Robust discretizations versus increase of the time step for the Lorenz system

Area of Science:

• Numerical Analysis
• Computational Physics
• Dynamical Systems

Background:

• Discretization of continuous systems leads to solutions dependent on the chosen time step.
• Standard numerical schemes can produce spurious solutions or parameter shifts with large time steps.
• Robust discrete equations are essential for accurate simulations, especially with coarse discretization.

Purpose of the Study:

• To investigate different discretizations of the Lorenz system.
• To evaluate the robustness of discrete schemes against variations in the time step.
• To identify nonstandard schemes that offer improved stability for larger time steps.

Main Methods:

• Discretization of the Lorenz system using various numerical schemes.
• Comparison of conventional methods (forward Euler, backward Euler, centered finite difference) with nonstandard schemes.
• Analysis of solution behavior and parameter space displacement as a function of the time step.

Main Results:

• Nonstandard schemes, specifically Mickens' and Monaco and Normand-Cyrot's schemes, demonstrate superior robustness.
• These nonstandard schemes maintain accuracy and stability with larger discretization time steps compared to standard schemes.
• Conventional discretizations exhibit increased sensitivity and potential for spurious solutions as the time step increases.

Conclusions:

• Nonstandard discretization schemes offer significant advantages in robustness for simulating continuous systems.
• The proposed nonstandard methods are more reliable than traditional schemes when large time steps are necessary.
• This research provides improved numerical tools for the accurate simulation of dynamical systems.