1.1 Basics of Computational Complexity

多场耦合流体力学的基础理论与模拟引言多场耦合流体力学是研究多个相互耦合的流体现象的学科领域。

它涉及流体动力学、传热学、质量传递等多个学科的交叉，是工程、地球科学和生命科学等领域中重要的研究内容。

本文将介绍多场耦合流体力学的基础理论和模拟方法，以帮助读者对该领域有更深入的了解。

1. 多场耦合流体力学的基本概念1.1 流体力学的基本方程流体力学是研究液体和气体运动以及与固体的相互作用的学科。

它基于质量守恒、动量守恒和能量守恒的基本方程，通过数学建模描述流体的运动规律。

在多场耦合流体力学中，我们需要考虑多个被耦合的流体场，如流速场、温度场、浓度场等。

1.2 多场耦合流体力学的耦合机制多场耦合流体力学的耦合机制主要包括热-动力耦合、热-声耦合、热-质量传递耦合等。

这些耦合机制的存在导致了多场耦合流体力学中的各个场之间相互影响，从而使得整个系统的行为表现出复杂的非线性特性。

2. 多场耦合流体力学的基础理论2.1 基于连续介质力学的多场耦合模型在多场耦合流体力学的研究中，广泛采用连续介质力学的基本假设，将流体视为连续不可压缩介质。

通过对连续介质的质量守恒、动量守恒和能量守恒方程进行耦合，可以建立多场耦合流体力学的数学模型。

2.2 多场耦合流体力学的守恒方程多场耦合流体力学的守恒方程是研究多个流体场相互作用的重要工具。

守恒方程可以描述不同流体场之间的质量、动量和能量传递过程，进而揭示系统的运动规律和耦合机制。

2.3 多场耦合流体力学的边界条件在多场耦合流体力学的数值模拟中，边界条件的设定是十分重要的。

合理的边界条件可以保证模拟结果的准确性和可靠性。

通过对不同流体场的物理性质和边界条件进行分析和建模，可以得到合适的边界条件。

2.4 多场耦合流体力学的数值求解方法多场耦合流体力学的数值求解方法是研究多场耦合流体问题的关键。

常用的求解方法包括有限元法、有限差分法和有限体积法等。

这些方法可以通过离散化流体力学方程，将其转化为代数方程组，再通过迭代求解数值解。

A central issue in assessing the practicality of an algorithm is the relative amount of time it takes to execute the algorithm. Typically, one cannot be sure that one has found the most efficient algorithm for a particular function. The most that one can say is that for a particular algorithm, the level of effort for execution is of a particular order of magnitude. One can then compare that order of magnitude to the speed of current or predicted processors to determine the level of practicality of a particular algorithm.A common measure of the efficiency of an algorithm is its time complexity. We define the time complexity of an algorithm to be f(n) if, for all n and all inputs of length n, the execution of the algorithm takes at most f(n) steps. Thus, for a given size of input and a given processor speed, the time complexity is an upper bound on the execution time.There are several ambiguities here. First, the definition of a step is not precise. A step could be a single operation of a Turing machine, a single processor machine instruction, a single high-level language machine instruction, and so on. However, these various definitions of step should all be related by simple multiplicative constants. For very large values of n, these constants are not important. What is important is how fast the relative execution time is growing.A second issue is that, generally speaking, we cannot pin down an exact formula for f(n). We can only approximate it. But again, we are primarily interested in the rate of change of f(n) as n becomes very large.There is a standard mathematical notation, known as the "big-O" notation, for characterizing the time complexity of algorithms that is useful in this context. The definition is as follows: f(n) = O(g(n)) if and only if there exist two numbers a and M such that|f(n)| ≤a× |g(n)|,n≥M(1)An example helps clarify the use of this notation. Suppose we wish to evaluate a general polynomial of the form:P(x ) = a n x n + a n –1x n –1 + . . . + a 1x + a 0Consider the following simple-minded algorithm from [POHL81]:algorithm P1;n, i, j: integer; x, polyval: real;a, S: array [0..100] of real;beginread(x, n);for i := 0 upto n dobeginS[i] := 1; read(a[i]);for j := 1 upto i do S[i] := x × S[i];S[i] := a[i] × S[i]end;polyval := 0;for i := 0 upto n do polyval := polyval + S[i];write ('value at', x, 'is', polyval)end.In