# Error in Numerical Solution of ODE by Euler Method - Patrick

• patrick1990
In summary, the conversation discusses the use of Euler method for numerical simulation of a PDE and the occurrence of errors with larger numerical steps, known as numerical instabilities or divergence. The speaker is not familiar with this phenomenon and is seeking the name for it. It is suggested to research "Euler region of absolute stability" for more information.

#### patrick1990

Hi,
I recently need to do some numerical simulation by Euler method to solve a PDE.
However, I noticed that there are some errors which are obtained with bigger numerical steps, when applying Euler scheme.
Since my major is not mathematics, I do not know what this phenomenon is called. I have read it somewhere else (from Wikipedia ?), but unfortunately I cannot recall it at all.
Anyone knows the name of this ?
Thank you so much !
patrick

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It is generally called numerical instabilities or divergence and occurs because the state of discrete system (the solver) does not converge to the solution of the continuous system. The Euler method is a very simple method with a very small region of absolute stability. In general one usually achieves much faster or more precise (or both) solutions by integrating using a more capable method.

Thank you so much !
:)

## 1. What is the Euler method for solving ODEs?

The Euler method is a numerical approach to solving ordinary differential equations (ODEs). It involves breaking down the continuous ODE into smaller discrete steps and using the derivative of the function to approximate the next step. This method is named after the mathematician Leonhard Euler.

## 2. What are the limitations of the Euler method?

The Euler method has several limitations, including the fact that it can only provide a rough approximation of the solution and may not be accurate for complex or nonlinear ODEs. Additionally, it can be unstable for certain types of ODEs and may not converge to the true solution.

## 3. How can the accuracy of the Euler method be improved?

The accuracy of the Euler method can be improved by using smaller step sizes, which will result in more points being calculated and a more accurate approximation of the solution. Additionally, higher-order methods such as the Runge-Kutta method can be used, which involve more complex calculations but provide a more accurate solution.

## 4. What is the significance of the error in numerical solutions of ODEs?

The error in numerical solutions of ODEs is important because it represents the difference between the true solution and the approximation obtained using a numerical method. This error can be used to assess the accuracy of the method and determine if it is a suitable approach for solving a particular ODE.

## 5. How can errors in numerical solutions of ODEs be minimized?

To minimize errors in numerical solutions of ODEs, it is important to choose an appropriate numerical method for the specific ODE being solved. Additionally, using smaller step sizes and higher-order methods can also help to reduce errors. It is also important to carefully consider the initial conditions and boundary conditions for the ODE, as these can greatly impact the accuracy of the solution.