数值数学：科学计算的方法与应用

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"Numerical mathematics (2nd Edition Springer, 2007)" 是一本关于数值数学的学术著作，该书深入探讨了科学计算中所使用的理论和方法，这些方法在多个数学领域如分析、线性代数、几何、近似理论、函数方程、优化和微分方程中都有应用。数值数学对于物理、自然科学、生物科学、工程、经济以及金融科学等领域的问题解决至关重要。本书是"TextsinAppliedMathematics"系列的一部分，由知名学者编辑，旨在为读者提供实用数学领域的深度见解。书中可能涵盖了以下核心主题： 1. 数值分析：这是数值数学的基础，涉及如何用数值方法近似解析解，包括求解微分方程、积分方程和线性系统的方法，例如欧拉方法、龙格-库塔法、高斯积分等。 2. 线性代数：讨论矩阵运算、特征值问题、特征向量、线性方程组的求解（如高斯消元法、迭代法）以及数值稳定性分析。 3. 几何与逼近理论：涉及曲线和曲面的表示、插值与拟合，如多项式插值、样条函数、最小二乘法等。 4. 微分方程：介绍了常微分方程和偏微分方程的数值解法，如有限差分、有限元素法、谱方法等。 5. 功能方程与优化：可能包含函数方程的数值解法以及最优化问题的求解策略，如梯度下降法、牛顿法、动态规划等。 6. 科学计算的应用：展示了数值方法在物理学、生物学、工程、医学和金融科学中的实际应用，强调了跨学科的重要性。书中列举的其他相关书籍则涵盖了非线性动力系统、流体动力学、控制理论、偏微分方程等多个专题，为读者提供了更广泛的参考资源，帮助他们深入理解和掌握数值方法在实际问题中的应用。 "Numerical mathematics (2nd Edition Springer, 2007)" 是一本全面且深入的教材，适合对数值方法感兴趣的学者和专业人士，以及需要使用数值技术解决复杂问题的学生和研究人员。通过学习本书，读者可以系统地掌握数值计算的原理和技巧，提高在相关领域的实践能力。

Contents XVII

12.3 The Spectral Collocation Method . . . . . . . . . . . . . . . . . . . . . . . . 550

12.4 The Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552

12.4.1 Integral Formulation of Boundary Value Problems . . 552

12.4.2 A Quick Introduction to Distributions . . . . . . . . . . . . 554

12.4.3 Formulation and Properties of the Galerkin

Method .......................................555

12.4.4 Analysis of the Galerkin Method . . . . . . . . . . . . . . . . . 556

12.4.5 The Finite Element Method . . . . . . . . . . . . . . . . . . . . . 558

12.4.6 Implementation Issues . . . . . . . . . . . . . . . . . . . . . . . . . . 564

12.4.7 Spectral Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566

12.5 Advection-Diﬀusion Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 568

12.5.1 Galerkin Finite Element Approximation . . . . . . . . . . . 569

12.5.2 The Relationship between Finite Elements and

Finite Diﬀerences; the Numerical Viscosity . . . . . . . . 572

12.5.3 Stabilized Finite Element Methods . . . . . . . . . . . . . . . 574

12.6 A Quick Glance at the Two-Dimensional Case . . . . . . . . . . . . 580

12.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583

12.7.1 Lubrication of a Slider . . . . . . . . . . . . . . . . . . . . . . . . . . 583

12.7.2 Vertical Distribution of Spore Concentration over

Wide Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584

12.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

13 Parabolic and Hyperbolic Initial Boundary Value

Problems .................................................589

13.1 The Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

13.2 Finite Diﬀerence Approximation of the Heat Equation . . . . . 591

13.3 Finite Element Approximation of the Heat Equation . . . . . . . 593

13.3.1 Stability Analysis of the θ-Method................595

13.4 Space-Time Finite Element Methods for the Heat

Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

13.5 Hyperbolic Equations: A Scalar Transport Problem . . . . . . . . 604

13.6 Systems of Linear Hyperbolic Equations . . . . . . . . . . . . . . . . . . 607

13.6.1 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608

13.7 The Finite Diﬀerence Method for Hyperbolic Equations . . . . 609

13.7.1 Discretization of the Scalar Equation . . . . . . . . . . . . . 610

13.8 Analysis of Finite Diﬀerence Methods . . . . . . . . . . . . . . . . . . . . 611

13.8.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612

13.8.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612

13.8.3 The CFL Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613

13.8.4 Von Neumann Stability Analysis . . . . . . . . . . . . . . . . . 615

13.9 Dissipation and Dispersion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618

13.9.1 Equivalent Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 619

13.10 Finite Element Approximation of Hyperbolic Equations . . . . 624

13.10.1 Space Discretization with Continuous and

DiscontinuousFinite Elements ...................625

4 1 Foundations of Matrix Analysis

Example 1.1 Remarkable instances of vector spaces are:

- V = R

(respectively V = C

): the set of the n-tuples of real (respectively

complex) numbers, n ≥ 1;

- V = P

: the set of polynomials p

(x)=



k=0

with real (or complex)

coeﬃcients a

having degree less than or equal to n, n ≥ 0;

- V = C

([a, b]): the set of real (or complex)-valued functions which are contin-

uous on [a, b]uptotheirp-th derivative, 0 ≤ p<∞. •

Deﬁnition 1.2 We say that a nonempty part W of V is a vector subspace of

V iﬀ W is a vector space over K. 

Example 1.2 The vector space P

is a vector subspace of C

∞

(R), which is the

space of inﬁnite continuously diﬀerentiable functions on the real line. A trivial sub-

space of any vector space is the one containing only the zero vector. •

In particular, the set W of the linear combinations of a system of p vectors

of V , {v

,...,v

}, is a vector subspace of V , called the generated subspace or

span of the vector system, and is denoted by

W = span {v

,...,v

}

= { v = α

+ ...+ α

with α

∈ K, i =1,...,p}.

The system {v

,...,v

} is called a system of generators for W .

If W

,...,W

are vector subspaces of V , then the set

S = {w : w = v

+ ...+ v

with v

∈ W

,i=1,...,m}

is also a vector subspace of V . We say that S is the direct sum of the subspaces

if any element s ∈ S admits a unique representation of the form s =

+ ...+ v

with v

∈ W

and i =1,...,m. In such a case, we shall write

S = W

⊕ ...⊕ W

Deﬁnition 1.3 A system of vectors {v

,...,v

} of a vector space V is called

linearly independent if the relation

+ α

+ ...+ α

= 0

with α

,α

,...,α

∈ K implies that α

= α

= ... = α

= 0. Otherwise,

the system will be called linearly dependent. 

We call a basis of V any system of linearly independent generators of V .If

,...,u

} is a basis of V , the expression v = v

+...+v

is called the

decomposition of v with respect to the basis and the scalars v

,...,v

∈ K are

the components of v with respect to the given basis. Moreover, the following

property holds.

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