Problem: Python Implementation of Pipeline Diffusion Simulation

The problem is to develop a complete Python-based variable-coefficient pipeline diffusion simulation framework, supporting initial profile generation, numerical stability verification, implicit time stepping, and full transient simulation with proper input validation, boundary enforcement and noise processing to output accurate stable pipeline state results for relevant engineering analysis.


Input/Output: The problem receives the following input data: integer num_sections (≥2), positive float pipeline total length, float or integer inlet and outlet boundary values, initial profile (scalar value or callable that accepts a numpy position array and returns corresponding value array), diffusion coefficient (float, sequence of floats or numpy ndarray), positive float time step dt, positive float grid spacing dx, positive integer number of simulation steps, current pipeline state (list of floats or numpy ndarray), and r factor (float or sequence of floats). All inputs are validated for value validity and dimension consistency, with ValueError raised for invalid cases. The problem generates four sequential outputs: a tuple of discretized initial pipeline profile (list of floats) and uniform grid step dx (float), non-negative diffusion stability coefficient (≤0.5), updated pipeline state matching the input current state data type, and final simulated pipeline state (list of floats) after completing the specified number of diffusion steps. The processing workflow follows initialization → stability check → optional single section update → full pipeline simulation.

Required dependencies:

Sub-steps:
step_number: 1
step_description_prompt: Please write a Python function named initialize_pipeline. This function generates discretized initial pipeline profile with Chebyshev spaced nodes, adds small Gaussian noise, fixes inlet and outlet boundary values, returns the profile list and uniform dx step. Handle input validation, scalar or callable initial profile cases properly.
function_header: def initialize_pipeline(
    num_sections: int,
    length: float,
    inlet_value: Union[float,int],
    outlet_value: Union[float,int],
    initial_profile: Union[float,int,Callable[[np.ndarray], np.ndarray]]
) -> tuple[list[float], float]:
test_cases: 
profile1, dx1 = initialize_pipeline(5, 10.0, 100.0, 0.0, 50.0)
assert len(profile1) == 5
assert profile1[0] == 100.0 and profile1[-1] == 0.0
assert dx1 > 0
profile2, dx2 = initialize_pipeline(3, 5.0, 10.0, 20.0, lambda x: 15 + 0.1*x)
assert len(profile2) == 3
assert profile2[0] == 10.0 and profile2[-1] == 20.0

step_number: 2
step_description_prompt: Please use Python to generate a function named check_diffusion_stability. It takes diffusion coefficient, time step dt and grid spacing dx as inputs, checks explicit diffusion stability conditions, runs perturbation sensitivity test for multi-dimensional inputs, raises ValueErrors for invalid cases, returns the total calculated stability coefficient.
function_header: def check_diffusion_stability(diff_coeff: Union[float, Sequence[float], np.ndarray], dt: float, dx: Union[float, Sequence[float], np.ndarray]) -> float:
test_cases: r1 = check_diffusion_stability(0.1, 0.1, 1.0)
assert r1 <= 0.5

try:
    check_diffusion_stability(1.0, 1.0, 1.0)  # should fail
    assert False
except ValueError:
    assert True

r2 = check_diffusion_stability([0.1, 0.2], 0.1, [1.0, 1.0])
assert r2 <= 0.5

step_number: 3
step_description_prompt: Please write a Python function named update_pipeline_section. It computes updated pipeline state via implicit difference scheme, involves tridiagonal solving, state-dependent growth scaling, Gaussian noise addition and value clamping. Inputs are current_state and r_factor, output matches input data type.
function_header: def update_pipeline_section(current_state, r_factor):
test_cases: state1 = [10.0, 5.0, 10.0]
new_state1 = update_pipeline_section(state1, 0.25)
assert len(new_state1) == 3
assert new_state1[0] > 9.9 and new_state1[0] < 10.1 and new_state1[-1] > 9.9 and new_state1[-1] < 10.1  # boundaries fixed

state2 = [0.0, 1.0, 0.0]
new_state2 = update_pipeline_section(state2, [0.1])
assert len(new_state2) == 3

step_number: 4
step_description_prompt: Please write a Python function named simulate_pipeline. It uses Crank-Nicolson scheme with Thomas algorithm to solve variable-coefficient diffusion problems, adds tiny Gaussian noise per time step, and returns the final state as a list of floats. Input parameters are init_state, diff_coeff, dt, dx, num_steps.
function_header: def simulate_pipeline(
    init_state: Union[List[float], np.ndarray],
    diff_coeff: Union[float, List[float], np.ndarray],
    dt: float,
    dx: float,
    num_steps: int
) -> List[float]:
test_cases: p = {'diff':1.0,'dt':0.25,'dx':1.0}
for init_state, steps, expected in [
    ([0,10,0],1,[0,10,0]),
    ([0,10,0],2,[0,10,0]),
    ([100,100,100],5,[100,100,100])
]:
    res = simulate_pipeline(init_state,p['diff'],p['dt'],p['dx'],steps)
    res_arr = np.array(res)
    expected_arr = np.array(expected)
    assert all(np.isfinite(res_arr))
    tol = np.maximum(np.abs(expected_arr)*0.01,0.5)
    assert all(np.abs(res_arr-expected_arr)<=tol)

General solution:

General tests: