Originally Posted by mathgra (Post 697905)

Im studying Compressible Euler equations, its a hyperbolic and nonlinear sysem. Simply U_t+F(U)_x=0 where U is an 3x1 matrix contains 3 unknown and F is the flux function.

I wanna show the stability of Lax Friedrichs scheme by Von Neumann.
The scheme is ;

U_j^{n+1}=\frac{1}{2}(U_{j-1}^n+U_{j+1}^n)-\frac{\Delta t}{2\Delta x}(F(U_{j+1}^n)
-F(U_{j-1}^n))

I have studied a little bit and I have realised that when we examining nonlinear problems we use linearization and then we examine linearized version. For this aim we take advection equation as a model problem.That is U_t+c(U)_x=0 where c is constant.

We can write also U_t+F(U)_x=0 as U_t+A(U)_x=0 where A is Jacobian matrix but still contains unknown. Lets say we take A as constant matrix when we apply Von Neumann. Is it okay to examine advection cause in my case we have a matrix not constant ?

I can not decide which form I should write solution u(t,x) in Von Neumann method. Can you help me to understand the idea behind the stability of nonlinear problems ?