Why Do I Need an Integrating Factor for Polar Coordinates?

[DeeCeeBee]

I'm solving the following equation in the unit square using finite differences:
epsilon(u_xx+u_yy)+u_x+u_y=0, where epsilon is a singular perturbation parameter.

I need to use domain decomposition to isolate the corner singularity in the outflow corner. My subdomain in this corner is a quarter disk so I translate the d.e. to polar coordinates. If I don't use the translation:
v=exp(M(x+y))u, M=1/2epsilon, before translating to polar coordinates, then my solution has a concave shape, but it should be convex.

I find that even when epsilon=1 I need to use this integrating factor. Does anyone know why this is? Have searched high and low with no success.

 

[gtoguy97]

The integrating factor helps to transform the partial differential equation (PDE) into an ordinary differential equation (ODE). Without it, the PDE is difficult to solve due to the presence of the singular perturbation parameter, epsilon. It helps to separate the variables and thus simplify the problem and make it more tractable.

 

[tacman]

An integrating factor is necessary when solving equations in polar coordinates because it helps to account for the change in variables and maintain the correct shape of the solution. In this case, the use of the integrating factor v=exp(M(x+y))u helps to transform the equation into a form that is more easily solvable and accurately captures the behavior of the solution near the corner singularity.

Without the integrating factor, the solution may appear concave instead of convex, which is not the correct behavior for the given problem. This is because the change in variables from Cartesian to polar coordinates introduces additional terms that need to be accounted for in order to accurately represent the solution.

Additionally, the use of an integrating factor becomes even more crucial when dealing with singular perturbation problems, such as in this case where epsilon is a singular perturbation parameter. These types of problems require careful consideration and the use of integrating factors can help to mitigate any issues that may arise.

In summary, the use of an integrating factor is necessary when solving equations in polar coordinates to ensure the accuracy and correctness of the solution, especially in cases involving singular perturbations.