right) \int _{{I_{i + 1}}} {u_h}\left( { \cdot ,y_j^{{l_1}}} \right) s\left( {{\xi ^{i + 1}}\left( x \right) } \right) dx} } \\&\quad \ge \frac{1}{{\Delta {x_{i + 1}}}}\sum \limits _{l = 1}^3 {\sum \limits _{{l_1} = 1}^3 {{w_{{l_1}}}\int _{{I_{i + 1}}} {u_h}\left( { \cdot ,y_j^{{l_1}}} \right) dx} } \mathop {\min }\limits _{\gamma ,l,{l_1} \in \left\{ {1,2,3} \right\} } \left\{ {\mathop {\min }\limits _{i,j} {\tilde{Z}_0}} \right\} \\&\quad = \sum \limits _{l = 1}^3 {\sum \limits _{{l_1} = 1}^3 {{w_{{l_1}}}\bar{u}_{i + 1,{l_1}}^n} } \mathop {\min }\limits _{\gamma ,l,{l_1} \in \left\{ {1,2,3} \right\} } \left\{ {\mathop {\min }\limits _{i,j} {\tilde{Z}_0}} \right\} = 3\bar{u}_{i + 1,j}^n\mathop {\min }\limits _{\gamma ,l,{l_1} \in \left\{ {1,2,3} \right\} } \left\{ {\mathop {\min }\limits _{i,j} {\tilde{Z}_0}} \right\} , \end{aligned}$$

where

$$\begin{aligned} {\tilde{Z}_0} = - \frac{{{\varphi ^\gamma }\left( {{x_{i + 1/2}}} \right) s\left( {{\eta _l}} \right) {\psi ^l}\left( {y_j^{{l_1}}} \right) }}{{\Delta {y_j}}}s\left( \xi \right) . \end{aligned}$$

Next, we proceed to estimate the rest terms in \(U_{i + 1,j}^\gamma \).

$$\begin{aligned}&{\varphi ^\gamma } \left( {{x_{i + 1/2}}} \right) \!\! \left[ {\frac{{{\alpha _{i + 1,j - 1/2}}}}{{\Delta y_j^2}}u_h^ + \left( {{x_{i + 1}},{y_{j - 1/2}}} \right) } \right. \left. { \!+ \frac{{{\alpha _{i + 1,j + 1/2}}}}{{\Delta y_{j + 1}^2}}u_h^ - \! \left( {{x_{i + 1}},{y_{j + 1/2}}} \right) } \right] \!\! \sum \limits _{l = 1}^3 {{w_l}} s\left( {{\eta _l}} \right) \\&\quad = {\varphi ^\gamma }\left( {{x_{i + 1/2}}} \right) \left[ {\int _{ - 1}^1 {{u_1}\left( {{x_{i + 1}},\eta } \right) \left( {\frac{{{\alpha _{i + 1,j - 1/2}}}}{{\Delta y_j^2}}z\left( { - 1,\eta } \right) + \frac{{{\alpha _{i + 1,j + 1/2}}}}{{\Delta y_{j + 1}^2}}z\left( {1,\eta } \right) } \right) d\eta } } \right] \\&\quad \ge 2\bar{u}_{L,j}^n\mathop {\min }\limits _{\gamma ,l,{l_1} \in \left\{ {1,2,3} \right\} } \left\{ {\mathop {\min }\limits _{i,j} {\tilde{Z}_2}} \right\} = \bar{u}_{i + 1,j}^n\mathop {\min }\limits _{\gamma ,l,{l_1} \in \left\{ {1,2,3} \right\} } \left\{ {\mathop {\min }\limits _{i,j} {\tilde{Z}_2}} \right\} , \end{aligned}$$

where

$$\begin{aligned} {\tilde{Z}_2} = \frac{{{\varphi ^\gamma }\left( {{x_{i + 1/2}}} \right) {\alpha _{i + 1,j - 1/2}}}}{{\Delta y_j^2}}z\left( { - 1,\eta } \right) + \frac{{{\varphi ^\gamma }\left( {{x_{i + 1/2}}} \right) {\alpha _{i + 1,j + 1/2}}}}{{\Delta y_{j + 1}^2}}z\left( {1,\eta } \right) . \end{aligned}$$

