方程：一维定常黏性 Burgers

The equation: steady one-dimensional viscous Burgers

本节实验：读图 1 的下图，扩散通量比对流通量小多少？再读图 2，同一个 $u^*$ 在不同 $\nu$ 下需要什么样的源项？

Experiment: In the lower panel of Figure 1, how much smaller is the diffusive flux than the convective one? Then read Figure 2: what source does the same $u^*$ need at different $\nu$?

从守恒律到定常方程

From a conservation law to the steady equation

黏性 Burgers 方程在其他一维页面的 Burgers 方程上增加了分子扩散。量 $u(x,t)$ 沿 $x$ 以自身的值为速度被输运，同时按系数 $\nu>0$ 扩散；写成守恒形式，总通量有两部分——对流通量 $u^2/2$ 与扩散通量 $-\nu u_x$——区间内还有源项 $s(x)$：

The viscous Burgers equation adds molecular diffusion to the Burgers equation of the other one-dimensional pages. A quantity $u(x,t)$ is carried along $x$ with its own value as the speed and spreads by diffusion with the coefficient $\nu>0$. In conservation form the total flux has two parts, the convective flux $u^2/2$ and the diffusive flux $-\nu u_x$, and a source $s(x)$ acts inside the interval:

本页只研究固定正黏性下的定常解。令 $u_t=0$，并记总通量为 $q(u,u_x)=\tfrac12u^2-\nu u_x$，得到要离散的方程；因为含二阶导数，两端各需要一个给定值（无黏 Burgers 页面只给入口条件，不需要出口条件）：

This page studies steady solutions at a fixed positive viscosity. Setting $u_t=0$ and writing $q(u,u_x)=\tfrac12u^2-\nu u_x$ for the total flux gives the equation to be discretised. Because it contains a second derivative it needs a prescribed value at each end (the inviscid Burgers pages, with an inlet condition only, needed none at the outlet):

$\nu>0$ 使解光滑：陡变被扩散成薄层，层宽与 $\nu$ 除以速度（或速度差）同量级。构造解本身没有层；$u>0$ 时这个问题的边界层只能出现在出口 $x=1$（默认设计下层宽 $\nu/u_{\rm out}=0.011$），入口 $x=0$ 处的层属于伴随 $\lambda$（图 4，第 11、16 节与附录 A）。在本页的格式里，网格 Péclet 数 $uh/\nu$ 远小于 1 时离散结果才进入渐近区（第 15 节，数值观察）；粗网格上的大误差主要来自两个不重构的端单元，而不是这个层（第 15、16 节）。激波属于单元中心的一维 Burgers 页的附录 B，$\nu\to0$ 与 $h\to0$ 的极限顺序见它的附录 D（D.4）；本页的 $\nu$ 始终固定。

With $\nu>0$ the solution is smooth: a steep change is spread into a thin layer whose width is of the order of $\nu$ divided by the speed (or the speed jump). The manufactured solution has no layer. With $u>0$ a boundary layer of this problem can only sit at the outlet end $x=1$ (width $\nu/u_{\rm out}=0.011$ at the default design), while the layer at the inlet end $x=0$ belongs to the adjoint $\lambda$ (Figure 4; Sections 11 and 16 and Appendix A). In this page's scheme the discrete results reach their asymptotic range only when the cell Péclet number $uh/\nu$ is well below 1 (Section 15, a numerical observation); the large errors on coarse meshes come mainly from the two unreconstructed end cells, not from the layer (Sections 15 and 16). Shocks belong to Appendix B of the cell-centred 1-D Burgers page and the orders of the limits $\nu\to0$, $h\to0$ to its Appendix D (D.4); here $\nu$ stays fixed.

一个构造解：先给解，再算源项

A manufactured solution: prescribe the solution, derive the source

要把每个导数都和精确答案比较，本页先规定解，再由解构造源项。取 $u^*$ 为在平均值 1 上叠加的正弦波，振幅为 $a$，相位为 $\varphi$；$u^*$ 对每个设计都是精确解。把它代入方程，就得到源项和两端的给定值：

To compare every derivative with an exact answer, the page prescribes the solution first and builds the source from it. Take $u^*$ to be a sine wave on the mean 1, with amplitude $a$ and phase $\varphi$; it is the exact solution for every design. Inserting it into the equation defines the source and the two end values:

解不依赖 $\nu$，源项则依赖：其中 $-\nu u^*_{xx}=4\pi^2\nu a\,\sin z$ 这一项就是源项必须补上的扩散（图 2）。$u^*$ 以 1 为周期，所以两端的给定值相等；默认值下 $u_{\rm in}=u_{\rm out}=0.9148$。

The solution does not depend on $\nu$, but the source does: the term $-\nu u^*_{xx}=4\pi^2\nu a\,\sin z$ is the diffusion that the source has to supply (Figure 2). Because $u^*$ has period 1 the two end values are equal; at the defaults $u_{\rm in}=u_{\rm out}=0.9148$.

