Which of the following methods is computationally more expensive?

Question

BlackTom AI · Accepted Answer

To evaluate which numerical method is computationally more expensive, we need to compare the typical number of function evaluations per integration step and the associated overhead of each method.
Option 1: Heun's method. This is a second-order Runge–Kutta method that generally requires two evaluations of the derivative f(t, y) per step (one for the predictor, one for the corrector). This makes it more expensive than a simple Euler step but still cheaper than methods with more evaluations per step.
Option 2: RK4. The classical fourth-order Runge–Kutta method uses four evaluations of f(t, y) per step (k1, k2, k3, k4) and additional arithmetic. This tends to be the most computationally intensive among standard one-step methods with the same step size, due to both the number of function evaluations and the complexity of the intermediate calculations.
Option 3: They are all equally computationally expensive. This is inaccurate because the per-step work varies: RK4 requires notably more function evaluations than the others listed, so it is not equally expensive.
Option 4: Forward Euler method. This is the simplest one-step method, using a single evaluation of f(t, y) per step. It is typically the cheapest in terms of computational cost per step, though it may require smaller step sizes for accuracy and stability.
Option 5: Midpoint method. Also known as a second-order Runge–Kutta method, it typically uses two evaluations of f(t, y) per step (one to estimate the midpoint, one to compute the final update). Its cost is higher than Forward Euler but lower than RK4, placing it in the middle of the range.
Overall, when comparing these methods, RK4 stands out as the most computationally intensive per step due to the four derivative evaluations and the associated arithmetic, while Forward Euler is the least expensive, with Heun’s and Midpoint methods occupying intermediate positions. The statement that RK4 is the most expensive is consistent with standard computational analysis of these common integrators.

Which of the following methods is computationally more expensive?单项选择题

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