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 同济大学学报(自然科学版)  2017, Vol. 45 Issue (8): 1243-1248.  DOI: 10.11908/j.issn.0253-374x.2017.08.021 0

### 引用本文

CHEN Rongsan, SU Meng, ZOU Min, XIAO Li. On Maximum-Principle-Satisfying Entropy Scheme for Linear Advection Equation[J]. Journal of Tongji University (Natural Science), 2017, 45(8): 1243-1248. DOI: 10.11908/j.issn.0253-374x.2017.08.021.

### 文章历史

On Maximum-Principle-Satisfying Entropy Scheme for Linear Advection Equation
CHEN Rongsan, SU Meng, ZOU Min, XIAO Li
School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China
Abstract: Mao Dekang et al developed an entropy scheme for computing one dimensional hyperbolic conservation equations, which has a super convergence property and is suitable for long time numerical computation. But the entropy scheme does not satisfy the maximum principle. Over-shooting or under-shooting may occur in the vicinity of maximum or minimum points. In this work, numerical simulations of one dimensional and two dimensional linear advection equations are carried out. The numerical results show that the proposed scheme does not lead to over-shooting or under-shooting, moreover, non-physical oscillations do not occur.
Key words: maximum principle    entropy scheme    linear advection equation

 $\left\{ \begin{array}{l} {u_t} + a{u_x} + b{u_y} = 0,\\ u\left( {x,y,0} \right) = {u_0}\left( {x,y} \right),\forall x \in \mathit{\Omega }\mathit{.} \end{array} \right.$ (1)

 $\left\{ \begin{array}{l} {u_t} + a{u_x} = 0,\\ u\left( {x,0} \right) = {u_0}\left( x \right),\forall x \in \mathit{\Omega }\mathit{.} \end{array} \right.$ (2)

1 一维线性传输方程满足最大值原理熵格式的描述

 $\left\{ \begin{array}{l} U{\left( u \right)_t} + aU{\left( u \right)_x} = 0,\\ u\left( {x,0} \right) = {u_0}\left( x \right),\forall x \in \mathit{\Omega }\mathit{.} \end{array} \right.$ (3)

 $u\left( {x,t} \right) = {u_0}\left( {x - at} \right)$ (4)

 $M = \mathop {\max }\limits_x {u_0}\left( x \right),$ (5)
 $m = \mathop {{\rm{Min}}}\limits_x {u_0}\left( x \right),$ (6)

 ${u_{j,n}} \approx \frac{1}{h}\int_{{x_{j - 1/2}}}^{{x_{j + 1/2}}} {u\left( {x,{t_n}} \right){\rm{d}}x}$ (7)

 ${U_{j,n}} \approx \frac{1}{h}\int_{{x_{j - 1/2}}}^{{x_{j + 1/2}}} {U\left( {u\left( {x,{t_n}} \right)} \right){\rm{d}}x}$ (8)

(1) 重构.首先给出tn时刻每一个网格Ij内熵重构解，它是一个线性函数，

 ${R_{\rm{E}}}\left( {x;{u_n},{U_n}} \right) = {u_{j,n}} + {s_{j,n,e}}\left( {x - {x_j}} \right),$ (9)

 $\frac{1}{h}\int_{{x_{j - 1/2}}}^{{x_{j + 1/2}}} {{R_{\rm{E}}}\left( {x;{u_n},{U_n}} \right){\rm{d}}x} = {u_{j,n}},$ (10)

 $\frac{1}{h}\int_{{x_{j - 1/2}}}^{{x_{j + 1/2}}} {U\left( {{R_{\rm{E}}}\left( {x;{u_n},{U_n}} \right)} \right){\rm{d}}x} = {U_{j,n}}$ (11)

 ${\left( {{s_{j,n,{\rm{e}}}}} \right)^2} = \frac{{12\left( {{U_{j,n}} - {{\left( {{u_{j,n}}} \right)}^2}} \right)}}{{{h^2}}}.$ (12)

