I'm having difficulty with numerically solving the inviscid burgers equation.Godunov's scheme is used in most of what I've found in literature . Now my question is if using a crank nicolson shceme is wrong or not? $\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x}=0$with this BC:at $t> 0 ,x=0$  :  $u=1$using a finite volume method: $\frac{\partial }{\partial t}\int udv + \int u\frac{\partial u}{\partial x}dv=0$$\frac{\partial u}{\partial t}\Delta v +(uuA) _{e}-(uuA) _{w}=0$$\frac{\partial u}{\partial t} + F_{e}u_{e}-F_{w}u_{w}=0$$F=\frac{u}{\Delta x}$and for discretization in time I used Crank-Nicolson $u_{p}^{n+1}-u_{p}^{n}=\frac{\Delta t}{2}(-F_{e}u_{e}+F_{w}u_{w})^{n+1}+\frac{\Delta t}{2}(-F_{e}u_{e}+F_{w}u_{w})^{n}$the final form of equation is as (upwind scheme) :$(1+\frac{\Delta t}{2}F_{e})u_{P}^{n+1}=(-\frac{\Delta t}{2}F_{w}) u_{W}^{n+1} +u_{P}^{n}+\frac{\Delta t}{2}(F_{w}u_{w}-F_{e}u_{e})^{n}$and for the first block :$(1+\frac{\Delta t}{2}F_{e})u_{P}^{n+1}= u_{P}^{n}+\frac{\Delta t}{2}(-F_{e}u_{e})^{n}+\Delta t F_{w}u_{w}$and after that I tried to solve a set of linear algebraic equation. The result in every time step seems to converge but with time passing the magnitude of velocity is also increasing which is an indication of a mistake.
Numerical solution of burgers equation with finite volume method and crank-nicolson
pde;finite volume;crank nicolson
If I understand correctly, you are using a centered finite difference in space and the implicit trapezoidal method in time.  That scheme is unconditionally absolutely stable, but will generate spurious oscillations.  So you should expect to see some increase in the maximum value of $u$, but it shouldn't blow up.  If it blows up, you have an implementation error (bug).I'll add that this is a lousy method for a purely hyperbolic PDE, since it generates oscillations and isn't very accurate for large time steps anyway.