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Chapter 5: Problem 127

What happens to the discretization and the round-off errors as the step size is decreased?

Short Answer

Expert verified
Answer: Decreasing step sizes generally reduce discretization errors by providing a more accurate representation of the continuous function. However, while the round-off errors initially decrease, they may eventually increase if the step size is too small, as the increased number of calculations can accumulate errors. The optimal step size strikes a balance between minimizing discretization errors and avoiding excessive round-off errors.

Step by step solution

01

Understand Discretization and Round-off Errors

Discretization error refers to the error that results from approximating a continuous function with a discrete one, usually in numerical methods. This error arises from the fact that we are taking a finite number of steps to approximate a function over a certain interval. Round-off error, on the other hand, is the difference between the exact value and its representation in the system. This error arises from calculating with finite precision, causing the least significant digits to be lost.
02

Understand the Role of Step Size

Step size is the distance between each discrete point in the approximation of a continuous function. Smaller step sizes lead to more points within a given interval, which can potentially lead to a better approximation of the function.
03

Analyze the Effect of Decreasing Step Size on Discretization Error

As the step size decreases, the number of discrete points increases. This leads to a better approximation of the continuous function, as the discrete points are closer together, creating a more accurate representation. Consequently, the discretization error will decrease as the step size decreases.
04

Analyze the Effect of Decreasing Step Size on Round-off Error

Decreasing the step size initially reduces round-off errors since the solution obtained is more accurate. However, when the step size is decreased beyond the optimal level, the round-off errors can accumulate because of the increased number of calculations the system must perform to reach the solution. As these errors accumulate, the overall round-off error may increase.
05

Conclusion

As the step size decreases, the discretization error usually decreases because the approximation of the continuous function becomes more accurate. However, the round-off error, while initially decreasing, may increase if the step size is too small due to the accumulation of errors in the increased number of calculations. The optimal step size should be a balance between minimizing discretization error and avoiding excessive round-off errors.

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Most popular questions from this chapter

Consider steady heat conduction in a plane wall whose left surface (node 0 ) is maintained at \(30^{\circ} \mathrm{C}\) while the right surface (node 8 ) is subjected to a heat flux of \(1200 \mathrm{~W} / \mathrm{m}^{2}\). Express the finite difference formulation of the boundary nodes 0 and 8 for the case of no heat generation. Also obtain the finite difference formulation for the rate of heat transfer at the left boundary.

What is the basis of the energy balance method? How does it differ from the formal finite difference method? For a specified nodal network, will these two methods result in the same or a different set of equations?

Consider a large plane wall of thickness \(L=0.3 \mathrm{~m}\), thermal conductivity \(k=2.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and surface area \(A=24 \mathrm{~m}^{2}\). The left side of the wall is subjected to a heat flux of \(\dot{q}_{0}=350 \mathrm{~W} / \mathrm{m}^{2}\) while the temperature at that surface is measured to be \(T_{0}=60^{\circ} \mathrm{C}\). Assuming steady one-dimensional heat transfer and taking the nodal spacing to be \(6 \mathrm{~cm},(a)\) obtain the finite difference formulation for the six nodes and (b) determine the temperature of the other surface of the wall by solving those equations.

Design a fire-resistant safety box whose outer dimensions are \(0.5 \mathrm{~m} \times 0.5 \mathrm{~m} \times 0.5 \mathrm{~m}\) that will protect its combustible contents from fire which may last up to \(2 \mathrm{~h}\). Assume the box will be exposed to an environment at an average temperature of \(700^{\circ} \mathrm{C}\) with a combined heat transfer coefficient of \(70 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and the temperature inside the box must be below \(150^{\circ} \mathrm{C}\) at the end of \(2 \mathrm{~h}\). The cavity of the box must be as large as possible while meeting the design constraints, and the insulation material selected must withstand the high temperatures to which it will be exposed. Cost, durability, and strength are also important considerations in the selection of insulation materials.

Is there any limitation on the size of the time step \(\Delta t\) in the solution of transient heat conduction problems using (a) the explicit method and \((b)\) the implicit method?
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