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 $\nu$?

本节的内容与离散格式无关：方程、构造解、设计和目标对格心页（adjoint_burgers_viscous_cell.html）同样成立，两页解的是同一个问题。区别只在未知量放在哪里：那里在单元中心，本页在网格节点上（第 2 节）。两种格式给出不同数值之处，各节会指出并量出来。

Nothing in this section depends on the discretisation: the equation, the manufactured solution, the design and the objective are those of the cell-centred page (adjoint_burgers_viscous_cell.html), which solves the same problem. The two pages differ only in where the unknowns live: at the cell centres there, at the mesh nodes here (Section 2). Where the two schemes give different numbers, the sections say so and measure it.

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）。网格 Péclet 数 $\mathrm{Pe}_h=u\,h/\nu$ 就是网格间距与这个层宽 $\nu/u$ 之比：$\mathrm{Pe}_h<1$ 表示一个网格间距比层还窄（第 3 节把它与格式的人工黏性联系起来）。激波属于一维无黏 Burgers 页（格点页与它的格心页）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). The mesh Péclet number $\mathrm{Pe}_h=u\,h/\nu$ is the mesh spacing in units of that layer width $\nu/u$: $\mathrm{Pe}_h<1$ means that one interval is narrower than the layer (Section 3 relates it to the artificial viscosity of the scheme). Shocks belong to Appendix B of the inviscid one-dimensional Burgers page (the node-centred page and its cell-centred partner) 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；本页里它是作用在节点值上的梯形公式，第 2 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; on this page it is the trapezoid rule applied to the nodal values, Section 2), 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：对流通量在 0.32 与 0.72 之间，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。上图里，每个节点一个点：8+1 个未知量中两端的两个恰好等于给定值，其余的点离空心圆的最远距离是 0.0278（$N=8$ 的粗网格；这个距离随网格加密怎样缩小，见第 15 节）。

The lower panel of Figure 1 answers the first question of the experiment: the convective flux lies between 0.32 and 0.72 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. In the upper panel there is one dot per node: of the 8+1 unknowns the two at the ends equal the prescribed values exactly, and the other dots lie at most 0.0278 from their rings (a coarse mesh, $N=8$; how this distance shrinks under refinement is measured in Section 15).

用页首的控件改变 $N$、重构和设计，下面的面板马上给出离散解、精确节点值、每个节点上的误差，以及离散目标的误差由哪两部分组成：

Change $N$, the reconstruction and the design with the controls at the top, and the panel below shows at once the discrete solution, the exact nodal values, the error at every node, and the two parts that the error of the discrete objective is made of:

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；再用下面的播放器，一步一步看源项是怎样由 $u^*$ 造出来的。

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; then use the player below to watch the source being built from $u^*$ step by step.

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 $\psi$（第 11 节）只定义在节点上：在内部节点上它近似 $\lambda(x_j)$（图中的圆圈），而只有节点落进层里，它才能表示这个层的形状。默认网格（$N=8$，$\nu=0.01$）上，第一个内部节点 $x_1=0.125$ 远在层外：$\psi_1=−1.709$，而 $\lambda(x_1)=−1.416$，相差 0.29；其余内部节点上的差都不超过 0.093。（两端的 $\psi_0$、$\psi_N$ 是两个约束行的乘子，不是 $\lambda$ 的值，图里不画；第 11 节说明它们是什么。）

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 $\psi$ (Section 11) is defined at the nodes only: at the interior nodes it approximates $\lambda(x_j)$ (the circles in the figure), and it can show the shape of such a layer only if nodes fall inside it. On the default mesh ($N=8$, $\nu=0.01$) the first interior node $x_1=0.125$ is far outside the layer: $\psi_1=−1.709$ while $\lambda(x_1)=−1.416$, a difference of 0.29; at the other interior nodes the difference is at most 0.093. (The end values $\psi_0$, $\psi_N$ are the multipliers of the two constraint rows, not values of $\lambda$, so they are not drawn; Section 11 explains what they are.)

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 9 个未知量（8+1 个节点值，其中两个由边界约束给定），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 9 unknowns (8+1 nodal values, two of which the boundary constraints fix), 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 $u_x$。面通量要用面两侧节点值的差商近似 $u_x$，间距在每个面上都是 $h$，边界面也不例外——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 the two nodal values on its sides to approximate $u_x$, with the distance $h$ on every face, the boundary faces included: 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。本页证明了的恒等式和性质，在它们出现的地方标明。

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 identities and properties it does prove are marked where they occur.

