conv-diff-mp

Solving 2D convection diffusion equation using julia multiprocessing

Problem

Numerically solving the 2D convection diffusion equation

$$ \frac{\partial{u}}{\partial{t}}+a\frac{\partial{u}}{\partial{x}}+b\frac{\partial{u}}{\partial{y}}=\kappa(\frac{\partial^2{u}}{\partial{x}^2}+\frac{\partial^2{u}}{\partial{y}^2}) $$

under the periodic boundary condition (PBC).

Finite Difference Method (FDM)

The discretization formula

$$ \frac{u_{i,j}^{n+1}-u_{i,j}^n}{\Delta t}+a \frac{u_{i,j}^n-u_{i-1,j}^n}{\Delta x}+b \frac{u_{i,j}^n-u_{i,j-1}^n}{\Delta y}=\kappa(\frac{u_{i+1,j}^n+u_{i-1,j}^n-2u_{i,j}^n}{\Delta x^2} +\frac{u_{i,j+1}^n+u_{i,j-1}^n-2u_{i,j}^n}{\Delta y^2}) $$

yields

$$ u_{i,j}^{n+1}=k_1 u_{i+1,j}^n+(k_1+k_3)u_{i-1,j}^n+k_2 u_{i,j+1}^n+(k_2+k_4)u_{i,j-1}^n+(1-2k_1-2k_2-k_3-k_4)u_{i,j}^n, $$

where

$$ k_1=\kappa \frac{\Delta t}{(\Delta x)^2}, k_2=\kappa \frac{\Delta t}{(\Delta y)^2}, k_3=a \frac{\Delta t}{\Delta x}, k_4=b \frac{\Delta t}{\Delta y}. $$

Results

Visualization

For $a=3,b=3,\kappa=0.001$

Run time

Testing under 4-core CPU Intel(R) Core(TM) i5-7300HQ CPU @ 2.50GHz

Configuration (nx * ny * dt)	single process	multi process (4)
81 * 81 * 0.001	1.229594 s	1.317900 s
8001 * 8001 * 0.00001	49.337162 s	15.908043 s