Convergence of implicit Euler method

In summary, the implicit Euler method is a numerical algorithm used to approximate the solution to a differential equation. It works by using a backward difference formula to approximate the derivative, making it more accurate than explicit methods. It has the advantage of stability and can handle stiff equations without requiring a small time step. However, it can be computationally expensive, introduces numerical damping, and may not be suitable for all types of differential equations. To improve convergence, a smaller time step, higher-order implicit methods, adaptive step size methods, and careful consideration of initial conditions and numerical parameters can be used.
  • #1
squenshl
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Homework Statement


The implicit Euler method is yn = yn-1 + hf(xn,yn).
Find the local truncation error and hence show that the method is convergent.


Homework Equations





The Attempt at a Solution


I found the error to be ln = (-h2/2)y''(xn-1) + O(h3).
For convergence I am up to using the Lipschitz condition, triangle inequality and ||ln|| = -Mh2/2 to get:
||en|| <= 1/(1-hL)||en-1|| - Mh2/(2(1-hL)) for hL <= 1/2 but I am stuck after this. Someone help please.
 
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  • #2
Since eo = y(x0) - y0 = 0
||en|| <= -Mh2/(2(1-hL))(1+(1-hL)+...+(1-hL)n-1) = -Mh2/(2(1-hL))*((1-hL)n-1 - 1)/hL
 

FAQ: Convergence of implicit Euler method

What is the implicit Euler method?

The implicit Euler method is a numerical algorithm used to approximate the solution to a differential equation. It is an implicit method because the next step in the solution is dependent on the current step, unlike explicit methods which only use previously calculated steps.

How does the implicit Euler method work?

The implicit Euler method uses a backward difference formula to approximate the derivative of the function at each step. This allows for a more accurate approximation of the solution compared to the forward difference formula used in explicit methods.

What are the advantages of using the implicit Euler method?

One advantage of using the implicit Euler method is its stability. Unlike explicit methods, it can handle stiff equations and does not require a small time step for accuracy. It is also unconditionally stable, meaning the step size can be chosen without worrying about the solution diverging.

What are the limitations of the implicit Euler method?

The implicit Euler method can be computationally more expensive compared to explicit methods, as it requires solving a system of equations at each step. It also introduces numerical damping, which can affect the accuracy of the solution. Additionally, it may not be suitable for all types of differential equations.

How can the convergence of the implicit Euler method be improved?

The convergence of the implicit Euler method can be improved by using a smaller time step, which reduces the error introduced by the backward difference formula. Additionally, using higher-order implicit methods or adaptive step size methods can also improve convergence. Careful consideration of the initial conditions and the choice of numerical parameters can also help improve convergence.

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