# Matrix-free iteration methods and implicit ODE solvers

• percolator
In summary, the speaker is discussing the implementation of the implicit Euler method for micromagnetic simulations. They explain that while the explicit method only requires the driving function, the implicit method requires solving a non-linear system of equations. They inquire about alternative methods that do not involve matrix multiplication. One possible option is the predictor corrector method.
percolator
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?

I am not really sure I understand your question but how about a predictor corrector method?

## 1. What are matrix-free iteration methods and how do they differ from traditional methods?

Matrix-free iteration methods are numerical techniques used to solve systems of linear equations without explicitly constructing the matrix of coefficients. In contrast, traditional methods such as Gaussian elimination require the matrix to be fully formed and stored. Matrix-free methods can be more efficient for large and sparse systems.

## 2. What are the benefits of using matrix-free iteration methods?

Matrix-free iteration methods have several advantages, including reduced memory requirements, improved efficiency for large and sparse systems, and the ability to handle systems with varying coefficients. They also eliminate the need to factorize the matrix, which can be a time-consuming process.

## 3. How do implicit ODE solvers differ from explicit ODE solvers?

Implicit ODE solvers use backward-difference formulas to compute the derivatives at future time steps, while explicit solvers use forward-difference formulas. This means that implicit solvers take into account the future behavior of the system, while explicit solvers only consider the current state.

## 4. What are the main challenges of using matrix-free iteration methods?

One of the main challenges of using matrix-free iteration methods is that they can be more complex and difficult to implement compared to traditional methods. They also require careful selection of numerical parameters and can be sensitive to rounding errors. Additionally, they may not be suitable for all types of systems.

## 5. What are the applications of matrix-free iteration methods and implicit ODE solvers?

Matrix-free iteration methods and implicit ODE solvers have numerous applications in scientific computing, particularly in solving large systems of linear equations and differential equations. They are commonly used in fields such as computational fluid dynamics, structural analysis, and numerical weather prediction.

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