设计、目标和它们的精确导数

The design, the objective and their exact derivatives

设计输入是 $D=(a,\varphi,\nu)$，共 3 个分量。$a$ 与 $\varphi$ 同时改变源项和两端的给定值；$\nu$ 改变物理通量本身，同时也改变源项。目标是解的一个加权积分，权重 $1+x$：

The design input is $D=(a,\varphi,\nu)$ with 3 components. Changing $a$ or $\varphi$ changes the source and the two end values together; changing $\nu$ changes the physical flux itself and, through $-\nu u^*_{xx}$, the source. The objective is a weighted integral of the solution with the weight $1+x$:

对构造解可以直接积分，目标和它的三个导数都有闭式（$\nu$ 不出现，因为 $u^*$ 不依赖它）：

For the manufactured solution the integral is exact, so the objective and its three derivatives have closed forms ($\nu$ does not appear because $u^*$ does not depend on it):

默认设计下 $J^*=1.471198$，梯度 $(−0.144008,\ 0.085156,\ 0)$。离散梯度随网格加密趋向的就是这些值（第 15 节的数据显示这一点；这是数值观察，不是证明）。有限网格上的离散目标 $J_h(D)$ 是另一个函数（第 7 节），切线和伴随算出的是它的梯度，不是这个精确梯度；两者的比较见第 10 至 12 节、第 15 节和第 16 节。

At the defaults $J^*=1.471198$ and the gradient is $(−0.144008,\ 0.085156,\ 0)$. These are the values that the discrete gradients approach as the mesh is refined, as the data of Section 15 show (a numerical observation, not a proof). On a finite mesh the discrete objective $J_h(D)$ is a different function (Section 7), and what the tangent and the adjoint compute is its gradient, not this exact one; the two are compared in Sections 10 to 12, 15 and 16.

图 1 的下图回答了实验的第一问：对流通量约 0.3–0.7，扩散通量最大只有 0.0126，不超过对流通量的 4%。黏性几乎不改变解的通量，却决定了第 11、16 节里伴随场的结构；先记住这一点。

The lower panel of Figure 1 answers the first question of the experiment: the convective flux is about 0.3–0.7 and the diffusive flux is at most 0.0126, no more than 4% of it. Viscosity hardly changes the flux of the solution, yet it decides the structure of the adjoint field in Sections 11 and 16; keep that in mind.

自己动手：改变 $a$、$\varphi$、$\nu$，看精确解、两部分通量和源项怎样变，并核对闭式的目标与梯度。

Try it yourself: change $a$, $\varphi$ and $\nu$, see how the exact solution, the two parts of the flux and the source change, and check the closed-form objective and gradient.

先看一眼后面要反复用到的结论：解几乎感觉不到黏性（图 1），但它的灵敏度感觉得到。目标的伴随场 $\lambda$（第 11、12、16 节与附录 A）在 $x=0$ 处有一个宽度约 $\nu/u$ 的边界层，$\nu$ 越小越陡；离散伴随只有在网格分辨这个层时才能重现它的形状（梯度需要的更少，见第 12 节）。

A look ahead at a result used again and again: the solution hardly feels the viscosity (Figure 1), but its sensitivity does. The adjoint field $\lambda$ of the objective (Sections 11, 12, 16 and Appendix A) has a boundary layer of width about $\nu/u$ at $x=0$, steeper for smaller $\nu$; the discrete adjoint reproduces the shape of such a layer only if the mesh resolves it (the gradient needs less, Section 12).