 $\begin{array}{l} {R_{{\rm{ME}}}}\left( {x;{u_n},{U_n}} \right) = \theta \left( {{R_{{\rm{ME}}}}\left( {x;{u_n},{U_n}} \right) - {u_{j,n}}} \right) + {u_{j,n}},\\ \theta = {\rm{min}}\left\{ {\left| {\frac{{{M_0} - {u_{j,n}}}}{{M' - {u_{j,n}}}}} \right|,\left| {\frac{{{m_0} - {u_{j,n}}}}{{m' - {u_{j,n}}}}} \right|,1} \right\}, \end{array}$ (13)

(2) 发展.以重构函数RME(x; un, Un)作为tn层的初值，求解以下初值问题:

 $\left\{ \begin{array}{l} {v_t} + a{v_x} = 0, - \infty < x < \infty ,{t_n} < t < {t_{n + 1}},\\ v\left( {x,{t_n}} \right) = {R_{{\rm{ME}}}}\left( {x;{u_n},{U_n}} \right), - \infty < x < \infty . \end{array} \right.$ (14)

(3) 网格平均.在t=tn+1时的数值解和数值熵分别计算为

 $u_j^{n + 1} = \frac{1}{h}\int_{{x_{j - 1/2}}}^{{x_{j + 1/2}}} {v\left( {x,{t_{n + 1}}} \right){\rm{d}}x} ,$ (15)
 $U_j^{n + 1} = \frac{1}{h}\int_{{x_{j - 1/2}}}^{{x_{j + 1/2}}} {U\left( {v\left( {x,{t_{n + 1}}} \right)} \right){\rm{d}}x} .$ (16)
2 二维线性传输方程满足最大值原理熵格式的描述

 $\left\{ \begin{array}{l} U{\left( u \right)_t} + aU{\left( u \right)_x} + bU{\left( u \right)_y} = 0,\\ u\left( {x,y,0} \right) = {u_0}\left( {x,y} \right),\forall x \in \mathit{\Omega }\mathit{.} \end{array} \right.$ (17)

 $M = \mathop {{\rm{Max}}}\limits_{x,y} {u_0}\left( {x,y} \right),$ (18)
 $m = \mathop {{\rm{Min}}}\limits_{x,y} {u_0}\left( {x,y} \right),$ (19)

 ${u_{j,n,k}} \approx \frac{1}{{{h^2}}}\int_{{x_{j - 1/2}}}^{{x_{j + 1/2}}} {\int_{{y_{k - 1/2}}}^{{y_{k + 1/2}}} {u\left( {x,y,{t_n}} \right){\rm{d}}x} } {\rm{d}}\mathit{y,}$ (20)
 ${U_{j,n,k}} \approx \frac{1}{{{h^2}}}\int_{{x_{j - 1/2}}}^{{x_{j + 1/2}}} {\int_{{y_{k - 1/2}}}^{{y_{k + 1/2}}} {U\left( {u\left( {x,y,{t_n}} \right)} \right){\rm{d}}x} } {\rm{d}}\mathit{y,}$ (21)

(1) 重构.在tn层上熵重构函数RE(x, y; un, Un)在网格Ij, k上取为xy的一次函数，

 ${R_{\rm{E}}}\left( {x,\mathit{y};{u_n},{U_n}} \right) = {u_{j,n,k}} + \alpha \left( {{{\left( {{s_x}} \right)}_{j,n,\mathit{k}}}\left( {x - {x_j}} \right) + {{\left( {{s_y}} \right)}_{j,n,\mathit{k}}}\left( {y - {y_k}} \right)} \right),$ (22)

 $\frac{1}{{{h^2}}}\int_{{x_{j - 1/2}}}^{{x_{j + 1/2}}} {\int_{{y_{k - 1/2}}}^{{y_{k + 1/2}}} {{R_{\rm{E}}}\left( {x,y;{\mathit{u}_n},{\mathit{U}_n}} \right){\rm{d}}x} } {\rm{d}}\mathit{y,} = {u_{j,n,k}},$ (23)