this algorithm, each subexpression is evaluated separately. Each S[i ] requires (i + 1)multiplications: i multiplications to compute S[i ] and one to multiply by a[i ]. Computing all n terms requiresi +1()i =0n ∑=n +2()n +1()2multiplications. There are also (n + 1) additions, which we can ignore relative to the much larger number of multiplications. Thus, the time complexity of this algorithm is f(n ) = (n + 2)(n + 1)/2.We now show that f(n ) = O(n 2). From the definition of Equation (1), we want to show that for a= 1 and M = 4, the relationship holds for g(n ) = n 2. We do this by induction on n . Therelationship holds for n = 4 because (4 + 2)(4 + 1)/2 = 15 < 42 = 16. Now assume that it holds for all values of n up to k [i.e., (k + 2)(k + 1)/2 < k 2]. Then, with n = k + 1:n +2()n +1()2=k +3()k +2()2=k +2()k +1()2+k +2≤k 2+k +2≤k 2+2k +1=k +1()2=n 2Therefore, the result is true for n = k + 1.In general, the big-O notation makes use of the term that grows the fastest. For example:1.O[ax 7 + 3x 3 + sin(x )] = O(ax 7) = O(x 7)2.O(e n + an 10) = O(e n )3.O(n ! + n 50) = O(n !)There is much more to the big-O notation, with fascinating ramifications. For the interested reader, two of the best accounts are in [GRAH94] and [KNUT97].An algorithm with an input of size n is said to be:1. Linear: if the running time is O(n )2. Polynomial: if the running time is O(n t ) for some constant t3. Exponential: if the running time is O(t h(n )) for some constant t and polynomial h(n )Generally, a problem that can be solved in polynomial time is considered feasible, whereas anything larger than polynomial time, especially exponential time, is considered infeasible. But you must be careful with these terms. First, if the size of the input is small enough, even very complex algorithms become feasible. Suppose, for example, that you have a system that can execute 1012 operations per unit time. Table 1 shows the size of input that can be handled in one time unit for algorithms of various complexities. For algorithms of exponential or factorial time, only very small inputs can be accommodated.The second thing to be careful about is the way in which the input is characterized. For example, the complexity of cryptanalysis of an encryption algorithm can be characterized equally well in terms of the number of possible keys or the length of the key. For the Advanced Encryption Standard (AES), for example, the number of possible keys is 2128, and the length of the key is 128 bits. If we consider a single encryption to be a "step" and the number of possible keys to be N = 2n, then the time complexity of the algorithm is linear in terms of the number of keys [O(N)] but exponential in terms of the length of the key [O(2n)].ReferencesGRAH94Graham, R.; Knuth, D.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science. Reading, MA: Addison-Wesley, 1994.KNUT97Knuth, D. The Art of Computer Programming, Volume 1: Fundamental Algorithms.Reading, MA: Addison-Wesley, 1997.POHL81Pohl, I., and Shaw, A. The Nature of Computation: An Introduction to Computer Science. Rockville, MD: Computer Science Press, 1981.Table 1 Level of Effort for Various Levels of ComplexityComplexity Size of Input Operations log2n21012=103×10111012 n10121012n21061012n610210122n391012n!151012。

