Is the Crank-Nicolson method appropriate for solving a system of nonlinear parabolic PDEs like $\partial u/\partial t - a\Delta u + u^4 = 0$ ?I tried to apply this method for solving such system but the solution was oscillating (maybe because of a small value of the coefficient of the time derivative) and the implicit Euler method calculates a correct solution.
Crank-Nicolson method for solving nonlinear parabolic PDEs
pde;finite difference;parabolic pde
This phenomenon is often called ringing and plagues methods that are not $L$-stable.  This can be seen in this motivating example from Hairer & Wanner (1999) Stiff differential equations solved by Radau methods.  Consider the equation$$ \dot y = -50 (y - \cos t) $$and apply explicit Euler with time step near the stability limit, implicit midpoint (or equivalently for this problem, trapezoid rule, aka. Crank-Nicolson), and implicit Euler.  The result, shown below, illustrates that the $A$-stable (but not $L$-stable) implicit midpoint method produces a poor-quality solution.To avoid this problem when solving stiff systems, you should use an $L$-stable method such as BDF-2, a suitable DIRK, or a Radau method.  See Hairer and Wanner's second volume for extensive discussion of this topic.