Numerical instability problems

The present drawback to the Van Trier and Symes algorithm is that for a given grid size the algorithm is numerically unstable. The problem is that presently one cannot determine a priori a stable grid spacing. The program will execute without any evidence of numerical instability. Only by examining the results one can determine if numerical instability has occurred. By reducing the grid spacing one can resolve this problem. Further work is necessary to detect automatically numerical instability or determine the stability conditions as a function of the velocity model and grid spacing. Detecting numerical instability and correcting it at run-time will be an important improvement. In 3-D the instability problem is more serious compared to 2-D. The velocity models I tested need to have smooth velocity variations in order to run in the presently available computer memory without presenting numerical instability.