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# 物理代写|空气动力学代写Aerodynamics代考|MAE351 Fundamental Stability Theory

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## 物理代写|空气动力学代写Aerodynamics代考|Fundamental Stability Theory

This chapter examines the stability of difference schemes for initial value problems defined by ordinary or partial differential equations. Three simple examples are examined first: an ordinary differential equation, the linear advection equation, which is the prototype for hyperbolic equations, and the diffusion equations. These serve to illustrate that in all three cases the numerical scheme can become unstable if the time step is too large, or in the case of a hyperbolic equation, if the difference scheme does not contain the proper region of dependence. The use of implicit schemes can remove the limit on the time step, at the expense of greater solution complexity.

Next, the general definitions of consistency, convergence, and stability are introduced in terms of an arbitrary norm, leading to the Lax equivalence theorem that consistent and stable schemes must converge to the true solution in the limit as the mesh interval and time step are reduced toward zero. Then stability in the Euclidean norm is examined, and the von Neumann stability test is introduced as a convenient way to deduce the stability of any linear scheme.

Nonlinear conservation laws generally admit solutions containing discontinuities, such as shock waves in a fluid flow. This motivates the need for difference schemes in conservation form. Moreover, the linear stability theory is no longer adequate, and schemes that pass a von Neumann test can easily admit highly oscillatory solutions in the neighborhood of shock waves. Thus alternative measures of stability are needed, leading to the introduction of concepts such as total variation diminishing (TVD) and local extremum diminishing (LED) schemes. Generally it proves necessary to use upwind-biased schemes to meet these criteria, either by direct construction or through the introduction of a controlled amount of artificial diffusion.

The simplest example of an initial value problem for a hyperbolic equation is provided by the linear advection equation
$$\frac{\partial u}{\partial t}+a \frac{\partial u}{\partial x}=0, \quad t>0, \quad-\infty \leq x \leq \infty,$$
which defines the solution $u(x, t)$ for a given initial condition
$$u(x, 0)=f(x) .$$
For convenience assume that a is positive. Then this equation describes a simple wave motion in which the solution is transported to the right at a speed $a$. Along a line $x-a t=\xi$, as illustrated in Figure 4.4, we have
$$\frac{d u}{d t}=\frac{d}{d t} u(a t+\xi, t)=a \frac{\partial u}{\partial x}+\frac{\partial u}{\partial t}=0 .$$
Thus $u$ is constant along this line, which is a characteristic – that is a line along which $\frac{\partial}{\partial t}=a \frac{\partial}{\partial x}$ is an internal operator – giving no information about derivatives normal to the line. This means that

1. $u$ cannot be arbitrarily specified along such a line, because $u$ is constant according to the equation
2. data given along the line is not sufficient to continue the solution into a larger region.
The general solution of (4.5) has the form
$$u(x, t)=f(\xi)=f(x-a t) .$$
This represents the solution uniquely in terms of the initial values. Conversely every $u$ of the form (4.6) is a solution of (4.5) provided it is differentiable, as can be verified by substituting $f(x-a t)$ for $u(x, t)$. Note that $u(x, t)$ depends only on previous values along the characteristic passing through the point $(x, t)$ and hence only on $f(x)$ at a single point on the axis.
Consider now a difference approximation with interval $\Delta x, \Delta t$ of the form
$$\frac{v(x, t+\Delta t)-v(x, t)}{\Delta t}+a \frac{v(x+\Delta x, t)-v(x, t)}{\Delta x}=0 .$$

## 物理代写|空气动力学代写空气动力学代考|线性平流方程

$$\frac{\partial u}{\partial t}+a \frac{\partial u}{\partial x}=0, \quad t>0, \quad-\infty \leq x \leq \infty,$$
，它定义了给定初始条件的解$u(x, t)$
$$u(x, 0)=f(x) .$$

$$\frac{d u}{d t}=\frac{d}{d t} u(a t+\xi, t)=a \frac{\partial u}{\partial x}+\frac{\partial u}{\partial t}=0 .$$

1. $u$不能沿着这条直线任意指定，因为根据
2. 公式，$u$是常数，沿着这条直线给出的数据不足以将解延续到更大的区域。
(4.5)的通解形式为
$$u(x, t)=f(\xi)=f(x-a t) .$$
这表示了根据初值唯一的解。相反，形式(4.6)的每个$u$都是(4.5)的解，只要它是可微的，这可以通过将$f(x-a t)$替换为$u(x, t)$来验证。注意，$u(x, t)$只依赖于经过$(x, t)$点的特征的先前值，因此只依赖于轴上单个点上的$f(x)$。现在考虑一个具有区间$\Delta x, \Delta t$的差分逼近形式
$$\frac{v(x, t+\Delta t)-v(x, t)}{\Delta t}+a \frac{v(x+\Delta x, t)-v(x, t)}{\Delta x}=0 .$$

## MATLAB代写

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