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Laws粘性守恒律解的稳定性Stability of Solutio ns for Viscous Con servation Laws微分方程数值解Numerical Methods for Differential Equations小波理论与应用Wavelet Theory and Application非线性方程组的数值解法Numerical Methods for No-li near System s of Eq uati ons网络算法与复杂性Network Algorithms and Complexity Graph Theory 60近世代数Modern Algebra高等量子力学Advaneed Quantum Mechanics统计力学Statistical Mechanics固体理论Solid State Theory薄膜物理Thin Film Physics计算物理学Computational Physics量子场论Quantum Field Theory非线性物理导论Introduction to Nonlinear Physics固体磁性理论Theory of Magnetism in SolidC语言科学计算方法Scientific Computation Method in C功能材料原理与技术Prin ciple and Tech no logy of Fun cti onal Materials超高真空科学与技术Scie nee and Techno logy of Ultrahigh Vacuum 60现代表面分析技术Modern Technology of Surface Analysis现代传感技术Moder n Sen sor Tech no logy数学模型与计算机模拟Mathematical Models and Computer Simulatio ns计算物理谱方法Spectral Method in Computational Physics蒙特卡罗方法在统计物理中的应用Applicatio ns of the Mon te Carlo Method in Statistical Physics理论物理Theoretical Physics固体物理Solid-State Physics近代物理实验Con temporary Physics Experime nts 计算物理基础Basics of Computational Physics真空与薄膜技术Vacuum & Thin Film Technology高等光学Advaneed Optics量子光学与统计光学Quantum Optics and Statistical Optics X光电子学与光电信息技术Optoelectr onics and Optoelectro nic In formatio n Tech no logy图像处理与分析Image Processing and Analysis光纤通信系统System of Fiber Communications计算机网络Computer Networks光电检测与信号处理Optoelectr onic Detect ion and Process ing物理光学与光电子技术实验Experime nts for Physical Optics and Optoelec tronic Tech no logy非线性光学Nonlinear Optics集成光学Integrated Optics光子学器件原理与技术Prin ciple and Tech no logy of Phot onics Devices物理光学与信息光子学实验Physical Optics & In formation Photo nics Expe rime nts现代激光医学Modern Laser Medicine生物医学光子学Biomedicine Photonics激光医学临床实践Clinical Practice for Laser Medicine光纤通信网络Networks of Fiber Communications光接入网技术Tech no logy of Light Access Network全光通信系统All-Optical Communication Systems计算机图形学Computer Graphics信息光学Information Optics光子学专题Special Topics on Photonics激光与近代光学Laser and Con temporary Optics光电子技术Photo electronic 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Application in An alytical Chemistry现代仪器分析技术Modern Instrument Analytical Technique分离科学Separation Scienee高等环境微生物Advaneed Environmentai Microorganism海洋资源利用与开发Utilization & Development of Ocean Resources立体化学Stereochemistry高等发光分析Advaneed Luminescenee Analysis激光光谱分析Laser Spectroscopy Analysis保健食品监督评价Evaluation and Supervision on Health Food s生物电化学Bioelectrochemistry现代技术与中药Moder n Tech no logy and Traditi onal Chin ese Medici ne 高等有机化学Advaneed organic Chemistry中药新药研究与开发Study and Exploitation of Traditional Chinese Medi cine 药物化学研究方法Pharmaceutical Chemical Research Methods废水处理工程Tech no logy of Wastewater Treatme nt生物与化学传感技术Biose nsors & Chemical Se nsors现代分析化学研究方法Research Methods of Modern An alytical Chemistry神经生物学Neurobiology动物遗传工程Animal Genetic Engineering动物免疫学Animal Immunology动物病害学基础Basis of Animal Disease受体生物化学Receptor Biochemistry动物生理与分子生物学Animal Physiology and Molecular Biochemistry分析生物化学Analytical Biochemistry学科前沿讲座Lectures on Frontiers of the Discipline微生物学Microbiology细胞生物学Cell 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Mathematics Course DescriptionMathematics course in middle school has two parts: compulsory courses and optional courses. Compulsory courses content lots of modern mathematical knowledge and conceptions, such as calculus,statistics, analytic geometry, algorithm and vector. Optional courses are chosen by students which is according their interests.Compulsory Courses:Set TheoryCourse content:This course introduces a new vocabulary and set of rules that is foundational to the mathematical discussions. Learning the basics of this all-important branch of mathematics so that students are prepared to tackle and understand the concept of mathematical functions. Students learn about how entities are grouped into sets and how to conduct various operations of sets such as unions and intersections(i.e. the algebra of sets). We conclude with a brief introduction to the relationship between functions and sets to set the stage for the next stepKey Topics:The language of set theorySet membershipSubsets, supersets, and equalitySet theory and functionsFunctionsCourse content:This lesson begins with talking about the role of functions and look at the concept of mapping values between domain and range. From there student spend a good deal of time looking at how to visualize various kinds of functions using graphs. This course will begin with the absolute value function and then move on to discuss both exponential and logarithmic functions. Students get an opportunity to see how these functions can be used to model various kinds of phenomena.Key Topics:Single-variable functionsTwo –variable functionsExponential functionLogarithmic functionPower- functionCalculusCourse content:In the first step, the course introduces the conception of limit, derivative and differential. Then students can fully understand what is limit of number sequence and what is limit of function through some specific practices. Moreover, the method to calculate derivative is also introduced to students.Key Topics:Limit theoryDerivativeDifferentialAlgorithmCourse content:Introduce the conception of algorithm and the method to design algorithm. Then the figures of flow charts and the conception of logical structure, like sequential structure, contracture of condition and cycle structure are introduced to students. Next step students can use the knowledge of algorithm to make simple programming language, during this procedure, student also approach to grammatical rules and statements which is as similar as BASIC language.Key Topics:AlgorithmLogical structure of flow chart and algorithmOutput statementInput statementAssignment statementStatisticsCourse content:The course starts with basic knowledge of statistics, such as systematic sampling and group sampling. During the lesson students acquire the knowledge like how to estimate collectivity distribution according frequency distribution of samples, and how to compute numerical characteristics of collectivity by looking at numerical characteristics of samples. Finally, the relationship and the interdependency of two variables is introduced to make sure that students mastered in how to make scatterplot, how to calculate regression line, and what is Method of Square.Key Topics:Systematic samplingGroup samplingRelationship between two variablesInterdependency of two variablesBasic Trigonometry ICourse content:This course talks about the properties of triangles and looks at the relationship that exists between their internal angles