While studying the finite difference method for solving partial differential equations, I came across this chapter of a book. On the first paragraph of page $2.27$ it says that the Von Neumann stability condition is only a necessary but not a sufficient condition for stable convergence of a variable coefficient partial differential equation solution. It did not say what is the sufficient condition.
When trying to solve a partial differential equation by using the finite difference method, what is the sufficient condition for stable convergence of a variable coefficient partial differential equation solution?