Predictor Corrector Scheme for Implicit Runge Kutta

Question

I want to solve an ODE system :

$$ \frac{dy}{dt} = f(y, t) $$ Since my application requires method to be symplectic, I am using an implicit runge kutta method.

$$ y_{n+1} = y_n + h\sum_{i=1}^s{b_iK_i} $$

$$ k_i = f(y_n + h\sum_{j=1}^s{a_{ij}k_j, t_n + c_ih}) $$

Since this is implicit, it will be required to solve the non linear system of equations.

I was thinking of using a predictor corrector scheme - use an explicit RK method for intermediate prediction, and then use this prediction as a starting point for solving the fixed point iteration for solving the system of equations in the implicit system.

I want to try many butcher tableau coefficients. Therefore I was thinking of implementing a generic code which accepts a butcher tableau, uses and explicit RK method, and then refines the prediction using the implicit system and fixed point iteration. The problem is, it is not possible everytime to analytically simplify implicit formula (getting rid of all $k$ by replacing them with $y_{n+1}$). How do I then use the predictor's output to solve the fixed point iterations? I was thinking of replacing $y_n$ in the second equation by the prediction - however, I am not sure if it is correct. I have added the code for this in python.

import numpy as np

def fixed_point_iteration_butcher_tableau(butcher_tableau, f, y_n, t_n, h, max_iterations=100, tolerance=1e-8):
    """
    Perform fixed-point iteration to solve the implicit Runge-Kutta equations.

    Parameters:
    - butcher_tableau: A dictionary containing the Butcher tableau coefficients.
    - f: The function defining the differential equation.
    - y_n: The current solution vector.
    - t_n: The current time.
    - h: The time step.
    - max_iterations: Maximum number of iterations.
    - tolerance: Convergence tolerance.

    Returns:
    - The updated solution vector.
    """

    c = np.array(butcher_tableau['c'])
    a = np.array(butcher_tableau['a'])
    b = np.array(butcher_tableau['b'])

    num_stages = len(c)
    y_n1 = y_n.copy()

    for _ in range(max_iterations):
        # Initialize stage values
        k = np.zeros((num_stages, len(y_n)))

        # Compute stage values using fixed-point iteration
        for i in range(num_stages):
            stage_sum = np.sum(a[i, :i] * k[:i, :], axis=0)
            k[i, :] = f(t_n + c[i] * h, y_n + h * stage_sum)

        # Update solution using the Butcher tableau coefficients
        y_n1_new = y_n + h * np.dot(b, k)

        # Check for convergence
        if np.linalg.norm(y_n1_new - y_n1) < tolerance:
            return y_n1_new

        y_n1 = y_n1_new

    # If max_iterations is reached without convergence, print a warning
    print("Warning: Fixed-point iteration did not converge within the specified iterations.")
    return y_n1

def predictor_corrector_butcher_tableau(butcher_tableau, f, y_n, t_n, h, max_iterations=100, tolerance=1e-8):
    """
    Perform predictor-corrector using fixed-point iteration.

    Parameters:
    - butcher_tableau: A dictionary containing the Butcher tableau coefficients.
    - f: The function defining the differential equation.
    - y_n: The current solution vector.
    - t_n: The current time.
    - h: The time step.
    - max_iterations: Maximum number of iterations for fixed-point iteration.
    - tolerance: Convergence tolerance for fixed-point iteration.

    Returns:
    - The updated solution vector.
    """

    # Predictor (explicit method)
    # for some explicit RK method
    predictor_y_n1 = some_explicit_method(f, y_n, h, t_n)

    # Corrector (implicit method using fixed-point iteration)
    corrected_y_n1 = fixed_point_iteration_butcher_tableau(butcher_tableau, f, predictor_y_n1, t_n, h,
                                                           max_iterations=max_iterations, tolerance=tolerance)

    return corrected_y_n1

# Example usage:
butcher_tableau = {
    'c': [0, 1],
    'a': [[0, 0], [1, 0]],
    'b': [1/2, 1/2]
}

def f(t, y):
    # Define the function for the differential equation
    # Modify this function based on your specific problem
    return np.array([y[0] + t, y[1]])

# Example initial values
t_n = 0
y_n = np.array([1, 2])
h = 0.1

# Perform predictor-corrector
updated_solution = predictor_corrector_butcher_tableau(butcher_tableau, f, y_n, t_n, h)

print("Updated Solution using Predictor-Corrector:", updated_solution)
```