and lengths of their sides. This leads to discussion of the most commonly used trigonometric functions that relate triangle properties to unit circles. This includes the sine, cosine and tangent functions. Students can use these properties and functions to solve a number of issues.Key Topics:Common AnglesThe polar coordinate systemTriangles propertiesRight trianglesThe trigonometric functionsApplications of basic trigonometryBasic Trigonometry IICourse content:This course will look at the very important inverse trig functions such as arcsin, arcos, and arctan, and see how they can be used to determine angle values. Students also learn core trig identities such as the reduction and double angle identities and use them as a means for deriving proofs. Key Topics:Derivative trigonometric functionsInverse trig functionsIdentities●Pythagorean identities●Reduction identities●Angle sum/Difference identities●Double-angle identitiesAnalytic Geometry ICourse content:This course introduces analytic geometry as the means for using functions and polynomials to mathematically represent points, lines, planes and ellipses. All of these concepts are vital in student’s mathematical development since they are used in rendering and optimization, collision detection, response and other critical areas. Students look at intersection formulas and distance formulas with respect to lines, points, planes and also briefly talk about ellipsoidal intersections. Key Topics:Parametric representationParallel and perpendicular linesIntersection of two linesDistance from a point to a lineAngles between linesAnalytic Geometry IICourse content:Students look at how analytic geometry plays an important role in a number of different areas of class design. Students continue intersection discussion by looking at a way to detect collision between two convex polygons. Then students can wrap things up with a look at the Lambertian Diffuse Lighting model to see how vector dot products can be used to determine the lighting and shading of points across a surface.Key Topics:ReflectionsPolygon/polygon intersectionLightingSequence of NumberCourse content:This course begin with introducing several conceptions of sequence of number, such as, term, finite sequence of number, infinite sequence of number, formula of general term and recurrence formula.Then, the conception of geometric sequence and arithmetic sequence is introduced to students. Through practices and mathematical games, students gradually understand and utilizethe knowledge of sequence of number, eventually students are able to solve mathematical questions.Key Topics:Sequence of numberGeometric sequenceArithmetic sequenceInequalityThis course introduces conception of inequality as well as its properties. In the following lessons students learn the solutions and arithmetic of one-variable quadratic inequality, two variables inequality, fundamental inequality as well how to solve simple linear programming problems. Key Topics:Unequal relationship and InequalityOne-variable quadratic inequality and its solutionTwo-variable inequality and linear programmingFundamental inequalityVector MathematicsCourse content:After an introduction to the concept of vectors, students look at how to perform various important mathematical operations on them. This includes addition and subtraction, scalar multiplication, and the all-important dot and cross products. After laying this computational foundation, students engage in games and talk about their relationship with planes and the plane representation, revisit distance calculations using vectors and see how to rotate and scale geometry using vector representations of mesh vertices.Key Topics:Linear combinationsVector representationsAddition/ subtractionScalar multiplication/ divisionThe dot productVector projectionThe cross productOptional CoursesMatrix ICourse content:In this course, students are introduced to the concept of a matrix like vectors, matrices and so on. In the first two lessons, student look at matrices from a purely mathematical perspective. The course talks about what matrices are and what problems they are intended to solve and then looks at various operations that can be performed using them. This includes topics like matrix addition and subtraction and multiplication by scalars or by other matrices. At the end, students can conclude this course with an overview of the concept of using matrices to solve system of linear equations.Key Topics:Matrix relationsMatrix operations●Addition/subtraction●Scalar multiplication●Matrix Multiplication●Transpose●Determinant●InversePolynomialsCourse content:This course begins with an examination of the algebra of polynomials and then move on to look at the graphs for various kinds of polynomial functions. The course starts with linear interpolation using polynomials that is commonly used to draw polygons on display. From there students are asked to look at how to take complex functions that would be too costly to compute in a relatively relaxed studying environment and use polynomials to approximate the behavior of the function to produce similar results. Students can wrap things up by looking at how polynomials can be used as means for predicting the future values of variables.Key Topics:Polynomial algebra ( single variable)●addition/subtraction●multiplication/divisionQuadratic equationsGraphing polynomialsLogical Terms in MathematicsCourse content:This course introduces the relationships of four kinds of statements, necessary and sufficient conditions, basic logical conjunctions, existing quantifier and universal quantifier. By learning mathematical logic terms, students can be mastered in the usage of common logical terms and can self-correct logical mistakes. At the end of this course, students can deeply understand the mathematical expression is not only accurate but also concise.Key Topics:Statement and its relationshipNecessary and sufficient conditionsBasic logical conjunctionsExisting quantifier and universal quantifierConic Sections and EquationCourse content:By using the knowledge of coordinate method which have been taught in the lesson of linear and circle, in this lesson students learn how to set an equation according the character of conic sections. Students is able to find out the property of conic sections during establishing equations. The aim of this course is to make students understand the idea of combination of number and shape by using the method of coordinate to solve simple geometrical problems which are related to conic sections.Key Topics:Curve and equation OvalHyperbolaParabola。

