Matrix-free iteration methods and implicit ODE solvers

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SUMMARY

The discussion centers on implementing the implicit Euler method for micromagnetic simulations using the Landau-Lifshitz equation. The challenge lies in solving a non-linear system of equations without relying on matrix multiplication, which is typical in Broyden family methods. Participants suggest exploring alternative approaches, such as predictor-corrector methods, to address the limitations of traditional iterative quasi-Newtonian techniques. The focus is on achieving high performance while adhering to the constraints of the implicit method.

PREREQUISITES
  • Understanding of the implicit Euler method in numerical analysis
  • Familiarity with the Landau-Lifshitz equation in micromagnetic simulations
  • Knowledge of non-linear systems of equations and iterative solving techniques
  • Experience with high-performance computing concepts
NEXT STEPS
  • Research predictor-corrector methods for solving ordinary differential equations
  • Explore advanced iterative methods for non-linear equations without matrix multiplication
  • Investigate the application of quasi-Newtonian methods in high-performance software
  • Study the implementation of the Landau-Lifshitz equation in computational physics
USEFUL FOR

Researchers and developers in computational physics, particularly those focused on micromagnetic simulations and numerical methods for solving ordinary differential equations.

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Im trying to implement the implicit Euler method in high-performance software for micromagnetic simulations, where I'm restricted to using the driving function of the ODE (Landau-Lifshitz equation) and the previous solution points. This obviously not a problem for an explicit method, since we only need the driving function to advance to the next timestep. However, when using an implicit method, a non-linear system of equations needs to be solved (f(x) = 0), where typically an iterative quasi-Newtonian method is used to find the solution when the derivative of f(x) can't be utilised.

Is there a method available that doesn't rely on some kind of matrix multiplication like the methods in the Broyden family?
 
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I am not really sure I understand your question but how about a predictor corrector method?
 

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