Radial symmetry occurs when a quantity (like concentration or temperature) depends only on the distance from a central point. In spherical coordinates, this means the quantity is a function of the radius 'r' alone, and not of the angles 'θ' or 'φ'.Essentially, the value is the same at all points on a sphere of a given radius.

Axisymmetry occurs when a quantity is independent of the azimuthal angle 'φ'.This means the quantity can vary with the radius 'r' and the polar angle 'θ', but it remains constant as you rotate around a central axis.

Radial Symmetry (1D Case: $u(r, t)$ )

The equation simplifies to:

$$ \frac{\partial u}{\partial t}=D\left[\frac{\partial^2 u}{\partial r^2}+\frac{2}{r} \frac{\partial u}{\partial r}\right] $$

Finite Difference Approximation

Using an explicit finite difference scheme:

• Time derivative: Forward difference

$$ \frac{u_i^{n+1}-u_i^n}{\Delta t} $$

• Second derivative in $r$ : Central difference

$$ \frac{\partial^2 u}{\partial r^2} \approx \frac{u_{i+1}^n-2 u_i^n+u_{i-1}^n}{\Delta r^2} $$

• First derivative in $r$ : Central difference

$$ \frac{\partial u}{\partial r} \approx \frac{u_{i+1}^n-u_{i-1}^n}{2 \Delta r} $$

Numerical Scheme

$$ u_i^{n+1}=u_i^n+\frac{D \Delta t}{\Delta r^2}\left[u_{i+1}^n-2 u_i^n+u_{i-1}^n\right]+\frac{2 D \Delta t}{r_i \Delta r}\left(u_{i+1}^n-u_{i-1}^n\right) $$

where:

• $i$ represents spatial index in $r$,
• $n$ represents time index.

Axisymmetric Case (2D: $u(r, \theta, t)$ )

The equation simplifies to: