MIT 18.086课程：Matlab程序mit18086 navierstokes在数值流体力学中的应用

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在MIT的课程18.086：计算科学与工程II中，"数值计算 mit18086 navierstokes.m" 是一个用于教学的Matlab程序，其核心内容是应用Navier-Stokes方程来解决流体力学问题，特别是与 Computational Science and Engineering 相关的主题。该程序可能是作为理论物理中的一个实用工具，让学生了解如何通过数值方法模拟复杂的流体动力学现象。 Navier-Stokes方程是描述流体运动的基本方程，它涉及速度场、压力和密度等物理量，广泛应用于航空航天、天气预报、生物流体动力学等领域。在数值计算中，这些方程通常通过有限差分、有限元或边界元素方法等数值求解技术来近似求解。在这个特定的课程中，学生们可能会学习到如何将这些理论概念转化为实际的代码实现，以及如何通过编程解决实际问题，如模拟黑洞附近的时空动力学或者宇宙尺度的流体运动。介绍部分提到了"Miguel Alcubierre"的工作，他是Univernidad Nacional Autónoma de México的研究员，主要研究领域包括数值相对论。他的文章"Introduction to 3+1 Numerical Relativity"（2006年）对理解数值模拟在广义相对论中的应用有重要意义，该文章介绍了如何将爱因斯坦场方程转化为数值形式，以便于计算机处理。Alcubierre还参与了名为"虫洞、曲速驱动与能量条件"的相关项目，展示了他对前沿物理学理论的贡献。课程参与者可能借此机会学习如何处理高维空间中的复杂几何结构，以及如何利用坐标系和协变导数的概念来处理时空坐标下的物理量变化。此外，他们还会接触到Bianchi恒等式和守恒定律在数值模拟中的应用，这些是确保数值解决方案符合物理原理的关键组成部分。总而言之，"数值计算 mit18086 navierstokes.m" 这个程序不仅涵盖了基础的流体力学和广义相对论概念，还强调了实际的数值技术在解决复杂问题上的重要性，对于理解现代计算科学在天体物理学、宇宙学和工程领域的应用具有重要价值。通过这个课程，学生能够提升他们的编程技能、数学建模能力和科学实践能力。

equation that expresses the vector ~v a s a linear combination of the basis vectors {~e

} with

coeﬃcients v

The coordinate basis, thoug h not unique, is the most commonly used, and the expres-

sions for the Riemann tensor given in the previous section are precisely those obta ined in

this basis.

Consider now the derivative of a vect or with respect to a given coordinate:

∂~v

∂x

∂

∂x





∂v

∂x

+ v

∂~e

∂x

. (1.30)

This equation shows that the derivative of a vector is more that just the derivative of its

components. One must also take into account the change in the basis vectors.

The derivative ∂~e

/∂x

is in itself also a vector, which means that it can be expressed

as a linear combination of the basis vectors. We int r oduce the symbols Γ

αβ

to denote the

coeﬃcients of such linear combination:

∂~e

∂x

= Γ

αβ

. (1.31)

The coeﬃcients Γ

αβ

turn out to be precisely the Christoﬀel symbols deﬁned above.

Using these coeﬃcients we ﬁnd

∂~v

∂x

∂v

∂x

+ v

αβ

. (1.32)

Changing now the name of the summed (dummy) indices, we can rewrite this as

∂~v

∂x

∂v

∂x

+ v

αµ

. (1.33)

This equation gives us the components of the vector ∂~v/∂x

. We now deﬁne the “covariant

derivative” o f the vector ~v as:

;β

≡ ∇

∂v

∂x

+ v

βµ

. (1.34)

Note that we have introduced two alternative notations for the covariant derivative: In

one case we use a semicolon, and in the other the nabla symbol. Both notat io ns are

common and are used indistinctively.

The covariant derivative tells us how the components of a vecto r change a s we move in

a general space, and includes the change in the basis vectors themselves. Notice that the

covariant derivative reduces to the partial derivative when the Christoﬀel symbols vanish,

which happens in ﬂat space with Cartesian coordinates but not with spherical coordinates.

