Error in Numerical Solution of ODE by Euler Method - Patrick

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SUMMARY

The discussion centers on the numerical instabilities encountered when using the Euler method for solving partial differential equations (PDEs). Patrick observed that larger numerical steps resulted in significant errors, which are attributed to the limited region of absolute stability inherent in the Euler method. The conversation emphasizes that more advanced integration methods can yield faster and more accurate solutions compared to the basic Euler scheme.

PREREQUISITES
  • Understanding of numerical methods, specifically the Euler method for solving differential equations.
  • Familiarity with the concept of numerical stability in computational mathematics.
  • Basic knowledge of partial differential equations (PDEs).
  • Experience with numerical simulations and error analysis.
NEXT STEPS
  • Research "Euler region of absolute stability" to understand the limitations of the Euler method.
  • Explore alternative numerical methods such as Runge-Kutta for improved accuracy in simulations.
  • Study the concept of numerical instabilities in greater detail to mitigate errors in simulations.
  • Learn about adaptive step size techniques to enhance the performance of numerical solvers.
USEFUL FOR

This discussion is beneficial for mathematicians, computational scientists, and engineers involved in numerical simulations, particularly those working with differential equations and seeking to improve solution accuracy.

patrick1990
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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.

For more information you may want to search your references for "Euler region of absolute stability".
 


Thank you so much !
:)
 

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