网格：节点与对偶控制体

The mesh: nodes and dual volumes

本节实验：数一数，$N=8$ 的网格有几个节点、几个对偶控制体、几个面？每个内部面和每个边界面各读哪些节点？哪两个控制体只有一半宽？先答，再对照图 5。

Experiment: Count the nodes, the dual volumes and the faces of the mesh with $N=8$ intervals. Which nodes does each interior face and each boundary face read, and which two dual volumes have only half the width? Answer first, then compare with Figure 5.

第 1 节给出了方程。本节把区间 $[0,1]$ 分成 $N$ 个等宽的区间，宽度 $h=1/N$，把未知量放在区间的端点（网格节点）上，让每个节点管着它周围的一段对偶控制体，再说明页面存哪些数据。

Section 1 gave the equation. This section divides $[0,1]$ into $N$ equal intervals of width $h=1/N$, puts the unknowns at the interval end points (the mesh nodes), gives each node the dual volume around it as its control volume, and says what the page stores.

节点、对偶控制体和面

Nodes, dual volumes and faces

节点 $j=0,\dots,N$ 在 $x_j=jh$。未知量 $U_j$ 是解在这个节点上的点值，一共 $N+1$ 个，两端的节点也在其中：

Node $j=0,\dots,N$ is at $x_j=jh$. The unknown $U_j$ is the point value of the solution at that node; there are $N+1$ of them, the two end nodes included:

每个节点管着它周围的一段，即对偶控制体 $V_j=[x_j-\tfrac h2,\,x_j+\tfrac h2]\cap[0,1]$，宽度是 $W_j=h$（$1\le j\le N-1$），两端的 $V_0$ 与 $V_N$ 只有 $W_0=W_N=h/2$。控制体之间的界面是面：面 $f$（$f=0,\dots,N-1$）在节点 $f$ 与节点 $f+1$ 的中点 $x_{f+1/2}=(f+\tfrac12)h$，它是 $V_f$ 的右面，也是 $V_{f+1}$ 的左面。一共 $N$ 个面，而且每个面两侧都有节点，最外面的两个面也不例外：没有面要去读给定值或半个间距的数据，第 3 节的面通量在边界面上没有特殊情形。

Each node controls the stretch around it, its dual volume $V_j=[x_j-\tfrac h2,\,x_j+\tfrac h2]\cap[0,1]$, of width $W_j=h$ ($1\le j\le N-1$); the two end volumes $V_0$ and $V_N$ have only $W_0=W_N=h/2$. The interfaces between the volumes are the faces: face $f$ ($f=0,\dots,N-1$) is the midpoint $x_{f+1/2}=(f+\tfrac12)h$ of the interval between node $f$ and node $f+1$; it is the right face of $V_f$ and the left face of $V_{f+1}$. There are $N$ faces and every one of them has a node on each side, the two outermost included: no face reads a prescribed value or half an interval of data, and the face flux of Section 3 has no special case at a boundary face.

两个端点节点的未知量在向量里，但它们那一行的方程不是通量平衡：第 $0$ 行和第 $N$ 行是边界约束 $U_0=u_{\rm in}$、$U_N=u_{\rm out}$（第 4 节）。所以 $N+1$ 个未知量对应 $N-1$ 个通量平衡（内部节点的对偶控制体）加 2 个约束。本页的格式不需要穿过 $x=0$ 与 $x=1$ 的通量，因为约束已经给出了端点值，它也从不计算这样的通量。

The two end nodes are unknowns of the vector, but the equations of their rows are not flux balances: row $0$ and row $N$ are the boundary constraints $U_0=u_{\rm in}$, $U_N=u_{\rm out}$ (Section 4). So the $N+1$ unknowns are matched by $N-1$ flux balances (the dual volumes of the interior nodes) plus 2 constraints. The scheme needs no flux through the ends $x=0$ and $x=1$ of the interval, because the constraints already fix the end values, and it never computes one.

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:

选一个面或一个节点，看它读哪些未知量；再换一种重构，数一数：

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

未知量的精确值：节点值，不是平均值

The exact values of the unknowns: nodal values, not means

对构造解，未知量 $U_j$ 的精确值就是解在节点上的取值，不需要任何平均：

For the manufactured solution the exact value of the unknown $U_j$ is simply the solution at the node, with no averaging:

状态的误差始终拿离散解和这些精确节点值比较（它们也是 Newton 迭代的初值）。格心页的未知量是另一种对象：单元 $i$ 上解的平均值。对这个正弦解，振荡部分的平均值是它在单元中心处的值乘以 $\operatorname{sinc}(h)=\sin(\pi h)/(\pi h)$，等于把振幅从 $a$ 缩小为 $a\,\operatorname{sinc}(h)$：