这个方程用来做什么

What this equation is for

黏性 Burgers 方程保留了流体动量方程的非线性对流项 $u\,u_x$，又带上分子黏性项 $-\nu u_{xx}$，是最简单的“对流加扩散”非线性方程。本课程把它放在无黏 Burgers 与一维 Navier–Stokes 之间，原因有五：

The viscous Burgers equation keeps the nonlinear convection term $u\,u_x$ of the momentum equation and adds the molecular viscosity term $-\nu u_{xx}$: it is the simplest nonlinear convection–diffusion equation. The course puts it between the inviscid Burgers pages and the 1-D Navier–Stokes pages, for five reasons:

• 小。默认网格只有 8 个未知量，每个数组和矩阵都能放在页面上，语句播放器也读得过来。Small. The default mesh has only 8 unknowns, so every array and matrix fits on the page and the statement players stay readable.
• 非线性。数值通量对迹的偏导 $c_L=q_L/2+1$ 随状态变化，所以线性化（Jacobian）依赖于它取在哪个状态；切线与伴随都取在收敛的原始解上。Nonlinear. The derivative of the numerical flux with respect to a trace, $c_L=q_L/2+1$, changes with the state, so the linearisation (the Jacobian) depends on where it is taken; the tangent and the adjoint are both evaluated at the converged primal solution.
• 有精确答案可比。构造解给出每个设计下的精确解、精确目标和精确梯度，所以每个导数、每个收敛阶都能和真值比较。Exact answers to compare with. The manufactured solution gives the exact solution, objective and gradient for every design, so every derivative and every convergence rate can be compared with the truth.
• 物理通量依赖导数 $u_x$。面通量要用相邻单元的差商近似 $u_x$，边界面要用半个单元的距离——这正是黏性 Navier–Stokes 页面每个面上要做的事，这里只有一个变量。The physical flux depends on the derivative $u_x$. A face flux needs a difference quotient of neighbouring cells to approximate $u_x$ and a boundary face needs half a cell of distance: exactly what each face of the viscous Navier–Stokes pages does, here with one variable.
• 设计进入物理通量和源项。$a,\varphi$ 改变源项和边界值，$\nu$ 改变通量本身；第 12 节逐项分清这三条路径。The design enters the physical flux and the source. $a,\varphi$ change the source and the boundary values, $\nu$ changes the flux itself; Section 12 separates the three paths one by one.

本页的数值、图和表来自本页的代码（模型在构建页面时与独立的 Python 实现逐项对照，第 15 节的大网格数据来自 Python 程序）。数值检查是验证证据，不是数学证明；本页不声称非线性方程或格式的存在性、唯一性或收敛定理（本页给出的证明列在附录 C 的 C.4）。

The numbers, figures and tables of this page come from the page's own code (the model is held to an independent Python implementation when the page is built, and the large-grid data of Section 15 come from the Python program). Numerical checks are verification evidence, not mathematical proofs; the page claims no existence, uniqueness or convergence theorem for the nonlinear equation or for the scheme (the proofs it does give are listed in C.4 of Appendix C).

网格：未知量与数组

The mesh: unknowns and arrays

本节实验：数一数，$N=8$ 个单元有几个面？每个内部面和每个边界面各读哪些数据？先答，再对照图 5。

Experiment: Count the faces of $N=8$ cells: which data does each interior face and each boundary face read? Answer first, then compare with Figure 5.

第 1 节给出了方程。本节把区间 $[0,1]$ 分成 $N$ 个等宽的单元，宽度 $h=1/N$，并说明页面存哪些数据。

Section 1 gave the equation. This section divides $[0,1]$ into $N$ equal control volumes (cells) of width $h=1/N$ and says what the page stores.

单元、面和未知量

Cells, faces and unknowns

单元 $i=0,\dots,N-1$ 的中心是 $x_i=(i+\tfrac12)h$，它夹在面 $i$ 与面 $i+1$ 之间；面 $f=0,\dots,N$ 在 $x_f=fh$。未知量是每个单元上解的平均值：

Cell $i=0,\dots,N-1$ has its centre at $x_i=(i+\tfrac12)h$ and lies between face $i$ and face $i+1$; face $f=0,\dots,N$ is at $x_f=fh$. The unknown is the average of the solution over each cell:

对构造解，精确单元平均有闭式：正弦在一个单元上的平均是 $\operatorname{sinc}(h)=\sin(\pi h)/(\pi h)$ 倍的中心值，所以

For the manufactured solution the exact cell average has a closed form: the average of the sine over a cell is $\operatorname{sinc}(h)=\sin(\pi h)/(\pi h)$ times its value at the centre, so

状态的误差始终拿离散解和这个精确平均比较，而不是和中心点的采样值比较（采样值只是 Newton 迭代的初值）。

The errors of the state are always measured between the discrete solution and this exact average, never against point samples at the centres (the samples are only the initial guess of the Newton iteration).