Finally, the result of estimating \(U_{i + 1,j}^\gamma \) is given by

$$\begin{aligned} U_{i + 1,j}^\gamma \ge \bar{u}_{i + 1,j}^n\mathop {\min }\limits _{\gamma ,l,{l_1} \in \left\{ {1,2,3} \right\} } \left\{ {\mathop {\min }\limits _{i,j} Z_1} \right\} , \end{aligned}$$

where \(Z_1 = 3{\tilde{Z}_0} + {\tilde{Z}_2} + 3\left( {\frac{1}{{\Delta {y_j}}} + \frac{1}{{\Delta {y_{j + 1}}}}} \right) {\tilde{Z}_1}\). Obviously, \(U_{i + 1,j}^\gamma \ge 0\) is equivalent to \(Z_1 \ge 0\). We denote \({\Gamma _{i,j}} = \frac{{\Delta {y_j}\Delta {y_{j + 1}} + \Delta y_j^2}}{{\Delta {x_{i + 1}}\Delta {y_{j + 1}}}}\). Notice that \({\Gamma _{i,j}} = 2/\tau \) for uniform meshes. Thus, the inequality \(Z_1 \ge 0\) further yields

$$\begin{aligned} {g_1}\left( {d{y_j},{\Gamma _{i,j}},{\alpha _{i + 1,j - 1/2}},{\alpha _{i + 1,j + 1/2}}} \right) = \Delta y_j^2{Z_1} \ge 0. \end{aligned}$$

1.4 The estimate of terms \(U_{i,j-1}^\gamma \)

The result of estimating \(U_{i,j-1}^\gamma \) is given by

$$\begin{aligned} U_{i,j-1}^\gamma \ge \bar{u}_{i,j-1}^n\mathop {\min }\limits _{\gamma ,l,{l_1} \in \left\{ {1,2,3} \right\} } \left\{ {\mathop {\min }\limits _{i,j} Z_2} \right\} , \end{aligned}$$

where

$$\begin{aligned} Z_2 = \left[ {9{\psi ^\gamma }\left( {x_i^l} \right) \!+\! \frac{{{\varphi ^\gamma }\left( {{x_{i - 1/2}}} \right) {\alpha _{i,j - 1/2}}}}{{\Delta {y_j}}} \!+\! \frac{{3{\varphi ^\gamma }\left( {{x_{i - 1/2}}} \right) s\left( {{\xi _{{l_1}}}} \right) {\varphi ^l}\left( {{y_{j - 1/2}}} \right) s\left( {{\eta _l}} \right) }}{{\Delta {x_i}}}} \right] \!z\left( {1,\eta } \right) , \end{aligned}$$

which further yields \({g_2}\left( {{\tau _{i,j}},{\alpha _{i,j - 1/2}}} \right) = \Delta {x_i}{Z_2} \ge 0\).

1.5 The estimate of terms \(U_{i+1,j-1}^\gamma \)

The result of estimating \(U_{i+1,j-1}^\gamma \) is given by

$$\begin{aligned} U_{i+1,j-1}^\gamma \ge \bar{u}_{i+1,j-1}^n\mathop {\min }\limits _{\gamma ,l,{l_1} \in \left\{ {1,2,3} \right\} } \left\{ {\mathop {\min }\limits _{i,j} Z_3} \right\} , \end{aligned}$$

where

$$\begin{aligned} Z_3 = - \left[ {\frac{{{\varphi ^\gamma }\left( {{x_{i + 1/2}}} \right) {\alpha _{i + 1,j - 1/2}}}}{{\Delta {y_j}}} + \frac{{3{\varphi ^\gamma }\left( {{x_{i + 1/2}}} \right) s\left( {{\xi _{{l_1}}}} \right) {\varphi ^l}\left( {{y_{j - 1/2}}} \right) s\left( {{\eta _l}} \right) }}{{\Delta {x_{i + 1}}}}} \right] z\left( {1,\eta } \right) , \end{aligned}$$

which further yields \({g_3}\left( {{\tau _{i + 1,j}},{\alpha _{i + 1,j - 1/2}}} \right) = \Delta {x_{i + 1}}{Z_3} \ge 0\).