The errors of the state are always measured between the discrete solution and these exact nodal values (which are also the initial guess of the Newton iteration). The unknown of the cell-centred page is a different object: the mean value of the solution over a cell $i$. For this sine wave the mean of the oscillating part over a cell is its value at the cell centre times $\operatorname{sinc}(h)=\sin(\pi h)/(\pi h)$, which reduces the amplitude from $a$ to $a\,\operatorname{sinc}(h)$:

默认网格上 $\operatorname{sinc}(h)=0.9745$，振幅缩小 2.6%，平均值与中心值最多差 $a(1-\operatorname{sinc}h)=0.0051$；$N=16$ 与 $32$ 时是 0.0013 与 0.00032，按 $h^2$ 缩小（图 6）。所以同一个 $u^*$ 在两页里给出的未知量向量并不相同，两页的状态误差是离各自精确向量（这里是采样值，那里是平均值）的距离：同一个 $N$ 下两个状态误差不是同一个量的误差。两页都定义的量，如目标 $J_h$ 和它的梯度，则可以直接比较。

At the defaults $\operatorname{sinc}(h)=0.9745$, a reduction of the amplitude by 2.6%, and a mean differs from the value at the centre by at most $a(1-\operatorname{sinc}h)=0.0051$; at $N=16$ and $32$ it is 0.0013 and 0.00032, shrinking like $h^2$ (Figure 6). So the same $u^*$ gives different unknown vectors on the two pages, and each page measures its state errors from its own exact vector (samples here, means there): two state errors at the same $N$ are not errors of one quantity. What both pages define, such as the objective $J_h$ and its gradient, can be compared directly.

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；图和文字除另有说明外都取默认设置。

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 $R(\mathbf U,D)\in\mathbb R^{N+1}$（两端两行是约束），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 $R(\mathbf U,D)\in\mathbb R^{N+1}$ (the two end rows are the constraints), 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 $\delta$ 只出现在连续推导里（附录 A 与第 16 节中的连续推导：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 continuous derivations (Appendix A and the continuous derivation of Section 16: 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：$j$ 总是节点（以及它的对偶控制体 $V_j$），$f$ 总是面；$x_j=jh$ 是节点 $j$ 的坐标，面 $f$ 在 $x_{f+1/2}=(f+\tfrac12)h$，夹在节点 $f$ 与 $f+1$ 之间（$N=8$ 时面 $4$ 在 $x_{9/2}=0.5625$）。伪代码里的数组 s[j] 存斜率 $\sigma_j$，它不是第 1 节的源项 $s(x)$。

Subscript convention: $j$ always names a node (and its dual volume $V_j$) and $f$ a face; $x_j=jh$ is the coordinate of node $j$, and face $f$ is at $x_{f+1/2}=(f+\tfrac12)h$, between nodes $f$ and $f+1$ (at $N=8$ face $4$ is at $x_{9/2}=0.5625$). In the pseudocode the array s[j] stores the slope $\sigma_j$; 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 ($N=8$)

第 $0$ 行和第 $N$ 行的 $U_j$ 等于给定值 $u_{\rm in}$、$u_{\rm out}$（约束行使它们在舍入误差内精确），其余是通量平衡的解。权重 $w_j$ 是第 7 节离散目标 $J_h=\sum_jw_jU_j$ 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 $-g$）。它们正是 $\int_0^1(1+x)u\,\mathrm dx$ 的梯形公式的权重：每个节点按它的对偶控制体的宽度计数，两端是半宽（图 7）。

The values $U_j$ in rows $0$ and $N$ equal the prescribed $u_{\rm in}$, $u_{\rm out}$ (the constraint rows make them exact up to rounding) and the others solve the flux balances. The weights $w_j$ are the coefficients of the discrete objective $J_h=\sum_jw_jU_j$ of Section 7 and also the state gradient $g=w$ (the right-hand side of the adjoint equation $A^T\psi=-g$ is $-g$). They are exactly the weights of the trapezoid rule for $\int_0^1(1+x)u\,\mathrm dx$: each node counts with the width of its dual volume, half of it at the two ends (Figure 7).