先看网格是怎样一步一步搭起来的：

First see how the mesh is built, step by step:

选一个面，看它读哪些单元；再换一种重构，数一数未知量的个数：

Select a face and see which cells it reads; then change the reconstruction and count the unknowns:

记号

Notation

默认设置：$N=8$，无限制器线性重构，$D=(a,\varphi,\nu)=(0.2,\ 0.07,\ 0.01)$，即页首控件的初始值；图和文字没有另注时都取默认设置。

Default settings: $N=8$, unlimited linear reconstruction and $D=(a,\varphi,\nu)=(0.2,\ 0.07,\ 0.01)$, the initial values of the controls at the top; the figures and the text use them unless another setting is stated.

按伴随优化文献的通常写法，设计变量记作 $D=(a,\varphi,\nu)$，分量 $D_0=a$、$D_1=\varphi$、$D_2=\nu$。残差是 $R(\mathbf U,D)\in\mathbb R^N$，状态 Jacobian 是 $A=R_{\mathbf U}$，设计 Jacobian 是 $R_D$，离散目标是 $J_h$，伴随变量是 $\psi$。

As is usual in the adjoint-optimization literature, the design variables are written $D=(a,\varphi,\nu)$, with components $D_0=a$, $D_1=\varphi$, $D_2=\nu$. The residual is $R(\mathbf U,D)\in\mathbb R^N$, the state Jacobian is $A=R_{\mathbf U}$, the design Jacobian is $R_D$, the discrete objective is $J_h$ and the adjoint variable is $\psi$.

切线（前向，tangent）与伴随（反向，adjoint）一律用自动微分的记号：程序变量 $x$ 的切线写 $\dot x$，伴随写 $\bar x$，伴随累加写 $\bar x\mathrel{\text{+=}}\dots$，伴随语句按逆序执行；$\dot x$ 与 $\bar x$ 一一对应 Tapenade 等工具里的变量 x_d 与 x_b，本页的伪代码就用这些名字。变分符号 $\delta$ 只出现在附录 A 的连续推导里（连续函数的变分）；Newton 与 GS 的迭代增量写 $\Delta$。

The tangent (forward) and the adjoint (reverse) always use the notation of automatic differentiation: the forward derivative of a program variable $x$ is $\dot x$, its adjoint is $\bar x$, an adjoint accumulation is written $\bar x\mathrel{\text{+=}}\dots$ and the statements run in reverse order. They correspond one to one to the variables x_d and x_b of AD tools such as Tapenade, and the pseudocode of this page uses these names. The variation symbol $\delta$ appears only in the continuous derivation of Appendix A (variations of continuous functions); the increments of the Newton and GS iterations are written $\Delta$.

有两个与黏性有关的量，不要混淆：物理黏性 $\nu$（设计分量 $D_2$，进入物理通量 $-\nu u_x$）；Rusanov 数值通量里的耗散（第 3 节），其波速 $\alpha=2$ 是固定常数，一阶格式里它等效于随网格变化的人工黏性 $h$，不是设计变量。

Two quantities that both look like a viscosity must not be confused: the physical viscosity $\nu$ (the design component $D_2$, which enters the physical flux $-\nu u_x$) and the dissipation of the Rusanov numerical flux (Section 3), whose speed $\alpha=2$ is a fixed constant and which acts at first order as an artificial viscosity $h$ that changes with the mesh; it is not a design variable.

本页的主要符号

The main symbols of this page

下标的约定：$i$ 总是单元，$f$ 总是面；数字下标写在 $x$ 上时指单元中心（$x_4=4.5\,h$ 是单元 4 的中心），面坐标写 $x_f$（面 4 在 $x_f=4h$）。伪代码里的数组 s[i] 存斜率 $\sigma_i$，它不是第 1 节的源项 $s(x)$。

Subscript convention: $i$ always names a cell and $f$ a face; a number written on $x$ names a cell centre ($x_4=4.5\,h$ is the centre of cell 4), and face coordinates are written $x_f$ (face 4 is at $x_f=4h$). In the pseudocode the array s[i] stores the slope $\sigma_i$; it is not the source $s(x)$ of Section 1.

