C++编程解决偏微分方程

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"Solving PDEs in C++是关于数值计算的经典著作，专注于C++在求解偏微分方程(PDEs)中的应用，旨在为理论应用学习者提供指导。该书适合对计算科学与工程(CS&E)感兴趣的学者和工程师阅读，尤其对依赖于计算科学和工程的专业人士具有很高的参考价值。该系列书籍由Omar Ghattas担任主编，邀请了来自世界各地知名大学和研究机构的专家组成编辑委员会，致力于提供深入的理论分析和实践应用的教材及专业读物。" 在计算科学与工程领域，偏微分方程是描述各种自然现象和工程问题的关键工具，如流体力学、热传导、电磁学、量子力学等。"Solving PDEs in C++"深入探讨如何利用编程语言C++来有效地数值求解这些复杂的数学模型。C++作为一门强类型、静态类型的编程语言，以其高效性、灵活性和面向对象的特性，成为科学计算领域的常用工具。本书可能会涵盖以下关键知识点： 1. **数值方法基础**：介绍有限差分法、有限元方法和有限体积法等常见的数值求解策略，以及如何在C++中实现这些方法。 2. **线性和非线性方程组的求解**：讲解如何使用迭代方法（如高斯-塞德尔、雅可比迭代）以及直接方法（如LU分解）解决线性系统，以及非线性方程组的处理方法，如牛顿-拉弗森法。 3. **矩阵运算和线性代数库**：讨论如何利用BLAS（基本线性代数子程序）和LAPACK（线性代数包）等库进行高效的矩阵操作。 4. **边界条件处理**：阐述如何在C++代码中设置和实施不同类型的边界条件，如Dirichlet、Neumann和Robin条件。 5. **并行计算**：探讨如何利用OpenMP或MPI进行多线程和分布式计算，以加速大规模PDE问题的求解。 6. **误差分析和收敛性**：解释数值解的误差来源，包括离散误差和舍入误差，并分析算法的收敛性质。 7. **软件设计和实践**：介绍良好的编程习惯，如模块化设计、错误处理和性能优化，以创建可维护和可扩展的代码。 8. **案例研究**：通过实际的工程和物理问题，演示如何应用所学方法解决实际的PDE问题，例如计算流体动力学(CFD)问题。 9. **C++编程技巧**：讲解C++的高级特性，如模板、STL容器、智能指针等，以及如何利用它们来提高代码的效率和可读性。 10. **软件工具和库**：可能介绍诸如Trilinos、PETSc、 deal.II等开源软件库，它们为PDE求解提供了便利的框架和工具。 "Solving PDEs in C++"是一本全面的指南，不仅教授如何用C++编程解决偏微分方程，还强调了数值方法背后的理论基础，以及在实际应用中的考虑因素，对想要在计算科学和工程领域深化理解和提升技能的读者来说，是一份宝贵的学习资源。

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xvi List of Figures

9.9 The characteristic line (along which F is integrated to contribute to

i+1/2,j+1/2

at the half time step) has slope dx/dt = u

i,j

> 0 if the state

i,j

on the left travels to the right (a), dx/dt = u

i,j+1

< 0 if the state

i,j+1

on the right travels to the left (b), or dx/dt = 0ifu

i,j

< 0 <u

i,j+1

(c). .....................................233

12.1 The original ﬁnite element e in the mesh. .................263

12.2 The reference element r that is mapped to e by M

.............263

12.3 The uniform ﬁnite-element mesh. .....................266

12.4 The original ﬁnite element e in the uniform mesh. The reference element

r is mapped to e by M

. ..........................267

12.5 The ﬁnite element t onto which the reference element r is mapped by

. .....................................268

12.6 The rotated ﬁnite element. .........................270

12.7 The diffusion coefﬁcients P and Q for the anisotropic diffusion equation. 273

12.8 The stretched ﬁnite-element mesh for the anisotropic diffusion equation. . 274

12.9 The initial coarse mesh that provides a poor approximation to the circular

boundary. ..................................275

12.10 The next, ﬁner, mesh that better approximates the circular boundary. . . . 275

13.1 The hierarchy of objects used to implement an unstructured mesh: the

"mesh" object isa connectedlist of"ﬁniteElement" objects, eachof which

is a list of (pointers to) "node" objects, each of which contains a "point"