作用在精确节点值上，$J_h(u^*)$ 就是 $(1+x)u^*$ 的梯形公式，它与 $J^*$ 的差可以写出来：梯形公式的误差是 $\tfrac{h^2}{12}\bigl[f'\bigr]_0^1+O(h^4)$，而 $f=(1+x)u^*$ 的导数在两端的差是 $f'(1)-f'(0)=u^*_x(0)$（$u^*$ 以 1 为周期，只剩权重 $1+x$ 的那一项）：

Applied to the exact nodal values, $J_h(u^*)$ is the trapezoid rule for $(1+x)u^*$, and its difference from $J^*$ can be written down: the error of the trapezoid rule is $\tfrac{h^2}{12}\bigl[f'\bigr]_0^1+O(h^4)$, and for $f=(1+x)u^*$ the jump of the derivative between the two ends is $f'(1)-f'(0)=u^*_x(0)$ ($u^*$ has period 1, so only the part from the weight $1+x$ is left):

默认设置下 $u^*_x(0)=1.1370$，$J_h(u^*)=1.4726944$，$J^*=1.4711985$，差 $1.496\times10^{−3}$，公式给出 $1.481\times10^{−3}$（比值 1.0104）。这是 $J_h$ 本身的离散误差：即使状态是精确的，离散目标也比 $J^*$ 大约差这么多，它在第 15、16 节里作为梯度误差的一部分重新出现。（这个公式是用 Euler–Maclaurin 公式推出的，本页对它做了数值核对，没有给出严格证明。）

At the defaults $u^*_x(0)=1.1370$, $J_h(u^*)=1.4726944$ and $J^*=1.4711985$; the difference is $1.496\times10^{−3}$ and the formula gives $1.481\times10^{−3}$ (ratio 1.0104). This is the discretisation error of $J_h$ itself: even with an exact state the discrete objective differs from $J^*$ by about this much, and it reappears in Sections 15 and 16 as part of the gradient error. (The formula follows from the Euler–Maclaurin formula; the page checks it numerically and gives no rigorous proof of it.)

后面各节要用到的所有数组和它们的形状都在图 8 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 $(N+1)\times(N+1)$ 的五对角矩阵，第 $0$ 行与第 $N$ 行是单位行（约束），$R_D$ 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 8: 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 $(N+1)\times(N+1)$ five-band matrix whose rows $0$ and $N$ are identity rows (the constraints), 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，这个面读哪几个节点？四个量 $q_L,q_R,U_L,U_R$ 里哪些恰好等于节点值？先预测，再用本节的“一个面的量”面板（以及第 2 节的“一个面、一个节点读哪些数据”面板）验证。

Experiment: Select a face and a reconstruction: which nodes does the face read, and which of the four quantities $q_L,q_R,U_L,U_R$ are nodal values? Predict first, then check with the panel “The quantities of one face” of this section (and “What a face and a node read” in Section 2).

第 2 节在区间上放好了节点和它们的对偶控制体。本节把方程写成对偶控制体上的通量差，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由节点值重构出面上的状态，以及这些公式精确满足的性质：耗散的恒等式、通量的相容性、导数与模板。

Section 2 placed the nodes and their dual volumes. This section writes the equation as a flux difference over a dual volume, turns the data on the two sides of a face into one face flux (a convective part and a viscous part), then shows how the second-order scheme reconstructs face states from the nodal values and what the formulas satisfy exactly: the identities of the dissipation, the consistency of the flux, the derivatives and the stencils.

从方程到通量差

From the equation to a flux difference

第 1 节的方程是 $q_x=s$，$q=\tfrac12u^2-\nu u_x$。在内部节点 $j$（$1\le j\le N-1$）的对偶控制体 $V_j$ 上积分，它的两个面在 $x_{j\pm1/2}$：由微积分基本定理，通量的导数变成两端通量之差，等于控制体内的总源项：

The equation of Section 1 is $q_x=s$ with $q=\tfrac12u^2-\nu u_x$. Integrate it over the dual volume $V_j$ of an interior node $j$ ($1\le j\le N-1$), whose two faces are at $x_{j\pm1/2}$: 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 volume:

对构造解，源项是 $s=q^*_x$，所以一个控制体里的总源项就是精确通量 $Q_f$ 在它两个面上的差，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。两个端点的控制体 $V_0$、$V_N$ 不做平衡：它们的行是边界约束（第 4 节），所以本页的格式没有 $x=0$、$x=1$ 处的通量。离散化要做的只剩一件事：用面两侧节点的数据近似每个面上的通量（图 9）。

For the manufactured solution the source is $s=q^*_x$, so the total source of a volume is exactly the difference of the exact flux $Q_f$ at its two faces, with no numerical quadrature; the residual of Section 5 uses this exact integrated source. The two end volumes $V_0$, $V_N$ do no balance: their rows are the boundary constraints (Section 4), so the scheme has no flux through $x=0$ or $x=1$. What remains for the discretisation is to approximate the flux at each face from the data of the nodes on its two sides (Figure 9).