默认网格上的数组（$N=8$）

The arrays on the default mesh ($N=8$)

权重 $w_i$ 是第 7 节离散目标 $J_h=\sum_iw_iU_i$ 的系数，也是状态梯度 $g=w$（目标对状态的导数；伴随方程 $A^T\psi=-g$ 的右端是 $-g$）。

The weights $w_i$ are the coefficients of the discrete objective $J_h=\sum_iw_iU_i$ of Section 7 and also the state gradient $g=w$ (the right-hand side of the adjoint equation $A^T\psi=-g$ is $-g$).

后面各节要用到的所有数组和它们的形状都在图 6 里：左边是向量，右边是矩阵（颜色表示正负与大小，蓝正橙负）。它们都是本页在默认设计 $D$ 和 $N=8$ 下真实算出的量；$A$ 是 $N\times N$ 的五对角矩阵，$R_D$ 只有 3 列，所以一次伴随求解（右端项 $-g$）给出梯度的全部 3 个分量，而切线求解要做 3 次。

All the arrays that the later sections use, with their shapes, are in Figure 6: vectors on the left, matrices on the right (colour shows sign and size: blue positive, orange negative). They are real quantities of this page at the default design $D$ and $N=8$: $A$ is an $N\times N$ five-band matrix and $R_D$ has only 3 columns, so one adjoint solve (right-hand side $-g$) gives all 3 design components while the tangent solves must be done 3 times.

面上的通量：对流、黏性与重构

The flux on a face: convection, viscosity and reconstruction

本节实验：选定一个面和一种重构，这个面读哪几个单元？两个迹 $q_L,q_R$ 哪个等于单元平均？先预测，再用本节末尾的面板验证。

Experiment: Select a face and a reconstruction: which cells does the face read, and which of the traces $q_L,q_R$ equals a cell average? Predict first, then check with the panel at the end of this section.

第 2 节把区间分成了单元。本节把方程写成单元上的通量差，用数值通量把面两侧的数据变成一个面通量（对流部分和黏性部分），再说明二阶格式怎样把单元平均重构成面上的状态。

Section 2 divided the interval into control volumes. This section writes the equation as a flux difference over a control volume, turns the data on the two sides of a face into one face flux (a convective part and a viscous part), and then shows how the second-order scheme reconstructs face states from the cell averages.

从方程到通量差

From the equation to a flux difference

第 1 节的方程是 $q_x=s$，$q=\tfrac12u^2-\nu u_x$。在单元 $i$（面 $i$ 到面 $i+1$）上积分，由微积分基本定理，通量的导数变成两端通量之差，等于单元内的总源项：

The equation of Section 1 is $q_x=s$ with $q=\tfrac12u^2-\nu u_x$. Integrate it over cell $i$ (face $i$ to face $i+1$): by the fundamental theorem of calculus the derivative of the flux becomes the difference of the end fluxes, which equals the total source in the cell:

对构造解，源项是 $s=q^*_x$，所以一个单元的总源项就是精确通量 $Q_f=q^*(x_f;D)$ 在这个单元两个面上的差，不需要数值积分；第 5 节的残差用的就是这个精确的积分源项。离散化要做的只剩一件事：用两侧单元的数据近似每个面上的通量。

For the manufactured solution the source is $s=q^*_x$, so the total source of a cell is exactly the difference of the exact flux $Q_f=q^*(x_f;D)$ at its two faces, with no numerical quadrature; the residual of Section 5 uses this exact integrated source. What remains for the discretisation is to approximate the flux at each face from the data of the cells on its two sides.

一个面上的数值通量：对流加黏性

The numerical flux on a face: convective plus viscous

面 $f$ 的通量分两部分，用不同的输入：对流部分用面两侧重构出的状态 $q_L,q_R$（下面叫“迹”），黏性部分用两侧的未重构状态 $U_L,U_R$（边界面上其中一个是给定值）和它们之间的距离 $d_f$：

The flux of face $f$ has two parts that use different inputs: the convective part uses the states $q_L,q_R$ reconstructed on its two sides (called traces below) and the viscous part uses the raw states $U_L,U_R$ (on a boundary face one of them is the prescribed value) and the distance $d_f$ between them:

本页其他地方（包括本节开头的公式），$q$ 与 $q^*$ 表示总通量；面上的重构状态沿用其他一维页面的写法 $q_L,q_R$。

Elsewhere on this page, including the first display of this section, $q$ and $q^*$ denote the total flux; for the face states the page follows the other 1-D pages and writes $q_L,q_R$.

