## Stability and CFL Restriction

The numerical solution of the partial differential equation (PDE) describing the front is very important to be accurate and stable. For simplicity, Taylor's series expansion is used to handle the partial derivatives of \$ as listed below,

\$(x, y, t + At) = \$(x, y, t) - AtF\V(9.57) \$x(x, y, t) = (<p(x + Ax, y, t) - \$(x, y, t))/Ax, (9.58)

Py(x, y, t) = (<p(x, y + Ay, t) - P(x, y, t))/Ay, (9.59)

pxx(x, y, t) = (p(x + 2Ax, y, t) - 2p(x, y, t) + p(x - 2Ax, y, t))/(2Ax2),

Pyy(x, y, t) = (4>(x, y + 2Ay, t) - 2p(x, y, t) + 4>(x, y - 2Ay, t))/(2Ay2).

There are different numerical techniques used for this problem and the details are given in . The solution is very sensitive to the time step. Time step is selected based on the Courant-Friedrichs-Levy (CFL) restriction. It requires the front to cross no more than one grid cell at each time step At. This calculation will give the maximum time step that guarantees stability. From Eq. 9.62, we maximize the denominator and minimize the nominator to get the best value of the time step. The time step is calculated at each iteration of the process to maintain the stability of the solution: