Shooting method vs. finite differences for BVP

In summary, when faced with solving a second order ODE with boundary conditions at plus/minus infinity, the standard techniques are the finite difference method and the shooting method. The shooting method is usually preferred for simpler boundary value problems with two endpoints, while the finite difference method is better suited for more complex problems with multiple endpoints. It is recommended to try both methods and compare results to determine the most effective approach.
  • #1
mordechai9
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I am considering a second order ODE of the form y''(x) + f(x) y(x) = 0, with boundary conditions that y(x) = 0 at plus/minus infinity. Note that f(x) is complex for my case.

It seems that the standard techniques for numerically solving this problem are (a.) the finite difference method and (b.) the shooting method. One book I'm looking at ("Numerical recipes", Press et. al.) indicates that the shooting method is the first approach to take, whereas another book ("Finite difference methods for ordinary and partial differential equations", LeVeque) advocates the finite difference method as the first approach.

The shooting method seems to be more complicated so I am a bit confused by this difference of opinion. Any comments would be appreciated.
 
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It ultimately depends on the type of ODE you are solving and the complexity of the problem. In general, the shooting method is preferred for boundary value problems (BVPs) with two endpoints, whereas the finite difference method is preferred for more complex BVPs with multiple endpoints. The shooting method is also simpler to implement for most cases, but the finite difference method can be more accurate in certain instances. Ultimately, it is best to try both methods and compare the results to determine which one yields the best solution.
 

What is the difference between the shooting method and finite differences for BVP?

The shooting method and finite differences are two numerical approaches used to solve boundary value problems (BVPs). The main difference between them lies in their approach to finding the solution. The shooting method solves the BVP by converting it into an initial value problem and using an iterative process to adjust the initial conditions until the desired boundary conditions are met. On the other hand, finite differences involve discretizing the BVP into a set of algebraic equations, which are then solved using numerical methods.

Which method is more accurate for solving BVPs?

Both methods have their advantages and limitations, and the accuracy depends on the specific BVP being solved. The shooting method is more accurate when the solution is smooth and has no discontinuities. Finite differences, on the other hand, can handle non-smooth solutions but may introduce numerical errors. In general, the accuracy of both methods can be improved by increasing the number of iterations or grid points used.

How do I choose between the shooting method and finite differences for a BVP?

The choice between the two methods depends on the nature of the BVP and the desired accuracy. If the solution is expected to be smooth, the shooting method may be a better choice. However, if the solution is known to have discontinuities, finite differences may be necessary. It is also recommended to try both methods and compare the results to determine which one yields a more accurate solution.

Is one method faster than the other?

The speed of convergence for both methods depends on the specific BVP being solved and the accuracy required. In general, the shooting method may require fewer iterations to reach a desired accuracy, but each iteration involves solving an initial value problem, which can be computationally expensive. Finite differences, on the other hand, involve solving a set of algebraic equations, which may be faster but require more grid points for accurate solutions.

Can these methods be applied to all types of BVPs?

Both methods can be applied to a wide range of BVPs, including linear and nonlinear problems. However, some BVPs may require specific modifications or additional techniques to be solved effectively. It is important to carefully consider the properties of the BVP and the limitations of each method before choosing one for the problem at hand.

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