openfoam mesh 解读OpenFOAM is a widely-used open-source software package for computational fluid dynamics (CFD) simulations. One crucial component of any CFD simulation is the mesh, which is used to discretize the computational domain into smaller elements for computation. In this article, we will explore the basics of OpenFOAM mesh and understand its significance in CFD simulations.The OpenFOAM mesh consists of cells, faces, and points. Cells are the three-dimensional volume elements that make up the computational domain. Faces are the two-dimensional surfaces that separate adjacent cells, and points represent the vertices of the mesh. Together, these components form the foundation of the mesh structure in OpenFOAM.The mesh plays a vital role in accurately representing the geometry and capturing the flow physics in a CFD simulation. It affects the accuracy, convergence, and stability of the simulation results. Therefore, it is essential to create a good quality mesh that adequately resolves complex flow features and boundary conditions.OpenFOAM offers various meshing techniques that can be used to generate high-quality meshes. Some popular meshing approaches include:1. Unstructured mesh: OpenFOAM supports the generation of unstructured meshes using methods like the finite volume method (FVM). This approach provides flexibility in mesh generation and allows for the representation of complex geometries with irregular shapes.2. Mesh refinement: OpenFOAM allows for mesh refinement in regions of interest. This can be done by locally refining the mesh to achieve higher resolution in specific areas, such as boundary layers or regions with strong flow gradients.3. Boundary conditions: OpenFOAM provides a wide range of boundary condition options that can be applied to the mesh faces. These conditions define how the fluid interacts with the boundaries of the computational domain, such as inflow, outflow, wall, and symmetry conditions.4. Mesh quality: OpenFOAM allows users to assess the quality of the mesh through various metrics like aspect ratio, skewness, and orthogonality. High-quality meshes ensure accurate results and prevent numerical instabilities during the simulation.When working with the OpenFOAM mesh, it is important to consider the mesh resolution, boundary conditions, and mesh quality to ensure reliable and accurate simulations. It is also recommended to validate the mesh by conducting sensitivity analyses and comparing the results with experimental or benchmark data.In conclusion, understanding the OpenFOAM mesh and its implications is crucial for successful CFD simulations. A well-designed and high-quality mesh is essential for obtaining accurate results and gaining insights into fluid flow behaviors. OpenFOAM provides various meshing techniques and options to create and refine meshes effectively, enabling researchers and engineers to simulate complex fluid flow phenomena in a wide range of applications.。