 $\frac{1}{{{h^2}}}\int_{{x_{j - 1/2}}}^{{x_{j + 1/2}}} {\int_{{y_{k - 1/2}}}^{{y_{k + 1/2}}} {U\left( {{R_{\rm{E}}}\left( {x,y;{\mathit{u}_n},{\mathit{U}_n}} \right)} \right){\rm{d}}x} } {\rm{d}}\mathit{y,} = {U_{j,n,k}}$ (24)

RE(x, y; un, Un)自动满足式(23).可通过式(24) 解出，

 ${\alpha ^2} = \frac{{12\left( {{U_{j,n}} - {{\left( {{u_{j,n}}} \right)}^2}} \right)}}{{{h^2}\left( {{{\left( {{{\left( {{s_x}} \right)}_{j,n,k}}} \right)}^2} + {{\left( {{{\left( {{s_y}} \right)}_{j,n,k}}} \right)}^2}} \right)}}$ (25)

 $\theta \left( {{R_{{\rm{ME}}}}\left( {x,y;{u_n},{U_n}} \right) - {u_{j,n,k}}} \right) + {u_{j,n,k}},$ (26)
 $\theta = {\rm{min}}\left\{ {\left| {\frac{{{M_0} - {u_{j,n,k}}}}{{M' - {u_{j,n,k}}}}} \right|,\left| {\frac{{{m_0} - {u_{j,n,k}}}}{{m' - {u_{j,n,k}}}}} \right|,1} \right\}$ (27)

(2) 发展.求解初值问题：

 $\left\{ \begin{array}{l} {v_t} + a{v_x} + b{v_y} = 0,\\ - \infty < x,y < \infty ,{t_n} < t < {t_{n + 1}},\\ v\left( {x,y,{t_n}} \right) = {R_{{\rm{ME}}}}\left( {x,y;{u_n},{U_n}} \right), - \infty < x,y < \infty . \end{array} \right.$ (28)

(3) 网格平均. t=tn+1时刻的数值解和数值熵的计算分别为

 ${u_{j,n + 1,k}} = \frac{1}{{{h^2}}}\int_{{x_{j - 1/2}}}^{{x_{j + 1/2}}} {\int_{{y_{k - 1/2}}}^{{y_{k + 1/2}}} {v\left( {x,y,{t_{n + 1}}} \right){\rm{d}}x{\rm{d}}y,} }$ (29)
 ${U_{j,n + 1,k}} = \frac{1}{{{h^2}}}\int_{{x_{j - 1/2}}}^{{x_{j + 1/2}}} {\int_{{y_{k - 1/2}}}^{{y_{k + 1/2}}} {{{\left( {v\left( {x,y,{t_{n + 1}}} \right)} \right)}^2}{\rm{d}}x{\rm{d}}y.} }$ (30)

 ${u_{j,n + 1,k}} = {u_{j,n,k}} - \lambda \left( {{{\hat f}_{j + 1/2,n,k}} + {{\hat f}_{j - 1/2,n,k}}} \right) - \lambda \left( {{{\hat g}_{j,n,k + 1/2}} + {{\hat g}_{j,n,k - 1/2}}} \right),$ (31)
 ${U_{j,n + 1,k}} = {U_{j,n,k}} - \lambda \left( {{{\hat F}_{j + 1/2,n,k}} + {{\hat F}_{j - 1/2,n,k}}} \right) - \lambda \left( {{{\hat G}_{j,n,k + 1/2}} + {{\hat G}_{j,n,k - 1/2}}} \right),$ (32)

