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== The energy method ==

== The energy method ==

The energy method is a mathematical procedure that can be used to verify well-posedness of initial-boundary-value-problems.<ref>{{cite book |title=High Order Difference Methods for Time Dependent PDE

The energy method is a mathematical procedure that can be used to verify well-posedness of initial-boundary-value-problems.<ref>{{cite book |title=High Order Difference Methods for Time Dependent PDE

|first=Bertil|last=Gustafsson|publisher=Springer|year=2008|isbn=978-3-540-74992-9|doi=10.1007/978-3-540-74993-6}}</ref> In the following example the energy method is used to decide where and which boundary conditions should be imposed such that the resulting IBVP is well-posed. Consider the one-dimensional hyperbolic PDE given by

|first=Bertil|last=Gustafsson|publisher=Springer|year=2008|isbn=978-3-540-74992-9|doi=10.1007/978-3-540-74993-6}}</ref> In the following example the energy method is used to decide where and which boundary conditions should be imposed such that the resulting IBVP is well-posed. Consider the one-dimensional hyperbolic PDE given by

| first Bertil | last Gustafsson | publisher | Springer | year 2008 | isbn 978-3-540-74992-9 | doi 10.1007 / 978-3-540-74993-6} / ref 在下面的例子中，使用能量方法来决定应该施加哪些边界条件，使得得到的 IBVP 适定。考虑一维双曲偏微分方程

| first Bertil | last Gustafsson | publisher | Springer | year 2008 | isbn 978-3-540-74992-9 | doi 10.1007 / 978-3-540-74993-6} / ref 在下面的例子中，使用能量方法来决定应该施加哪些边界条件，使得得到的 IBVP适定。考虑一维双曲偏微分方程

  <math>\frac{\partial u}{\partial t} + \alpha \frac{\partial u}{\partial x} = 0, \quad x \in [a,b], \operatorname t > 0,</math>

  <math>\frac{\partial u}{\partial t} + \alpha \frac{\partial u}{\partial x} = 0, \quad x \in [a,b], \operatorname t > 0,</math>

where <math>\alpha \neq 0</math> is a constant and <math>u(x,t)</math> is an unknown function with initial condition <math>u(x,0) = f(x)</math>. Multiplying with <math>u</math> and integrating over the domain gives

where <math>\alpha \neq 0</math> is a constant and <math>u(x,t)</math> is an unknown function with initial condition <math>u(x,0) = f(x)</math>. Multiplying with <math>u</math> and integrating over the domain gives

其中 math alpha neq 0 / math 是常数，math u (x，t) / math 是未知函数，初始条件是 math u (x，0) f (x) / math。使用 math u / math 进行乘法，并在域上进行集成

其中 <math>\alpha \neq 0</math> 是常数，并且 <math>u(x,t)</math> 是未知函数，初始条件是 <math>u(x,0) = f(x)</math>。乘以 <math>u</math> 并在域上进行积分。

  <math>\int_a^b u \frac{\partial u}{\partial t} \operatorname dx + \alpha \int _a ^b u \frac{\partial u}{\partial x} \operatorname dx = 0.</math>

  <math>\int_a^b u \frac{\partial u}{\partial t} \operatorname dx + \alpha \int _a ^b u \frac{\partial u}{\partial x} \operatorname dx = 0.</math>

  <math>\int _a ^b u \frac{\partial u}{\partial t} \operatorname dx = \frac{1}{2} \frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \quad \text{and} \quad \int _a ^b u \frac{\partial u}{\partial x} \operatorname dx = \frac{1}{2} u(b,t)^2 - \frac{1}{2} u(a,t)^2,.</math>

  <math>\int _a ^b u \frac{\partial u}{\partial t} \operatorname dx = \frac{1}{2} \frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \quad \text{and} \quad \int _a ^b u \frac{\partial u}{\partial x} \operatorname dx = \frac{1}{2} u(b,t)^2 - \frac{1}{2} u(a,t)^2,.</math>

 

where integration by parts has been used for the second relationship, we get

where integration by parts has been used for the second relationship, we get

where integration by parts has been used for the second relationship, we get

where integration by parts has been used for the second relationship, we get

 

  <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 + \alpha u(b,t)^2 - \alpha u(a,t)^2 = 0.</math>

  <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 + \alpha u(b,t)^2 - \alpha u(a,t)^2 = 0.</math>

 

Here <math>\vert \vert \cdot \vert \vert</math> denotes the standard L2-norm.

Here <math>\vert \vert \cdot \vert \vert</math> denotes the standard L2-norm.

在这里，math  vert vert cdot vert math 表示标准的 L2-norm。

在这里，<math>\vert \vert \cdot \vert \vert</math> 表示标准的 L2-正则。

For well-posedness we require that the energy of the solution is non-increasing, i.e. that <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math>, which is achieved by specifying <math>u</math> at <math>x = a</math> if <math>\alpha > 0</math> and at <math>x = b</math> if <math>\alpha < 0</math>. This corresponds to only imposing boundary conditions at the inflow. Note that well-posedness allows for growth in terms of data (initial and boundary) and thus it is sufficient to show that <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math> holds when all data is set to zero.

For well-posedness we require that the energy of the solution is non-increasing, i.e. that <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math>, which is achieved by specifying <math>u</math> at <math>x = a</math> if <math>\alpha > 0</math> and at <math>x = b</math> if <math>\alpha < 0</math>. This corresponds to only imposing boundary conditions at the inflow. Note that well-posedness allows for growth in terms of data (initial and boundary) and thus it is sufficient to show that <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math> holds when all data is set to zero.

For well-posedness we require that the energy of the solution is non-increasing, i.e. that <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math>, which is achieved by specifying <math>u</math> at <math>x = a</math> if <math>\alpha > 0</math> and at <math>x = b</math> if <math>\alpha < 0</math>. This corresponds to only imposing boundary conditions at the inflow. Note that well-posedness allows for growth in terms of data (initial and boundary) and thus it is sufficient to show that <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math> holds when all data is set to zero.

For well-posedness we require that the energy of the solution is non-increasing, i.e. that <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math>, which is achieved by specifying <math>u</math> at <math>x = a</math> if <math>\alpha > 0</math> and at <math>x = b</math> if <math>\alpha < 0</math>. This corresponds to only imposing boundary conditions at the inflow. Note that well-posedness allows for growth in terms of data (initial and boundary) and thus it is sufficient to show that <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math> holds when all data is set to zero.

对于适定性，我们要求解的能量是不增加的，即。部分 t-vert u-vert 2-leq 0 / math 通过在 math x a / math if math alpha 0 / math 和 math x b / math if math alpha 0 / math 中指定数学 u / math 来实现。这只相当于在进水口附加边界条件。注意，适定性允许在数据(初始和边界)方面的增长，因此它足以证明当所有数据设置为零时 math frac { partial t } vert u vert 2 leq 0 / math 持有。

对于适定性，我们要求解的能量是不增加的，即 <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math> ，这种关系可以通过在<math>x = a</math>处 （如果 <math>\alpha > 0</math>） 以及 <math>x = b</math>处 （如果 <math>\alpha < 0</math>）指定<math>u</math>的值来实现。这只相当于在入流处附加边界条件。注意，适定性允许在数据(初始和边界)上的增长，因此它足以表明当所有数据设置为零时应有 <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math>。

==Notes==

==Notes==