object to indicate its location in the Cartesian plane. ...........283

13.2 Schematic representation of inheritance from the base class "connect-

edList" to the derived class "mesh". ....................291

14.1 The initial coarse mesh that approximates the square poorly. .......298

14.2 The second, ﬁner, mesh resulting from one step of local reﬁnement. It is

assumed here that extra accuracy is needed only at the origin. ......298

14.3 The third, yet ﬁner, mesh resulting from the second local-reﬁnement step.

It is assumed here that extra accuracy is needed only at the origin. ....299

14.4 The adaptive-reﬁnement algorithm: the numerical solution obtained by

the multigridpreconditioner at a particular mesh is usedto reﬁne it further

and produce the next level of reﬁnement. .................300

14.5 The original coarse ﬁnite-element mesh. ..................301

14.6 The adaptively reﬁned ﬁnite-element mesh. ................301

14.7 The coarse triangle with vertices A, nI, and nJ (a) is divided into two

smaller triangles by the new line leading from A to nIJ (b). In order to

preserve conformity, its neighbor on the upper right is also divided by a

new line leading from nIJ to B (c) in the "reﬁneNeighbor()" function. . . 303

14.8 Thecoarse meshthatservesasinputfortheadaptive-reﬁnementalgorithm

with automatic boundary reﬁnement. ...................308

14.9 The ﬁner mesh, in which the upper-right triangle is reﬁned and two extra

triangles are also added to better approximate the upper-right part of the

circular boundary. .............................308

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List of Figures xvii

14.10 The nonconvex domain in which the PDE is deﬁned. ...........309

14.11 The original coarse mesh that gives a poor approximation to the noncon-

vex domain. .................................310

14.12 The reﬁned mesh that gives a better approximation to the nonconvex

domain. ...................................310

15.1 The reference triangle in the quadratic ﬁnite-element method. The nodes

are numbered 1, 2, 3, 4, 5, 6. A typical quadratic nodal function assumes

the value 1 at one of these nodes and 0 at the others. ...........315

15.2 The reference triangle in the cubic ﬁnite-element method. The nodes

are numbered 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. A typical cubic nodal function

