The most precise computations using Euler's method in standard floating-point arithmetic applied to modelling of...

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Differential Equations

Background:

• The explicit Euler's method is a foundational numerical technique for solving differential equations.
• Existing methods often face challenges with accuracy and efficiency, particularly for complex systems.
• Previous work focused on linear systems with constant coefficients.

Purpose of the Study:

• To extend the explicit Euler's method to arbitrary systems of ordinary differential equations.
• To introduce an optimal step size calculation at each iteration to minimize total error.
• To demonstrate the method's effectiveness on challenging stiff systems.

Main Methods:

• Modification of the explicit Euler's method algorithm.
• Development of a formula for calculating optimal step size dynamically.
• Application to various systems of ordinary differential equations, including stiff ones.

Main Results:

• The enhanced Euler's method provides a significant reduction in total error compared to standard implementations.
• Optimal step size calculation leads to improved accuracy without compromising computational efficiency.
• Successful application to stiff systems, demonstrating robustness and broad applicability.

Conclusions:

• The proposed optimal step size strategy enhances the explicit Euler's method for arbitrary ODE systems.
• This approach offers a more accurate and reliable numerical solution, especially for stiff problems.
• The method is easy to implement and effective across a range of applications.