Project 3 & 4

Wave Equation in one dimension

Implicit Methods (5 and 9 point)

Convergance factors for fixed boundary conditions

• 5-point implicit scheme
• 9-point implicit scheme

Convergance Factors
• 5 point Convergance factors for the 5-point implicit scheme. CFL = 1.1 and CFL = 0.3
• 9 point Convergance factors for the 9-point implicit scheme. CFL = 1.1 and CFL = 0.3
• All four convergance factors plotted together

Wave at boundary The wave passing through the right side of the boundary only to reappear at the left. Here CFL = 1.1, time = 0.99, 45 iterations. The solutions are plotted with the 9- and 5-point schemes.

The following plots are all at time = 3.3. When CFL = 1.1, the program performs 150 iterations, and when CFL = 0.3 550 iterations are required.

• CFL = 1.1 The wave calculated with both the 9- and 5-point schemes.
  (2 solutions plotted together.)
• CFL = 1.1, and CFL = 0.3 The wave calculated with both the 9- and 5-point schemes.
  (4 solutions plotted together.)
• CFL = 1.1 The 9-point scheme with solutions from 100, 200, and 400 point spatial grids.
  (3 solutions plotted together.)
• CFL = 1.1 The 5-point scheme with solutions from 100, 200, and 400 point spatial grids.
  (3 solutions plotted together.)
• CFL = 0.3 The 9-point scheme with solutions from 100, 200, and 400 point spatial grids.
  (3 solutions plotted together.)
• CFL = 0.3 The 5-point scheme with solutions from 100, 200, and 400 point spatial grids.
  (3 solutions plotted together.)