 ${{\hat f}_{j \pm 1/2,n,k}} = \frac{1}{{h\tau }}\int_{{t_n}}^{{t_{n + 1}}} {\int_{{y_{k - 1/2}}}^{{y_{k + 1/2}}} {av\left( {{x_{j \pm 1/2}},y,t} \right){\rm{d}}y{\rm{d}}t,} }$ (33)
 ${{\hat g}_{j,n,k \pm 1/2}} = \frac{1}{{h\tau }}\int_{{t_n}}^{{t_{n + 1}}} {\int_{{x_{j - 1/2}}}^{{x_{j + 1/2}}} {bv\left( {x,{y_{k \pm 1/2}},t} \right){\rm{d}}\mathit{x}{\rm{d}}t,} }$ (34)
 ${{\hat F}_{j \pm 1/2,n,k}} = \frac{1}{{h\tau }}\int_{{t_n}}^{{t_{n + 1}}} {\int_{{y_{k - 1/2}}}^{{y_{k + 1/2}}} {a{{\left( {v\left( {{x_{j \pm 1/2}},y,t} \right)} \right)}^2}{\rm{d}}y{\rm{d}}t,} }$ (35)
 ${{\hat G}_{j,n,k \pm 1/2}} = \frac{1}{{h\tau }}\int_{{t_n}}^{{t_{n + 1}}} {\int_{{x_{j - 1/2}}}^{{x_{j + 1/2}}} {b{{\left( {v\left( {x,{y_{k \pm 1/2}},t} \right)} \right)}^2}{\rm{d}}\mathit{x}{\rm{d}}t,} }$ (36)

3 数值实验

 $u\left( {x,0} \right) = \left\{ \begin{array}{l} \exp \left\{ { - \frac{1}{{1 - 16{{\left( {x - \frac{1}{2}} \right)}^2}}}} \right\},x \in \left[ {\frac{1}{4},\frac{3}{4}} \right),\\ 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \notin \left[ {\frac{1}{4},\frac{3}{4}} \right). \end{array} \right.$ (37)

 图 1 算例1, 200网格的数值解 Fig.1 Example 1, numerical solution on a grid of 200 cells

 $u\left( {x,0} \right) = \exp \left\{ { - 100 \cdot {{\left( {x - \frac{1}{2}} \right)}^2}} \right\}\sin \left( {80x} \right),0 \le x < 1.$ (38)

 图 2 算例2, 200网格的数值解 Fig.2 Example 2, numerical solution on a grid of 200 cells

 $u\left( {x,y,0} \right) = \left\{ \begin{array}{l} \exp \left\{ { - \frac{1}{{1 - 16{{\left( {x - \frac{1}{2}} \right)}^2}}}} \right\} \times \exp \left\{ { - \frac{1}{{1 - 16{{\left( {y - \frac{1}{2}} \right)}^2}}}} \right\},\\ x \in \left[ {\frac{1}{4},\frac{3}{4}} \right) \times \left[ {\frac{1}{4},\frac{3}{4}} \right),\\ 0,x \notin \left[ {\frac{1}{4},\frac{3}{4}} \right) \times \left[ {\frac{1}{4},\frac{3}{4}} \right). \end{array} \right.$ (39)

 图 3 算例3, 200网格t=20的数值解 Fig.3 Example 3, numerical solution on a grid of 200 cells at t=20

 $\begin{array}{l} u\left( {x,y,0} \right) = \exp \left\{ { - 1000 \cdot {{\left( {x - \frac{1}{2}} \right)}^2}} \right\}\\ \begin{array}{*{20}{l}} {\sin \left( {80x} \right) \cdot \exp \left\{ { - 100 \times {{\left( {y - \frac{1}{2}} \right)}^2}} \right\}\sin \left( {80y} \right),}\\ {\left( {x,y} \right) \in \left[ {0,1} \right) \times \left[ {0,1} \right).} \end{array} \end{array}$ (40)

 图 4 算例4, 200网格t=10的数值解 Fig.4 Example 4, numerical solution on a grid of 200 cells at t=10
4 结论

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