“LET’S do it again,” calls a ten-year-old. Once more, pupils clasping printed numbers follow tangled lines marked with white tape on the floor of their school hall. When two meet, the one holding the higher number follows the line right; the other goes left. Afterwards they line up—and the numbers are in ascending order. “The idea is to show how a computer sorts data,” explains their teacher, Claire Lotriet.This was the scene at a recent event in London to promote “Hour of Code”, an initiative organised by , a non-profit, aimed at rousing interest in computer programming—or “coding”in the language of the digital cognoscenti. In September, when computer science becomes part of England’s primary-school curriculum, such games are likely to become a common sight in the country’s classrooms. Many other places are beefing up computer-science teaching, too. Israel was an early adopter, updating its high-school syllabus a decade ago; New Zealand and some German states recently did the same. Australia and Denmark are now following suit. And the coding craze goes far beyond the classroom: more than 24m people worldwide have signed up to free tutorials from Codecademy, an educational website.When computer science was first taught in some American and European schools in the 1970s, generally as an optional subject for older pupils, computers did little unless given instructions in a specialist language. So classes focused on programming. But the advent of ready-made applications and graphical user interfaces in the 1980s saw a shift to teaching “ICT” (information and communications technology): how to use computers for word-processing, creating presentations and the like. The result was that pupils left school with little idea how computers work. England’s ICT curriculum has come in for particular criticism. It “focuses on teaching how to use software, but gives no insight into how it’s made,”said Eric Schmidt, Google’s chairman (and a director of The Economist’s parent company), in a lecture in 2011. “That is just throwing away [England’s] great computing heritage.”Digital technology is now so ubiquitous that many think a rounded education requires a grounding in this subject just as much as in biology, chemistry or physics. That is one reason that the pendulum is swinging back towards teaching coding. Employers’ moans are another. The shortage of skilled programmers is clear from the high salaries they command. The shallower the pool of people who know the basics, the smaller the number of potential tech entrepreneurs. According to Judith Gal-Ezer of the Open University of Israel, who was part of the team that put computer science on the country’s high-school curriculum, a growing share of jobs in developed countries require “computational thinking”: the ability to formulate problems in such a way that they can be tackled by computers. But bosses complain that workers lack such skills—and politicians are listening. Mr Schmidt’s public criticism is credited with prod ding Britain’s government into action.In many places, enthusiasts have moved faster than governments. The sorting game described above comes from “Computer Science Unplugged”, a collection of activities compiled by Tim Bell of the University of Canterbury in New Zealand that teach programming concepts using “cards, string, crayons and lots of running around”. Scratch, a simple language created by MIT’s Media Lab, allows youngsters to develop computer games out of on-screen building blocks that have been compared to “virtual Lego”. In America, where computer science features on the regular curriculum in just a tenth of high schools, the Computer Science Teachers Association produces teaching guides and supports several foreign groups, including the one that wrote Britain’s new curriculum.But the subject is so young that teachers and curriculum designers have little pedagogical research to guide them, says Peter Hubwieser of the Technical University of Munich. How best to explain a concept as abstract as recursion to non-specialists? (Your correspondent has tried several ways, without success.) Which programming language should come first? How to teach mixed-ability groups that will range from self-taught app developers to those who struggle to find the “on” switch?Even basic matters, such as striking the right balance between conceptual exercises like the sorting game and actually writing programs, are still not settled. Doing some coding is essential, says Michael Kölling, a specialist in computing education at the University of Kent: it motivates pupils and means they find out whether their algorithms work. But should pupils start with programming and leave principles till later, or the other way round?How a country answers such questions depends partly on what its economy needs. Estonia is short of programmers for its burgeoning tech industry; it puts great emphasis on programming, with some schools teaching it to pupils as young as six. Denmark devotes more time to topics such as the design of user interfaces, which interests its big firms, and the impact of digital technology on society. “We wanted the curriculum to be meaningful for all pupils, not just those who go on to study computer science,”says Michael Caspersen of Aarhus University.Above all, the new subject will require teachers who know what they are doing. Only a fewplaces take this seriously: Israel has about 1,000 trained computer-science teachers, and Bavaria more than 700. Mathematics and computer-science graduates generally choose more lucrative trades; the humanities and social-science graduates who will find themselves teaching coding will need plenty of support. Britain is skimping: it is introducing its new curriculum in a rush, and preparing teachers has mostly been left to industry groups such as Computing at School, which helped put together the syllabus. If coding is to take its rightful place in the classroom, it cannot be done on the cheap.。