[mathgra]

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 ?

[FMDenaro]

Originally Posted by mathgra

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 ?

You have to consider a linearization such that the error is lineraly added to the solution. This way, it is supposed that the evolution of the error obey to the same scheme of the solution. Now you can just consider the error as sum of Fourier components to study the amplification of the error. The advection velocity is prescribed.