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# 物理代写|空气动力学代写Aerodynamics代考|ASC4551 Weak Form of a Conservation Law

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## 物理代写|空气动力学代写Aerodynamics代考|Weak Form of a Conservation Law

The derivatives appearing in the differential form of a conservation law are not defined at discontinuities. This difficulty can be circumvented by introducing the weak form, in which the differential equation is multiplied by a smooth test function and integrated by parts over space and time to transfer the derivative from the solution to the test function.
Consider the general nonlinear scalar conservation law,
\begin{aligned} \frac{\partial u}{\partial t}+\frac{\partial}{\partial x} f(u) &=0 \ u(x, 0) &=u_0(x) . \end{aligned}
This has the form of a divergence and represents a conservation law for a vector with components $u, f$. Multiply by any smooth function $w(x, t)$, which vanishes for large $x, t$, and integrate over $x$ and $t$ to obtain
$$0=\int_0^{\infty} \int_{-\infty}^{\infty}\left(\frac{\partial u}{\partial t}+\frac{\partial f}{\partial x}\right) w d x d t$$
or
$$0=\int_0^{\infty} \int_{-\infty}^{\infty}\left(u \frac{\partial w}{\partial t}+f \frac{\partial w}{\partial x}\right) d x d t+\int_{-\infty}^{\infty} u_0 w d x$$

## 物理代写|空气动力学代写Aerodynamics代考|Shock Waves

Consider again the example of the inviscid Burgers’ equation (4.10), for which $f(u)=\frac{u^2}{2}$, and
$$[f]=\frac{1}{2}\left(u_R^2-u_L^2\right) .$$
Thus, we obtain the jump condition that a discontinuity propagates at a speed
$$S=\frac{1}{2}\left(u_L+u_R\right),$$
where $u_L$ and $u_R$ are the values to the left and right of the discontinuity yielding the solution illustrated in Figure 4.10. Note that weak solutions satisfying this relationship are not necessarily unique.
In the case that
\begin{aligned} u_0(x) &=0, \quad x \leq 0, \ &=1, \quad x>0, \end{aligned}

one finds that
$u=0, \quad x \leq \frac{t}{2}$,
$=1, \quad x>\frac{t}{2}$
satisfies the differential equation and the jump conditions. This solution, illustrated in Figure 4.12, is an alternative to the expansion fan. To restore uniqueness, we need the additional condition that a discontinuity will only be permitted if the characteristics on both sides converge on the discontinuity, in this case that
$$u_L>S>u_R,$$
where $S$ is the speed of the discontinuity. Such a jump is called a shock. The condition that the characteristics must converge on the jump is called an entropy condition, because in the case of fluid dynamics, it corresponds to the condition that entropy cannot decrease and hence that discontinuous expansions are impossible.

## 物理代写|空气动力学代写空气动力学代考|守恒定律的弱形式

\begin{aligned} \frac{\partial u}{\partial t}+\frac{\partial}{\partial x} f(u) &=0 \ u(x, 0) &=u_0(x) . \end{aligned}

$$0=\int_0^{\infty} \int_{-\infty}^{\infty}\left(\frac{\partial u}{\partial t}+\frac{\partial f}{\partial x}\right) w d x d t$$

$$0=\int_0^{\infty} \int_{-\infty}^{\infty}\left(u \frac{\partial w}{\partial t}+f \frac{\partial w}{\partial x}\right) d x d t+\int_{-\infty}^{\infty} u_0 w d x$$

## 物理代写|空气动力学代写空气动力学代考|冲击波

$$[f]=\frac{1}{2}\left(u_R^2-u_L^2\right) .$$

$$S=\frac{1}{2}\left(u_L+u_R\right),$$

\begin{aligned} u_0(x) &=0, \quad x \leq 0, \ &=1, \quad x>0, \end{aligned}

one发现
$u=0, \quad x \leq \frac{t}{2}$，
$=1, \quad x>\frac{t}{2}$

$$u_L>S>u_R,$$
，其中$S$是不连续的速度。这样的跳跃被称为震动。特性必须在跳跃上收敛的条件被称为熵条件，因为在流体动力学的情况下，它对应的条件是熵不能减少，因此不连续膨胀是不可能的

## Matlab代写

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