1.6 The estimate of terms \(U_{i,j+1}^\gamma \)

The result of estimating \(U_{i,j+1}^\gamma \) is given by

$$\begin{aligned} U_{i,j+1}^\gamma \ge \bar{u}_{i,j+1}^n\mathop {\min }\limits _{\gamma ,l,{l_1} \in \left\{ {1,2,3} \right\} } \left\{ {\mathop {\min }\limits _{i,j} Z_4} \right\} , \end{aligned}$$

where

$$\begin{aligned} Z_4&= 3\left( {{\psi ^\gamma }\left( {x_i^l} \right) + \frac{{{\varphi ^\gamma }\left( {{x_{i - 1/2}}} \right) s\left( {{\xi _{{l_1}}}} \right) {\varphi ^l}\left( {{y_{j + 3/2}}} \right) s\left( {{\eta _l}} \right) }}{{\Delta {x_i}}}} \right) z\left( {1,\eta } \right) \\&\quad + 2{\psi ^\gamma }\left( {x_i^l} \right) \left( {3 - 15\eta } \right) + \left( {\frac{{{\alpha _{i,j + 1/2}}}}{{\Delta {y_{j + 1}}}}z\left( { - 1,\eta } \right) - 3s\left( {{\eta _l}} \right) {\psi ^l}\left( {y_{j + 1}^{{l_1}}} \right) s\left( \xi \right) } \right) {\varphi ^\gamma }\left( {{x_{i - 1/2}}} \right) , \end{aligned}$$

which further yields \({g_4}\left( {{\tau _{i,j + 1}},{\alpha _{i,j + 1/2}}} \right) = \Delta {x_i}{Z_4} \ge 0\).

1.7 The estimate of terms \(U_{i+1,j+1}^\gamma \)

The result of estimating \(U_{i+1,j+1}^\gamma \) is given by

$$\begin{aligned} U_{i+1,j+1}^\gamma \ge \bar{u}_{i+1,j+1}^n\mathop {\min }\limits _{\gamma ,l,{l_1} \in \left\{ {1,2,3} \right\} } \left\{ {\mathop {\min }\limits _{i,j} Z_5} \right\} , \end{aligned}$$

where

$$\begin{aligned} Z_5&= {\varphi ^\gamma }\left( {{x_{i + 1/2}}} \right) \left( {3s\left( {{\eta _l}} \right) {\psi ^l}\left( {y_{j + 1}^{{l_1}}} \right) s\left( \xi \right) - \frac{{{\varphi ^\gamma }\left( {{x_{i + 1/2}}} \right) {\alpha _{i + 1,j + 1/2}}}}{{\Delta {y_{j + 1}}}}z\left( { - 1,\eta } \right) } \right) \\&\quad - \frac{{3{\varphi ^\gamma }\left( {{x_{i + 1/2}}} \right) s\left( {{\xi _{{l_1}}}} \right) {\varphi ^l}\left( {{y_{j + 3/2}}} \right) s\left( {{\eta _l}} \right) }}{{\Delta {x_{i + 1}}}}z\left( {1,\eta } \right) , \end{aligned}$$

which further yields \({g_5}\left( {{\tau _{i + 1,j + 1}},{\alpha _{i + 1,j + 1/2}}} \right) = \Delta {x_{i + 1}}{Z_5} \ge 0\).

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Bi, H., Zhao, F. Bound-preserving schemes for \(P^2\) local discontinuous Galerkin discretizations of KdV-type equations. Comp. Appl. Math. 44, 46 (2025). https://doi.org/10.1007/s40314-024-03002-z

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