Project 5

Wave Equation in one dimension

Implicit Methods (5 and 9 point)

Part A

Properties of the 5-point implicit scheme. Here the 5-point implicit scheme is studied with different CFL numbers and with different grid spacings.

General Results

Dissipation was the only observed error.
The dissipation error increases as the CFL number increases
The dissipation error is larger for coarser grids.

The following plots compare the solutions for different CFL numbers. Each plot shows three different solutions, generated on grids with 150, 300 and 600 points and a plot of the initial data. (The central peak with amplitude = 1.0 is the inital pulse.)

These plots compare on the same graph three solutions each with different CFL numbers. The grid spacing is constant in these plots at 600 points. (Unfortunately, these plots compare waves at different times, so the waves are not centered at the same place. Ideally, their amplitudes and widths would be the same, however, deviations from the ideal and each other are obvious.)

The whole range CFL = 0.3, 1.0, 1.6.
The lower range CFL = 0.3, 0.5, 0.8.
The upper range CFL = 1.0, 1.2, 1.6.

These results can be best understood by examining the truncation error for this scheme. The largest order term in the truncation error is proportional to the grid spacing and the CFL number. This term is also dissipative. This gives two obvious conclusions. First, as (/\x) is increased the error increases linearly. Second, the error is smaller as the CFL number approaches zero.

The error terms are not given here, but they are in the readme file for projects 3 and 4. Click here to see the project 4 web page.

Part B

The one dimensional wave equation with a constant shift vector (beta) was solved using the 9-point fully averaved implicit method of Alcubierre and Schutz with periodic boundary conditions.

General Results

The FAI scheme: The FAI (9-point) scheme was stable for all CFL numbers and all shift values. However, the coefficient matrix is singular when beta = 1.0, for periodic boundary conditions, and results could not be obtained at this time.
Truncation Error: The solutions display a little diffusion over time. Dispersion errors dominate when beta < 0. (When the grid moves in the opposite direction of the wave.) The dispersion can be minimized and almost eliminated by using a small CFL number.

The following plots were generated with


   CFL = 0.9
  
 Grids with 150, 300, and 450 elements

200 iterations

Convergance Factors

When beta is positive, the convergance factors are quite boring. Here the convergance factors for beta = 0, 0.5, 0.9, 1.3 are plotted on one convenient graph.
When beta is negative, dispersion degrades the solutions on the coursest grids, causing the convergance factor to decrease in time. Here is the convergance factor for beta = -0.5, and here is the factor for beta = -1.00001.

What happens as beta changes? Below several solutions are given with different values for beta. In each case two solutions are plotted together. One solution is generated on a grid with 600 points, and the other on a grid with 150 points. This allows one to compare discretization effects.

Also notice that dispersion errors increase dramatically when beta simply changes sign (with constant magnitude). For example compare solutions for beta = +0.5 to solutions for beta = -0.5, etc. Below we show how to compensate for the increased dispersion by using a smaller CFL number.

In the last section it was demonstrated that dispersion gets out of hand when beta < 0. The dispersion can be minimized by using smaller CFL numbers.

Here we compare beta = -1.3 at three different CFL numbers

CFL = 0.3: The solutions, The convergance factor.
CFL = 0.9: The solutions, The convergance factor.
CFL = 1.3 The solutions, The convergance factor.