对流部分是 Rusanov 通量：两侧物理通量 $F(u)=u^2/2$ 的平均，减去与状态跳跃成比例的耗散。耗散项里的波速 $\alpha=2$ 取固定值，不随状态或设计变化，所以求导时没有隐藏的波速项（通常的波速 $\max(|q_L|,|q_R|)$ 在 $|q_L|=|q_R|$ 处不可导，用它构成的 Rusanov 通量在 $q_L=-q_R\ne0$ 处不可导）；控制面板允许的所有设计与网格上，收敛状态的迹都满足 $|q|<2$。Rusanov 通量满足相容性 $H^c(u,u)=F(u)$，并且同一个 $H_f$ 离开左边的单元、进入右边的单元，使内部通量在全局求和中抵消。黏性部分是物理通量 $-\nu u_x$ 的中心差分近似：内部面 $d_f=h$，边界面 $d_f=h/2$（第 4 节）。

The convective part is the Rusanov flux: the average of the physical fluxes $F(u)=u^2/2$ of the two sides, minus a dissipation proportional to the jump of the state. The dissipation speed $\alpha=2$ is a fixed number that depends on neither the state nor the design, so differentiation hides no wave-speed term (the usual wave speed $\max(|q_L|,|q_R|)$ is not differentiable where $|q_L|=|q_R|$, and the Rusanov flux built with it is not differentiable where $q_L=-q_R\ne0$); on all designs and meshes that the controls allow, the converged states have traces below 2 in magnitude, $|q|<2$. It is consistent, $H^c(u,u)=F(u)$, and the same $H_f$ leaves the cell on the left and enters the cell on the right, so the interior fluxes cancel in a global sum. The viscous part is the central-difference approximation of the physical flux $-\nu u_x$: $d_f=h$ on interior faces and $d_f=h/2$ on boundary faces (Section 4).

耗散项 $-(\alpha/2)(q_R-q_L)$ 是人工黏性，它的作用可以写成精确的恒等式（下面的面板在当前状态上再验算一遍）。一阶时迹等于单元平均，内部行 $j$ 的通量差 $H_{j+1}-H_j=R_j+S_j$ 是

The dissipation term $-(\alpha/2)(q_R-q_L)$ is artificial viscosity, and what it does can be written as exact identities (the panel below checks them again on the current state). At first order the traces equal the cell averages and the flux difference of an interior row $j$, $H_{j+1}-H_j=R_j+S_j$, is

这正是物理黏性 $\nu$ 再加人工黏性 $h$ 的中心格式（除以 $h$ 后是 $u\,u_x-(\nu+h)\,u_{xx}$ 的中心差分）：一阶格式在内部行上就是黏性 $\nu+h$ 的中心格式。默认 $N=8$ 时 $h=0.125$，是 $\nu=0.01$ 的 12.5 倍；$h$ 要小到与 $\nu$ 同量级（这里 $N\approx100$），人工黏性才不再主导。无限制器线性重构时，在两侧单元都做了重构的面上（$2\le f\le N-2$），耗散项是三阶差分：

This is exactly the central scheme for the physical viscosity $\nu$ plus an artificial viscosity $h$ (divided by $h$ it is the central difference of $u\,u_x-(\nu+h)\,u_{xx}$): on interior rows the first-order scheme is the central scheme for the viscosity $\nu+h$. At the default $N=8$ one has $h=0.125$, that is 12.5 times $\nu=0.01$; $h$ has to become comparable with $\nu$ (about $N\approx100$ here) before the artificial viscosity stops dominating. With unlimited linear reconstruction, on a face whose two neighbouring cells are both reconstructed ($2\le f\le N-2$) the dissipation of the face is a third difference:

行 $j$（$2\le j\le N-3$，即两个面之差）的耗散是四阶差分，对光滑数据是 $O(h^4)$：

and that of a row $2\le j\le N-3$ (the difference of two faces) is a fourth difference, $O(h^4)$ for smooth data:

