Test Matrices

Given equation to solve is following Burger’s equation with having both convection and diffusion terms:

\[\frac{\partial \phi}{\partial t} + U \frac{\partial \phi}{\partial x} = \frac{\partial}{\partial x}\left ( \Gamma \frac{\partial \phi}{\partial x} \right )\]

One dimensional domain is set to 40 as size with \(5 \leqslant x\leqslant 45\). Boundary conditions at both left and right node points are set to 0.

Consider two cases:

  • CASE1: Euler equation without diffusion term: \(\Gamma = 0\)
  • CASE2: Burger’s equation with diffusion term: \(\Gamma = 001\)

Euler equation (Pure convection problem)

  • Grid setups: N = 1001, 2001, 4001, 6001
  • Different Couran’t number conditions: Cr = 0.25, 0.5, 0.75, 0.1 for N = 4001 only
  • Test methods: Euler Explicit (EE), Euler Implicit (EI), and Crank-Nicolson (CR)
  • No diffusivity: \(\Gamma\) = 0

Stability check

  1. Test on different time step

  • Stability was checked on grid setup with N = 4001 with different time step conditions.

  • If unstable solution appears with any inappropriate values, stability is NOT considered to be achieved.

    Courant no. (dt) EE EI CR
    0.25 (0.0025) X O O
    0.5 (0.005) X O O
    0.75 (0.0075) X O O
    0.1 (0.01) X O O

    (O: stable, X: unstable)


  1. Test on different grid resolution

  • Stability was checked on different grid setup with fixed time step of dt = 0.005 [sec]

  • Note that Courant number must vary for different grid size to fix the time step: Following Courant numbers in table were chosed to set the ‘dt’ constant.

    Number of nodes (Cr) EE EI CR
    1001 (0.125) X O O
    2001 (0.25) X O O
    4001 (0.5) X O O
    6001 (0.75) X O O

    (O: stable, X: unstable)


Burger’s equation (Convection + Diffusion problem)

  • Grid setups: N = 1001, 2001, 4001, 6001
  • Different Couran’t number conditions: Cr = 0.25, 0.5, 0.75, 0.1 for N = 4001 only
  • Test methods: Euler Explicit, Euler Implicit, and Crank-Nicolson
  • Diffusivity: \(\Gamma\) = 0.01

Stability check

  1. Test on different time step

  • Stability was checked on grid setup with N = 4001 with different time step conditions.

  • If unstable solution appears with any inappropriate values, stability is NOT considered to be achieved.

    Courant no. (dt) EE EI CR
    0.25 (0.0025) O O O
    0.5 (0.005) O O O
    0.75 (0.0075) X O O
    0.1 (0.01) X O O

    (O: stable, X: unstable)


  1. Test on different grid resolution

  • Stability was checked on different grid setup with fixed time step of dt = 0.005 [sec]

  • Note that Courant number must vary for different grid size to fix the time step: Following Courant numbers in table were chosed to set the ‘dt’ constant.

  • Peclet number are specifed in the following table. (Note: Peclet number can only be defined in diffusion term contained solution)

    Number of nodes (Cr / Pe) EE EI CR
    1001 (0.125 / 4.0) O O O
    2001 (0.25 / 2.0) O O O
    4001 (0.5 / 1.0) O O O
    6001 (0.75 / 0.667) X O O

    (O: stable, X: unstable)