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2

Here $u$ is complex so the energy is $u^* u$ where $u^*$ is complex conjugate. Then you must compute $$ (u^* u)_t = u^* u_t + u^*_t u= \frac{i}{2} u^* u_{xx} - \frac{i}{2} u^*_{xx} u = \frac{i}{2}( (u^* u_x)_x - (u^*_x u)_x ) $$ Integrating over $x$ and using zero boundary conditions on $u$ $$ \frac{d}{dt}\int_0^1 u^* u dx = \frac{d}{dt}\int_0^1 |u|^2 dx = 0 ...


4

The explanation in the book does not use von Neumann analysis at all but the absolute stability regions and the eigenvalues of the discrete Laplacian operator. For the result you specifically mentioned we use the fact that the maximum eigenvalues is $$ \lambda_m \approx -\frac{4}{h^2} $$ from the expression given. We then want this eigenvalue to lie inside ...


3

I believe Von Neumann's stability analysis would give you the answer here. Consider the heat transfer equation: $$\frac{\partial \mathcal{T}}{\partial t} = \alpha \frac{\partial^{2} \mathcal{T}}{\partial x^{2}}$$ By using Forward Euler time integration and central difference in space discretization: $$\mathcal{T}^{t+\Delta t}_{x} = \mathcal{T}^{t}_{x} + \...


3

There is an entire research field on system theory. The problem is well known but that doesn't mean the answer is simple. Some basics First, let me write your problem in a more common notation. In general, a discrete linear system is described as: $x_{k+1} = Ax_k + Bu_k $ $y_{k} = Cx_k + Du_k $ We call $x$ the internal states and $y$ the observable output. I ...


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