所以远离端点时二阶格式没有二阶导数形式的人工黏性，只有四阶的超黏性；但在面 $0,1,N-1,N$ 上，有一侧的迹没有重构，耗散约为 $-\tfrac h2u_x$，相当于 $h/2$ 的人工黏性（第 4 节）。一阶格式的人工黏性 $h$ 是两种阶数的主要差别，第 15 节（状态与梯度的误差）和第 16 节（伴随）把这个差别量出来。光滑限制器时权重随状态变化，没有这样的闭式。

So away from the ends the second-order scheme has no artificial viscosity of second-derivative form, only a fourth-order hyperviscosity; on faces $0,1,N-1,N$ one trace is not reconstructed and the dissipation is about $-\tfrac h2u_x$, an artificial viscosity $h/2$ (Section 4). The artificial viscosity $h$ of the first-order scheme is the main difference between the two orders, and Section 15 (the errors of the state and the gradient) and Section 16 (the adjoint) put that difference in numbers. With the smooth limiter the weights change with the state and there is no such closed form.

重构：从单元平均到面上的状态

Reconstruction: from cell averages to face states

每个单元用一个斜率 $\sigma_i$ 重构成一条线段，线段在单元两端的值就是相邻面的迹。斜率由两个单侧差 $l_i=U_i-U_{i-1}$、$r_i=U_{i+1}-U_i$ 决定（$\sigma_i$ 是穿过整个单元的变化量，不除以 $h$）；两端的单元不重构，$\sigma_0=\sigma_{N-1}=0$：

Each cell is reconstructed as a line segment with one slope $\sigma_i$, and the values of the segment at the two ends of the cell are the traces of the neighbouring faces. The slope is determined by the two one-sided differences $l_i=U_i-U_{i-1}$ and $r_i=U_{i+1}-U_i$ ($\sigma_i$ is the change across the whole cell, not divided by $h$); the two end cells are not reconstructed, $\sigma_0=\sigma_{N-1}=0$:

公式中的 first、none、smooth 是三种重构在代码里的名字，分别对应控制面板中的“一阶”“无限制器线性（二阶）”“光滑限制器”。“none”是无限制器的中心斜率，二阶精度，但不保证单调；“smooth”是 van Albada 型的光滑有理斜率：把它写成中心斜率乘一个因子，$\sigma_i=\dfrac{l_i+r_i}2\cdot\dfrac{2l_ir_i}{l_i^2+r_i^2+\epsilon^2}$，因子严格落在 $(-1,1)$ 内。$l_i,r_i$ 同号、相近且远大于 $\epsilon$ 时因子接近 1，斜率回到中心斜率；有一个差分为零时因子为 0；在极值处（$l_ir_i<0$）因子为负，斜率很小并且与中心斜率反向，而不是被截成零。这个限制器没有硬开关，$\epsilon$ 使它光滑，所以斜率处处可微；本页不声称它具有 TVD 或保正性。

In the formula, “first”, “none” and “smooth” are the code names of the three reconstructions (First order, Unlimited linear (second order) and Smooth limiter in the control panel). The slope “none” is the unlimited central slope: second-order accurate but not monotone. The slope “smooth” is the van Albada form, smooth because of $\epsilon$: written as the central slope times a factor, $\sigma_i=\dfrac{l_i+r_i}2\cdot\dfrac{2l_ir_i}{l_i^2+r_i^2+\epsilon^2}$, where the factor lies strictly inside $(-1,1)$. It is close to 1 when $l_i,r_i$ have the same sign, are close to each other and are much larger than $\epsilon$, so the slope returns to the central one; it is 0 when one difference vanishes; at an extremum ($l_ir_i<0$) it is negative, so the slope is small and points against the central slope instead of being cut to zero. There is no hard switch, so the slope is differentiable everywhere; the page claims no TVD or positivity property for it.

无限制器线性重构时，迹是未知量的固定线性组合：内部面 $q_L=-\tfrac14U_{f-2}+U_{f-1}+\tfrac14U_f$、$q_R=\tfrac14U_{f-1}+U_f-\tfrac14U_{f+1}$，所以一个内部面依赖 4 个未知量（紧邻边界的面是 3 个，边界面是 1 个）；一阶时只依赖 2 个。光滑限制器时这些权重随状态变化。

With unlimited linear reconstruction the traces are fixed linear combinations of the unknowns: on an interior face $q_L=-\tfrac14U_{f-2}+U_{f-1}+\tfrac14U_f$ and $q_R=\tfrac14U_{f-1}+U_f-\tfrac14U_{f+1}$, so an interior face depends on 4 unknowns (3 next to a boundary, 1 on a boundary face) and only on 2 at first order. With the smooth limiter these weights change with the state.