assumes the value 1 at one of these nodes and 0 at the others. .......316

16.1 Schematic representation of inheritance from the base class "connect-

edList<rowElement>" to the derived class "row". .............329

16.2 Thehierarchyof objects usedtoimplement thesparsematrix: the"sparse-

Matrix" object is a list of "row" objects, each of which is a connected list

of "rowElement" objects, which use the template ’T’ to refer to the type

of value of the matrix elements. ......................334

16.3 Schematic representation of inheritance from the base class "list<row>"

to the derived class "sparseMatrix". ....................335

17.1 The multigrid iteration has the shape of the letter V: ﬁrst, relaxation is

used at the ﬁne grid; then, the residual is transferred to the next, coarser,

grid, where a smaller V-cycle is used recursively to produce a correction

term, which is transferred back to the ﬁne grid; ﬁnally, relaxation is used

again at the ﬁne grid. ............................354

17.2 The "multigrid" object contains the square matrices A, U, and L and the

rectangular matrices R and P , which are used to transfer information to

and from the coarse grid. ..........................356

17.3 The domain decomposition. The bullets denote nodes in the coarse

grid c.....................................363

17.4 The ﬁrst prolongation step, in which the known values at the bullets

are prolonged to the line connecting them by solving a homogeneous

subproblem in the strip that contains this line. ...............363

17.5 The second prolongation step, in which the known values at the bullets

and edges are prolonged to the interior of the subdomain by solving a

homogeneous Dirichlet subproblem. ....................364

17.6 Prolonging to node i by solving a homogeneous Dirichlet–Neumann sub-

problem in the “molecule” of ﬁnite elements that surround it. .......365

18.1 Thecache-oriented relaxation fora tridiagonalsystem: ﬁrstpart, in which

the unknowns in each chunk are relaxed over and over again using data

from the previous relaxation. ........................372

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xviii List of Figures

18.2 The cache-oriented relaxation for a tridiagonal system: second part, in

which the unknowns in each intermediate (interface) chunk are relaxed

over and over again using data from the previous relaxation and the ﬁrst

part described above. ............................372

18.3 Shared-memory architecture: each processor has direct access to the

entire memory. ...............................374

18.4 Distributed-memory architecture: each processor has access only to its

own local memory but not to the memories of other processors. If a

processor needs data from the memory of another processor, then it must

send a request through the communication wires, indicated by straight

lines. ....................................375

18.5 A connectionmachine with four processors: everyprocessor isconnected

to every other processor by a wire. .....................376

18.6 Four processors connected in a line: each processor is connected physi-

cally only to its immediate neighbors. ...................376

18.7 A three-dimensional hypercube (cube) with eight processors in its nodes,

numbered by the binary integers 000 to 111. Each node is connected only

to the three nodes that differ from it by one binary digit only. .......377

18.8 Different kinds of digital computers and architectures. ..........378

18.9 The implementation of the "sparseMatrix" and "dynamicVector" objects

in the low-level code is completely hidden from the algorithms that use

them, so it can be parallelized with no need to change the high-level

code, provided that the low-level interface functions still take and return

the same types of arguments as before. ..................380

18.10 Fair comparison between parallel and sequential computers for a partic-

ular problem: the performance of each computer is estimated with the

algorithm that is most suitable for it in terms of efﬁcient implementation. 382

18.11 Symmetric Gauss–Seidel relaxation: schematic description of the trian-

gular preconditioners L and U as windows through which elements of A

are seen. ..................................384

19.1 Thedomain, diffusioncoefﬁcients,andboundaryconditionsforthepresent

diffusion example. .............................396

19.2 The initial (coarse) mesh that contains only 11 nodes and 9 triangles and

is reﬁned further in the adaptive-reﬁnement algorithm both in the interior

of the domain and at the circular boundary. ................397

19.3 Distribution of nodes in the ﬁrst ﬁve levels of reﬁnement in the diffusion

example. The adaptive reﬁnement uses a threshold of 0.01 and automatic

boundary reﬁnement. The strong diffusion in the lower-left quarter of the

domain prevents large variation, so no extra reﬁnement is needed. ....398

20.1 The staircase in  that leads from 

to (x, y) ∈  in the coercivity

proof. ....................................406

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List of Figures xix

20.2 The adaptive-reﬁnement algorithm for the linear elasticity equations: the

numerical approximations to u and v obtained by the modiﬁed multigrid

preconditionerata particularmesh areusedtoreﬁne it furtherandproduce

the next level of reﬁnement. ........................409

20.3 The boundary segments 

(where Dirichlet boundary conditions are

given) and 

(where Neumann boundary conditions are given) in the

linear elasticity example. ..........................410

20.4 Distribution of nodes at the ﬁrst ﬁve levels of reﬁnement (with automatic

boundary reﬁnement). The Poisson ratio is ν = 1/3. ...........411

21.1 BoththeStokesandthe linear elasticityequationscanbeviewedasspecial

cases of general linear elasticity. ......................419

21.2 Distribution of nodes at the ﬁrst ﬁve levels of reﬁnement (with automatic

boundary reﬁnement). The Poisson ratio increases gradually from 0.9at

the ﬁrst level to 0.99 at the seventh level (11000 nodes). .........420

22.1 The bilinear ﬁnite-element mesh. .....................429

22.2 The reference element for the bilinear ﬁnite-element scheme. .......429

22.3 The measuring problem: ﬁnd the depth of the slit in the object on the left

from the wave reﬂected from it. ......................430

22.4 The bilinear ﬁnite-element mesh for the measuring problem. .......431

22.5 The nearly circular model for the measuring problem. ..........432

22.6 The nonlinear ﬁnite-element mesh in the half-circular domain. ......433

22.7 The wave and Maxwell equations can be reduced to the Helmholtz equa-

tion by a Fourier transform in the time variable t and the assumption

∇·V =−



, respectively. ........................437

A.1 Hierarchy of objects for the convection-diffusion equation in two spatial

dimensions: the "domain2" object uses an "xytGrid" object, which uses

"dynamicVector2" and "difference2" objects. ...............444

A.2 Schematic representation of inheritance from the base class "dynam-

icVector" to the derived class "dynamicVector2". .............445

A.3 Schematic representation of inheritance from the base class "list" (list of

"dynamicVector2" objects) to the derived classes "xytGrid" and "differ-

ence2". ...................................452

A.4 The coarse mesh that contains only 11 nodes and 9 triangles and serves

as input for the adaptive-reﬁnement algorithm to solve the diffusion

problem. ..................................479

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