Laplacian in Polar Coordinates

Problem

In polar coordinates the Laplacian is Laplacian(f) = f_rr + (1/r)*f_r + (1/r^2)*f_theta_theta.

Compute each term with central differences (step h = 1e-5) for the function given by the coefficient grid coeffs[i][j] * r^i * theta^j and evaluate at (r, theta). Ragged rows are allowed.

Hint

The 1/r and 1/r^2 weights distinguish this from the Cartesian Laplacian.

Test cases

case 1

in: [[[0,0],[0,0],[1,0]],2,0.5]

expected: 4.000000165493844

case 2

in: [[[0,1]],2,0.5]

expected: -1.3877787807814454e-7

case 3

in: [[[0,0],[0,0],[0,1]],2,0.5]

expected: 1.99999952763541

case 4

in: [[[],[0,0,1]],2,1]

expected: 1.500001192966671

Starter codePython

def polar_laplacian(coeffs, r, theta, h=1e-5):
    # Your code here
    pass

4 test cases · run your code against them in the editor