However, it is always possible to ﬁnd coordinates such that the Christoﬀel symbols vanish

at a given point (but not at other points unless the space is ﬂat). This is because every

curved space is “locally ﬂat”: in an inﬁnitesimal region around an arbitrary point the

geometry can be approximated by the ﬂat one (that is why ﬂat maps of a city are usually

very accurate, while ﬂat maps of t he whole planet introduce serious distortions).

The covariant derivative of a vector with the indices down v

(a “covariant” vector or

simply “co-vector”) can be shown to be given by:

α;β

∂v

∂x

− Γ

αβ

. (1.35)

The same concept of covariant derivative can also be applied to tensors with many

indices, the rule is to add a term with Christoﬀel symbols for each free index, with the

adequate sign depending on whether the index is down or up. For example:

µν

;α

= ∂

µν

+ Γ

αβ

βν

+ Γ

αβ

µβ

µν; α

= ∂

µν

− Γ

αµ

βν

− Γ

αν

µβ

ν;α

= ∂

+ Γ

αβ

− Γ

αν

Using this rule one can show that the covaria nt derivative o f the metric tensor is in

fact always zero:

µν; α

= 0 , g

µν

;α

= 0 , (1.36)

which implies that the operation of raising and lowering indices commutes with the co-

variant derivative:

;α

= (g

µν

)

;α

= g

µν

ν;α

) . (1.37)

1.5 The Einstein ﬁeld equations

The element we are still missing from Einstein’s theory of gravity is the one that relates

the geometry of space-time with the distribution of mass and energy. This ﬁnal element

is contained in the Einstein ﬁeld equations, or simply Einstein’s equations. These equa-

tions can be derived in several diﬀerent ways, either looking for a consistent relativistic

generalization of Newton’s theory (the road followed by Einstein), or deriving them in a

formal way from a variational principle staring with an adequate Lagrangian (the road

followed by Hilbert). Here I will limit myself to writing the equations down. In their most

compact form, the Einstein equations are:

µν

= 8πT

µν

, (1.38)

where G

µν

is the so-called “ Einstein tensor” that is related with the Ricci curvature t ensor

(see below), and T

µν

is the “stress-energy tensor” of matter. That is, the left hand side

represents the geometry of space-time, and the right hand side the distribution of mass-

energy. The factor 8π is simply an adequate normalization necessary to obtain the correct

Newtonian limit. Note that t hese are in fact 10 equations, as the indices µ and ν take

values from 0 to 3, that is 4 × 4 = 16, and the Einst ein and stress-energy tensors are

symmetric, reducing the number of independent equations to only 10.

The Einstein equations that we have just int r oduced could not appear to be more

simple. This simplicity, however , is only apparent since each term is in fact a short-hand

for considerably more complex objects. Written in their most general way, in an ar bitrar y

coordinate system, and with all the terms expanded out, the Einstein equations become

a system of 10 coupled non-linear partial diﬀerent ia l equations with thousands of terms.

Let us consider each term in Einstein’s equations separately. The Einstein tensor is

deﬁned in terms of the Ricci tensor as:

µν

:= R

µν

−

µν

R , (1.39)

with R := g

µν

the trace of the Ricci tensor, also know as the “scalar curva ture”.

The other side of the Einstein equations, the stress-energy tensor, describes the energy

density, momentum density and momentum ﬂux of matter (i, j = 1, 2, 3):

= energy density , (1.40)

= momentum density , (1.41)

= ﬂux of i momentum in direction j. (1.42)

For example, for a perfect ﬂuid with no pressure (known as “dust” in relativity) and in

ﬂat space we have:

= ρ/(1 − v

) , (1.43)

= ρ v

/(1 − v

) , (1.44)

= ρ v

/(1 − v

) , (1.45)

where ρ is the energy density measured in the rest frame of the ﬂuid, v

is the velocity

ﬁeld of the ﬂuid, and v

= v

+ v

It is important to mention that in r elativity, all ﬁelds expect gravity itself are con-

sidered as a type of matter. For example, an electromagnetic ﬁeld has an associated

stress-energy tensor that a cts as the source of gravity, i.e. electric and magnetic ﬁelds

produce gravity.

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