先用滑块看斜率：给定两个单侧差，三种重构在中间单元里画出什么线段。

First look at the slope with the sliders: given the two one-sided differences, what segment does each reconstruction draw in the middle cell.

下面的播放器把这个面的计算拆成七步，每一步写出当时的数值；这七步就是第 5 节、第 8 节的四条语句（取值、重构、通量、分发），只是把迹和通量的两个部分分开显示。按“播放”或用步进按钮，换面、换重构再看一遍。

The player below splits the computation of this face into seven steps and writes the numbers of each step; the seven steps are the four statements gather, reconstruct, flux and scatter of Section 5 and Section 8, with the traces and the two parts of the flux shown separately. Press Play or use the step buttons, and change the face and the reconstruction to see it again.

一个面上的局部导数

Local derivatives on one face

第 8、9 节的切线与伴随，在面上只用到三类局部导数：数值通量对迹的偏导、对未重构状态的偏导，以及迹对未知量的偏导（此外还有通量对设计 $\nu$ 的偏导，以及边界值和精确通量对设计的偏导）。数值通量是多项式，逐项求导：

On a face, the tangent and the adjoint of Sections 8 and 9 use only three kinds of local derivative: those of the numerical flux with respect to the traces, those with respect to the raw states, and those of the traces with respect to the unknowns (plus the derivative of the flux with respect to the design $\nu$ and those of the boundary values and the exact fluxes with respect to the design). The numerical flux is a polynomial, so it is differentiated term by term:

有两个性质可以检验这些公式。相容性 $H^c(u,u)=u^2/2$ 沿 $q_L=q_R=u$ 求导，得 $c_L+c_R=u$：两个偏导把波速分成上游和下游两份。当 $|q_L|,|q_R|<2$ 时 $c_L>0>c_R$：通量随左状态增加、随右状态减小。默认设计的各面上（无限制器线性）$c_L$ 在 1.374 与 1.565 之间，$c_R$ 在 −0.625 与 −0.426 之间。黏性偏导 $\pm\kappa_f$ 是常数，内部面 $\kappa_f=\nu/h=0.080$。

Two properties check these formulas. Consistency $H^c(u,u)=u^2/2$ differentiated along $q_L=q_R=u$ gives $c_L+c_R=u$: the two partial derivatives split the wave speed into an upstream and a downstream share. For $|q_L|,|q_R|<2$ one has $c_L>0>c_R$: the flux increases with the left state and decreases with the right one. On the faces of the default design (unlimited linear) $c_L$ lies between 1.374 and 1.565 and $c_R$ between −0.625 and −0.426. The viscous partials $\pm\kappa_f$ are constants, $\kappa_f=\nu/h=0.080$ on interior faces.

迹对未知量的导数在无限制器线性重构下就是上面的权重，与状态无关；光滑限制器时多一项：$q_L=U_{f-1}+\tfrac12\sigma_{f-1}$ 的导数含 $\partial\sigma/\partial l,\ \partial\sigma/\partial r$，它们随状态变化（线性重构时它们是常数 $\tfrac12,\tfrac12$）。图 8 左：$c_L,c_R$ 随状态变化，灰带是本页各面迹的范围；右：每个面的通量对每个未知量的导数 $\partial H_f/\partial U_j$，一阶时每个内部面涉及 2 个未知量，无限制器线性时涉及 4 个。

The derivative of a trace with respect to the unknowns is the weight above and does not depend on the state for unlimited linear reconstruction; with the smooth limiter there is one more term: the derivative of $q_L=U_{f-1}+\tfrac12\sigma_{f-1}$ contains $\partial\sigma/\partial l,\ \partial\sigma/\partial r$, which change with the state (for linear reconstruction they are the constants $\tfrac12,\tfrac12$). Figure 8 left: $c_L,c_R$ against the state, the grey band being the range of the traces of this page; right: the derivative of every face flux with respect to every unknown, $\partial H_f/\partial U_j$; an interior face touches 2 unknowns at first order and 4 with unlimited linear reconstruction.