... understanding ofthe lesson.PART III CONCLUSION 1. SummariesIn summary, the study deals with the theories ofthe role of grammar, students’ motivation, and theapplicationof games in teaching ... win. 3. 81.1% ofthe students find that the games guided by their teacher are easy to understand, 18.4% ofthe students sometimes don’t understand the rule ofthe games, and 0.5% of the students ... tense or the form of verb in each clause of that sentence, guess the meaning and the usage of this condition. The group which gives the clear and correct answer will be the winner. The teacher...
... In order to safeguard the reliable operation ofthe steering gear, two motors are to be provided, with any one of them in service and the other in the state of stand-by. The hand steering system ... that, the power supply for the outdoor illumination shall originate from the distribution box fitted in the wheel-house so that it can be easily cut off from the wheel house for the sake of convenient ... automatically if the main network or emergency network fails, or the voltage drops to 40 % ofthe rated value. On the contrary, the circuit will be open automatically as well when the main network...
... 19. Partial Differential Equations19.0 Introduction The numerical treatment ofpartialdifferential equations is, by itself, a vastsubject. Partialdifferential equations are at the heart of ... are:• What are the variables?• What equations are satisfied in the interior ofthe region of interest?• What equations are satisfied by points on the boundary ofthe region of interest? (Here ... points on the boundary ofthe spatial region of interest? Examples:Dirichletconditionsspecify the values ofthe boundarypoints as a function of time; Neumann conditionsspecify the values ofthe normal...
... approximate the fluid by alarge number of cells of uniform states, and piece them together using the Riemannsolution. There have been many generalizations of Godunov’s approach, of which the most ... time. In that case, the independent solutions, or eigenmodes, ofthe differenceequations are all ofthe formunj= ξneikj∆x(19.1.12)846Chapter 19. PartialDifferential EquationsSample ... Taylorseries and includes the nonlinearity ofthe equations explicitly. There is an analyticsolutionforthe evolutionof two uniformstates of a fluid separated by a discontinuity, the Riemann shock problem....
... software n. phần mềm 30. program n. chương trình 31. the system software phần mềm hệ thống 32. theapplication software phần mềm ứng dụng 33. the operation system(OS) hệ điều hành 34. the ... Ngôn ngữ Assemble 36. the data base management system(DBMS) hệ quản lý cơ sở dữ liệu 37. the office automation(OA) tự động hóa văn phòng 38. the graphic processing software phần mềm xử lý ... diệu,thành công rực rỡ 2. as early as ngay từ sớm 3. take the form of dưới hình dạng của 4. let sth. pass cho đi qua 5. vast quantities of lượng lớn của 6. in a word nói tóm lại 7. do one's...
... Integration of Ordinary Differential Equations16.0 IntroductionProblems involving ordinary differential equations (ODEs) can always bereduced to the study of sets of first-order differential equations. ... variables. The generic problem in ordinary differential equations is thus reduced to the study of a set of N coupled first-order differential equations for the functionsyi,i=1,2, ,N, having the general ... and (ii) neither is it very stable(see §16.6 below).Consider, however, the use of a step like (16.1.1) to take a “trial” step to the midpoint ofthe interval. Then use the value of both x and...
... and (ii) neither is it very stable(see §16.6 below).Consider, however, the use of a step like (16.1.1) to take a “trial” step to the midpoint ofthe interval. Then use the value of both x and ... different coefficients of higher-ordererror terms. Adding up the right combination of these, we can eliminate the errorterms order by order. That is the basic idea ofthe Runge-Kuttamethod. ... obtained by using the initial derivative ateach step to find a point halfway across the interval, then using the midpoint derivative across the fullwidth ofthe interval. In the figure, filled...
... parabolic equation, the diffusion equation in one spacedimension,∂u∂t=∂∂xD∂u∂x(19.2.1)where D is the diffusion coefficient. Actually, this equation is a flux-conservative equation ofthe ... pieces like the right-hand side of (19.2.1) in many other situations.Consider first the case when D is a constant. Then the equation ∂u∂t= D∂2u∂x2(19.2.3)can be differenced in the obvious ... nasty set of coupled nonlinear equations to solve at each timestep. Oftenthere is an easier way: If the form of D(u) allows us to integratedz = D(u)du (19.2.23)analytically for z(u), then the right-hand...
... increase the order of a differencing method to greater than the order ofthe original PDEs, you introducespurious solutions to the difference equations. This does not create a problem if theyall ... ∆t/m)(19.3.22) The timestep for each fractional step in (19.3.22) is now only 1/m of thefulltimestep,because each partial operation acts with all the terms ofthe original operator. Equation( 19.3.22) ... differencingschemefor the operator L. In fact, as a rule of thumb, it is often sufficient to have stableUi’sonly for the operator pieces having the highest number of spatial derivatives — the other Ui’s...
... level of CR, we have reduced the number of equations by a factor of two. Since the resulting equations are ofthe same form as the original equation, wecan repeat the process. Taking the number of ... dimension and solve the tridiagonal equations by the usual algorithm in the other dimension) gives about afactor of two gain in speed. The optimal FACR with r =2gives another factor of two gain in ... constantin space. The cyclic reduction method is somewhat more general; its applicabilityis related to the question of whether the equations are separable (in the sense of “separation of variables”)....
... North America). The beauty of Chebyshev acceleration is that the norm ofthe error always decreaseswith each iteration. (This is the norm ofthe actual error in uj,l. The norm of the residual ... is the diagonal part of A, L is the lower triangle of A with zeros on the diagonal, and U is the upper triangle of A with zeros on the diagonal.In the Jacobi method we write for the rth step of ... < 1 for the relaxation to work at all! The rate of convergence ofthe method is set by the rate for the slowest-decaying eigenmode, i.e., the factor with largest modulus. The modulus of this...
... willbe that all equations are within their respective allowed errors. In other words, wewill rescale the stepsize according to the needs ofthe “worst-offender” equation. How is ∆0, the desired ... to fold these considerations into a generally useful stepperroutine is this: One ofthe arguments ofthe routine will of course be the vector of dependent variables at the beginning of a proposed ... on the other hand, then theequation tells how much we can safelyincrease the stepsize for the next step. Local extrapolation consists in accepting the fifth order value yn+1, even though the...
... evaluationper step h instead ofthe two required by second-order Runge-Kutta. Perhaps thereare applications where the simplicity of (16.3.2), easily coded in-line in some otherprogram, recommends ... use ofthe modified midpoint methodby itself will be dominated by the embedded Runge-Kutta method with adaptivestepsize control, as implemented in the preceding section. The usefulness ofthe ... practice, the method finds its most important application as a part of the more powerful Bulirsch-Stoer technique, treated in §16.4. You can thereforeconsider this section as a preamble to §16.4.The...
... integration ofdifferential equations. The scaling “trick” suggested in the discussion following equation (16.2.8) is a good general purpose choice, butnot foolproof. Scaling by the maximum values ofthe ... Akis the work to obtain row k ofthe extrapolation tableau,so Ak+1is the work to obtain column k. We will assume the work is dominated by the cost of evaluating the functions defining the right-hand ... functionseven after the various terms in powers of h all have comparable magnitudes. Inother words, h can be so large as to make the whole notion ofthe “order” of the method meaningless — and the method...